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3.23.45 \(\int \frac {A+B x}{(d+e x)^2 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=746 \[ \frac {e \sqrt {a+b x+c x^2} \left (-8 b c \left (B \left (5 a^2 e^4+3 a c d^2 e^2+2 c^2 d^4\right )+2 A c d e \left (9 a e^2+4 c d^2\right )\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )\right )-2 b^3 e^2 \left (-3 a B e^2-10 A c d e+9 B c d^2\right )+4 b^2 c e \left (25 a A e^3-14 a B d e^2+3 A c d^2 e+10 B c d^3\right )+3 b^4 e^3 (3 B d-5 A e)\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac {2 \left (c x \left ((2 c d-b e) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {e^3 \left (5 A e (2 c d-b e)-B \left (8 c d^2-e (2 a e+3 b d)\right )\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}} \]

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Rubi [A]  time = 1.56, antiderivative size = 744, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {822, 806, 724, 206} \begin {gather*} \frac {e \sqrt {a+b x+c x^2} \left (-8 b c \left (B \left (5 a^2 e^4+3 a c d^2 e^2+2 c^2 d^4\right )+2 A c d e \left (9 a e^2+4 c d^2\right )\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )\right )-2 b^3 e^2 \left (-3 a B e^2-10 A c d e+9 B c d^2\right )+4 b^2 c e \left (25 a A e^3-14 a B d e^2+3 A c d^2 e+10 B c d^3\right )+3 b^4 e^3 (3 B d-5 A e)\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac {2 \left (c x \left ((2 c d-b e) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {e^3 \left (-B e (2 a e+3 b d)-5 A e (2 c d-b e)+8 B c d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*x))/(3*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2)^(3/2)) + (2*(6*a*c*e*(2*c*d - b*e)*(b*B*d - 2*A*c*d + A
*b*e - 2*a*B*e) + (b*c*d - b^2*e + 2*a*c*e)*(b^2*e*(3*B*d - 5*A*e) + 8*c*(A*c*d^2 - a*B*d*e + 2*a*A*e^2) - 2*b
*(2*B*c*d^2 + A*c*d*e - a*B*e^2)) + c*(6*c*e*(b*d - 2*a*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + (2*c*d - b*e)
*(b^2*e*(3*B*d - 5*A*e) + 8*c*(A*c*d^2 - a*B*d*e + 2*a*A*e^2) - 2*b*(2*B*c*d^2 + A*c*d*e - a*B*e^2)))*x))/(3*(
b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(3*b^4*e^3*(3*B*d - 5*A*e) - 2*
b^3*e^2*(9*B*c*d^2 - 10*A*c*d*e - 3*a*B*e^2) + 4*b^2*c*e*(10*B*c*d^3 + 3*A*c*d^2*e - 14*a*B*d*e^2 + 25*a*A*e^3
) - 16*c^2*(a*B*d*e*(2*c*d^2 - 13*a*e^2) - A*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)) - 8*b*c*(2*A*c*d*e*(4*c*
d^2 + 9*a*e^2) + B*(2*c^2*d^4 + 3*a*c*d^2*e^2 + 5*a^2*e^4)))*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^2*(c*d^2
- b*d*e + a*e^2)^3*(d + e*x)) - (e^3*(8*B*c*d^2 - B*e*(3*b*d + 2*a*e) - 5*A*e*(2*c*d - b*e))*ArcTanh[(b*d - 2*
a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)^(7/2
))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
 IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (-2 b B d \left (2 c d-\frac {3 b e}{2}\right )-2 a B e (4 c d-b e)+2 A \left (4 c^2 d^2-\frac {5 b^2 e^2}{2}-c e (b d-8 a e)\right )\right )-3 c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (6 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )+c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} e \left (6 c e \left (b^2 d-8 a c d+2 a b e\right ) (b (B d+A e)-2 (A c d+a B e))-\left (2 b c d-3 b^2 e+8 a c e\right ) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )\right )+\frac {1}{2} c e \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (6 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )+c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (3 b^4 e^3 (3 B d-5 A e)-2 b^3 e^2 \left (9 B c d^2-10 A c d e-3 a B e^2\right )+4 b^2 c e \left (10 B c d^3+3 A c d^2 e-14 a B d e^2+25 a A e^3\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )\right )-8 b c \left (2 A c d e \left (4 c d^2+9 a e^2\right )+B \left (2 c^2 d^4+3 a c d^2 e^2+5 a^2 e^4\right )\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (e^3 \left (8 B c d^2-B e (3 b d+2 a e)-5 A e (2 c d-b e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (6 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )+c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (3 b^4 e^3 (3 B d-5 A e)-2 b^3 e^2 \left (9 B c d^2-10 A c d e-3 a B e^2\right )+4 b^2 c e \left (10 B c d^3+3 A c d^2 e-14 a B d e^2+25 a A e^3\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )\right )-8 b c \left (2 A c d e \left (4 c d^2+9 a e^2\right )+B \left (2 c^2 d^4+3 a c d^2 e^2+5 a^2 e^4\right )\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\left (e^3 \left (8 B c d^2-B e (3 b d+2 a e)-5 A e (2 c d-b e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (6 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )+c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (3 b^4 e^3 (3 B d-5 A e)-2 b^3 e^2 \left (9 B c d^2-10 A c d e-3 a B e^2\right )+4 b^2 c e \left (10 B c d^3+3 A c d^2 e-14 a B d e^2+25 a A e^3\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )\right )-8 b c \left (2 A c d e \left (4 c d^2+9 a e^2\right )+B \left (2 c^2 d^4+3 a c d^2 e^2+5 a^2 e^4\right )\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {e^3 \left (8 B c d^2-B e (3 b d+2 a e)-5 A e (2 c d-b e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 4.71, size = 754, normalized size = 1.01 \begin {gather*} \frac {2 \left (\frac {-4 b c \left (A c \left (a e^2 (9 d-7 e x)+2 c d^2 (d-3 e x)\right )-B \left (-4 a^2 e^3+a c d e (d+3 e x)+2 c^2 d^3 x\right )\right )-8 c^2 \left (a^2 e^2 (4 A e-5 B d+3 B e x)-a c d e (A d-7 A e x+2 B d x)+2 A c^2 d^3 x\right )+b^3 e \left (2 a B e^2+A c e (3 d-5 e x)+B c d (3 e x-7 d)\right )+2 b^2 c \left (16 a A e^3+a B e^2 (e x-5 d)+A c d e (5 d+e x)+2 B c d^2 (d-4 e x)\right )+b^4 e^2 (3 B d-5 A e)}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+x (b+c x)} \left (e (b d-a e)-c d^2\right )}+\frac {e \sqrt {a+x (b+c x)} \left (-8 b c \left (B \left (5 a^2 e^4+3 a c d^2 e^2+2 c^2 d^4\right )+2 A c d e \left (9 a e^2+4 c d^2\right )\right )+16 c^2 \left (A \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )+a B d e \left (13 a e^2-2 c d^2\right )\right )+2 b^3 e^2 \left (3 a B e^2+10 A c d e-9 B c d^2\right )+4 b^2 c e \left (25 a A e^3-14 a B d e^2+3 A c d^2 e+10 B c d^3\right )+3 b^4 e^3 (3 B d-5 A e)\right )}{2 \left (b^2-4 a c\right ) (d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac {3 e^3 \left (b^2-4 a c\right ) \left (-B e (2 a e+3 b d)+5 A e (b e-2 c d)+8 B c d^2\right ) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{4 \left (e (a e-b d)+c d^2\right )^{5/2}}+\frac {-2 A c (a e+c d x)+a B (2 c (d-e x)-b e)+A b^2 e+A b c (e x-d)+b B c d x}{(d+e x) (a+x (b+c x))^{3/2}}\right )}{3 \left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*((e*(3*b^4*e^3*(3*B*d - 5*A*e) + 2*b^3*e^2*(-9*B*c*d^2 + 10*A*c*d*e + 3*a*B*e^2) + 4*b^2*c*e*(10*B*c*d^3 +
3*A*c*d^2*e - 14*a*B*d*e^2 + 25*a*A*e^3) + 16*c^2*(a*B*d*e*(-2*c*d^2 + 13*a*e^2) + A*(2*c^2*d^4 + 9*a*c*d^2*e^
2 - 8*a^2*e^4)) - 8*b*c*(2*A*c*d*e*(4*c*d^2 + 9*a*e^2) + B*(2*c^2*d^4 + 3*a*c*d^2*e^2 + 5*a^2*e^4)))*Sqrt[a +
x*(b + c*x)])/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + (A*b^2*e + b*B*c*d*x - 2*A*c*(a*e + c
*d*x) + A*b*c*(-d + e*x) + a*B*(-(b*e) + 2*c*(d - e*x)))/((d + e*x)*(a + x*(b + c*x))^(3/2)) + (b^4*e^2*(3*B*d
 - 5*A*e) + 2*b^2*c*(16*a*A*e^3 + 2*B*c*d^2*(d - 4*e*x) + a*B*e^2*(-5*d + e*x) + A*c*d*e*(5*d + e*x)) + b^3*e*
(2*a*B*e^2 + A*c*e*(3*d - 5*e*x) + B*c*d*(-7*d + 3*e*x)) - 8*c^2*(2*A*c^2*d^3*x - a*c*d*e*(A*d + 2*B*d*x - 7*A
*e*x) + a^2*e^2*(-5*B*d + 4*A*e + 3*B*e*x)) - 4*b*c*(A*c*(a*e^2*(9*d - 7*e*x) + 2*c*d^2*(d - 3*e*x)) - B*(-4*a
^2*e^3 + 2*c^2*d^3*x + a*c*d*e*(d + 3*e*x))))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d + e*x)*Sqrt[a + x*(
b + c*x)]) + (3*(b^2 - 4*a*c)*e^3*(8*B*c*d^2 - B*e*(3*b*d + 2*a*e) + 5*A*e*(-2*c*d + b*e))*ArcTanh[(-(b*d) + 2
*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*(c*d^2 + e*(-(b*d) + a*e
))^(5/2))))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))

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IntegrateAlgebraic [F]  time = 180.70, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

$Aborted

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fricas [B]  time = 103.12, size = 12690, normalized size = 17.01

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(3*(8*(B*a^2*b^4*c - 8*B*a^3*b^2*c^2 + 16*B*a^4*c^3)*d^3*e^3 - (3*B*a^2*b^5 + 160*A*a^4*c^3 + 16*(3*B*a^
4*b - 5*A*a^3*b^2)*c^2 - 2*(12*B*a^3*b^3 - 5*A*a^2*b^4)*c)*d^2*e^4 - (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(2*B*a^5
- 5*A*a^4*b)*c^2 - 8*(2*B*a^4*b^2 - 5*A*a^3*b^3)*c)*d*e^5 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d^2*
e^4 - (3*B*b^5*c^2 + 160*A*a^2*c^5 + 16*(3*B*a^2*b - 5*A*a*b^2)*c^4 - 2*(12*B*a*b^3 - 5*A*b^4)*c^3)*d*e^5 - (1
6*(2*B*a^3 - 5*A*a^2*b)*c^4 - 8*(2*B*a^2*b^2 - 5*A*a*b^3)*c^3 + (2*B*a*b^4 - 5*A*b^5)*c^2)*e^6)*x^5 + (8*(B*b^
4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d^3*e^3 + (13*B*b^5*c^2 - 160*A*a^2*c^5 + 16*(13*B*a^2*b + 5*A*a*b^2)*c^
4 - 2*(52*B*a*b^3 + 5*A*b^4)*c^3)*d^2*e^4 - (6*B*b^6*c + 16*(2*B*a^3 + 15*A*a^2*b)*c^4 + 40*(2*B*a^2*b^2 - 3*A
*a*b^3)*c^3 - (46*B*a*b^4 - 15*A*b^5)*c^2)*d*e^5 - 2*(16*(2*B*a^3*b - 5*A*a^2*b^2)*c^3 - 8*(2*B*a^2*b^3 - 5*A*
a*b^4)*c^2 + (2*B*a*b^5 - 5*A*b^6)*c)*e^6)*x^4 + (16*(B*b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c^4)*d^3*e^3 + 2*
(B*b^6*c - 10*A*b^5*c^2 + 32*(4*B*a^3 - 5*A*a^2*b)*c^4 - 16*(3*B*a^2*b^2 - 5*A*a*b^3)*c^3)*d^2*e^4 - (3*B*b^7
- 14*B*a*b^5*c + 320*A*a^3*c^4 + 160*(B*a^3*b - A*a^2*b^2)*c^3 - 4*(8*B*a^2*b^3 - 5*A*a*b^4)*c^2)*d*e^5 - (2*B
*a*b^6 - 5*A*b^7 + 32*(2*B*a^4 - 5*A*a^3*b)*c^3 - 6*(2*B*a^2*b^4 - 5*A*a*b^5)*c)*e^6)*x^3 + (8*(B*b^6*c - 6*B*
a*b^4*c^2 + 32*B*a^3*c^4)*d^3*e^3 - (3*B*b^7 - 160*B*a^3*b*c^3 + 320*A*a^3*c^4 + 4*(32*B*a^2*b^3 - 15*A*a*b^4)
*c^2 - 2*(17*B*a*b^5 - 5*A*b^6)*c)*d^2*e^4 - (8*B*a*b^6 - 5*A*b^7 + 32*(2*B*a^4 + 5*A*a^3*b)*c^3 + 32*(3*B*a^3
*b^2 - 5*A*a^2*b^3)*c^2 - 10*(6*B*a^2*b^4 - 5*A*a*b^5)*c)*d*e^5 - 2*(2*B*a^2*b^5 - 5*A*a*b^6 + 16*(2*B*a^4*b -
 5*A*a^3*b^2)*c^2 - 8*(2*B*a^3*b^3 - 5*A*a^2*b^4)*c)*e^6)*x^2 + (16*(B*a*b^5*c - 8*B*a^2*b^3*c^2 + 16*B*a^3*b*
c^3)*d^3*e^3 - 2*(3*B*a*b^6 - 32*(2*B*a^4 - 5*A*a^3*b)*c^3 + 80*(B*a^3*b^2 - A*a^2*b^3)*c^2 - 2*(14*B*a^2*b^4
- 5*A*a*b^5)*c)*d^2*e^4 - (7*B*a^2*b^5 - 10*A*a*b^6 + 160*A*a^4*c^3 + 16*(7*B*a^4*b - 15*A*a^3*b^2)*c^2 - 2*(2
8*B*a^3*b^3 - 45*A*a^2*b^4)*c)*d*e^5 - (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(2*B*a^5 - 5*A*a^4*b)*c^2 - 8*(2*B*a^4*
b^2 - 5*A*a^3*b^3)*c)*e^6)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*
c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*
e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(2*(4
*(2*B*a^2 - 3*A*a*b)*c^5 + (2*B*a*b^2 + A*b^3)*c^4)*d^7 - 8*(4*A*a^2*c^5 + (4*B*a^2*b - 11*A*a*b^2)*c^4 + (2*B
*a*b^3 + A*b^4)*c^3)*d^6*e + 4*(4*(10*B*a^3 - A*a^2*b)*c^4 - 2*(2*B*a^2*b^2 + 13*A*a*b^3)*c^3 + 3*(2*B*a*b^4 +
 A*b^5)*c^2)*d^5*e^2 - 8*(32*A*a^3*c^4 + 4*(9*B*a^3*b - 8*A*a^2*b^2)*c^3 - (7*B*a^2*b^3 + 3*A*a*b^4)*c^2 + (2*
B*a*b^5 + A*b^6)*c)*d^4*e^3 + (4*B*a*b^6 + 2*A*b^7 + 8*(4*B*a^4 + 45*A*a^3*b)*c^3 + 2*(86*B*a^3*b^2 - 155*A*a^
2*b^3)*c^2 - (31*B*a^2*b^4 - 32*A*a*b^5)*c)*d^3*e^4 + (7*B*a^2*b^5 - 16*A*a*b^6 - 176*A*a^4*c^3 - 16*(B*a^4*b
+ A*a^3*b^2)*c^2 - (48*B*a^3*b^3 - 91*A*a^2*b^4)*c)*d^2*e^5 - (11*B*a^3*b^4 - 11*A*a^2*b^5 + 112*(B*a^5 - A*a^
4*b)*c^2 - 4*(20*B*a^4*b^2 - 19*A*a^3*b^3)*c)*d*e^6 + 3*(A*a^3*b^4 - 8*A*a^4*b^2*c + 16*A*a^5*c^2)*e^7 + (16*(
B*b*c^6 - 2*A*c^7)*d^6*e - 8*(7*B*b^2*c^5 - 4*(B*a + 3*A*b)*c^6)*d^5*e^2 + 2*(29*B*b^3*c^4 - 88*A*a*c^6 + 2*(2
*B*a*b - 19*A*b^2)*c^5)*d^4*e^3 - (27*B*b^4*c^3 + 176*(B*a^2 - 2*A*a*b)*c^5 + 8*(B*a*b^2 + A*b^3)*c^4)*d^3*e^4
 + (9*B*b^5*c^2 - 16*A*a^2*c^5 + 16*(17*B*a^2*b - 16*A*a*b^2)*c^4 - (44*B*a*b^3 - 35*A*b^4)*c^3)*d^2*e^5 - (16
*(13*B*a^3 - A*a^2*b)*c^4 - 16*(B*a^2*b^2 + 5*A*a*b^3)*c^3 + 3*(B*a*b^4 + 5*A*b^5)*c^2)*d*e^6 + (128*A*a^3*c^4
 + 20*(2*B*a^3*b - 5*A*a^2*b^2)*c^3 - 3*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2)*e^7)*x^4 + 2*(8*(B*b*c^6 - 2*A*c^7)*d^7
 - 8*(2*B*b^2*c^5 - (2*B*a + 3*A*b)*c^6)*d^6*e - (13*B*b^3*c^4 + 88*A*a*c^6 - 2*(14*B*a*b + 17*A*b^2)*c^5)*d^5
*e^2 + (36*B*b^4*c^3 + 4*(2*B*a^2 + 11*A*a*b)*c^5 - (46*B*a*b^2 + 61*A*b^3)*c^4)*d^4*e^3 - 2*(12*B*b^5*c^2 + 6
4*A*a^2*c^5 + 2*(32*B*a^2*b - 49*A*a*b^2)*c^4 - (19*B*a*b^3 + 2*A*b^4)*c^3)*d^3*e^4 + (9*B*b^6*c - 16*(2*B*a^3
 - 11*A*a^2*b)*c^4 + 2*(106*B*a^2*b^2 - 121*A*a*b^3)*c^3 - 6*(8*B*a*b^4 - 5*A*b^5)*c^2)*d^2*e^5 - (56*A*a^3*c^
4 + 2*(74*B*a^3*b + 19*A*a^2*b^2)*c^3 - 15*(B*a^2*b^3 + 6*A*a*b^4)*c^2 + 3*(B*a*b^5 + 5*A*b^6)*c)*d*e^6 - 3*(4
*(2*B*a^4 - 13*A*a^3*b)*c^3 - 7*(2*B*a^3*b^2 - 5*A*a^2*b^3)*c^2 + (2*B*a^2*b^4 - 5*A*a*b^5)*c)*e^7)*x^3 + 3*(8
*(B*b^2*c^5 - 2*A*b*c^6)*d^7 - 2*(13*B*b^3*c^4 + 8*A*a*c^6 - 2*(6*B*a*b + 11*A*b^2)*c^5)*d^6*e + 2*(12*B*b^4*c
^3 + 4*(6*B*a^2 - 5*A*a*b)*c^5 - (18*B*a*b^2 + 13*A*b^3)*c^4)*d^5*e^2 - 4*(B*b^5*c^2 + 32*A*a^2*c^5 + 2*(6*B*a
^2*b - 17*A*a*b^2)*c^4 - 2*(B*a*b^3 - 2*A*b^4)*c^3)*d^4*e^3 - (5*B*b^6*c + 68*A*a*b^3*c^3 + 64*(B*a^3 - 2*A*a^
2*b)*c^4 - 14*(B*a*b^4 + A*b^5)*c^2)*d^3*e^4 + (3*B*b^7 - 48*A*a^3*c^4 + 4*(22*B*a^3*b + 7*A*a^2*b^2)*c^3 + 2*
(11*B*a^2*b^3 - 21*A*a*b^4)*c^2 - (14*B*a*b^5 - 5*A*b^6)*c)*d^2*e^5 - (B*a*b^6 + 5*A*b^7 + 8*(14*B*a^4 + A*a^3
*b)*c^3 + 2*(2*B*a^3*b^2 + 9*A*a^2*b^3)*c^2 - 2*(B*a^2*b^4 + 15*A*a*b^5)*c)*d*e^6 - (2*B*a^2*b^5 - 5*A*a*b^6 -
 16*A*a^3*b^2*c^2 - 64*A*a^4*c^3 - 6*(2*B*a^3*b^3 - 5*A*a^2*b^4)*c)*e^7)*x^2 + 2*(3*(B*b^3*c^4 - 8*A*a*c^6 + 2
*(2*B*a*b - A*b^2)*c^5)*d^7 - (12*B*b^4*c^3 - 4*(2*B*a^2 + 15*A*a*b)*c^5 + (22*B*a*b^2 - 19*A*b^3)*c^4)*d^6*e
+ 2*(9*B*b^5*c^2 - 56*A*a^2*c^5 + 4*(13*B*a^2*b - 8*A*a*b^2)*c^4 - 2*(5*B*a*b^3 + 4*A*b^4)*c^3)*d^5*e^2 - 2*(6
*B*b^6*c + 4*(2*B*a^3 - 11*A*a^2*b)*c^4 + 2*(44*B*a^2*b^2 - 23*A*a*b^3)*c^3 - 3*(8*B*a*b^4 - A*b^5)*c^2)*d^4*e
^3 + (3*B*b^7 - 152*A*a^3*c^4 + 2*(6*B*a^3*b + 61*A*a^2*b^2)*c^3 + (91*B*a^2*b^3 - 102*A*a*b^4)*c^2 - (23*B*a*
b^5 - 14*A*b^6)*c)*d^3*e^4 + (5*B*a*b^6 - 5*A*b^7 - 4*(14*B*a^4 - 39*A*a^3*b)*c^3 + (74*B*a^3*b^2 - 125*A*a^2*
b^3)*c^2 - (38*B*a^2*b^4 - 43*A*a*b^5)*c)*d^2*e^5 - (4*B*a^2*b^5 + 5*A*a*b^6 + 64*A*a^4*c^3 + 20*(4*B*a^4*b +
A*a^3*b^2)*c^2 - 2*(15*B*a^3*b^3 + 16*A*a^2*b^4)*c)*d*e^6 - 2*(2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(B*a^5 - 4*A*a^4
*b)*c^2 - (14*B*a^4*b^2 - 37*A*a^3*b^3)*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a
^4*c^6)*d^9 - 4*(a^2*b^5*c^3 - 8*a^3*b^3*c^4 + 16*a^4*b*c^5)*d^8*e + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^
4*b^2*c^4 + 32*a^5*c^5)*d^7*e^2 - 4*(a^2*b^7*c - 5*a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 48*a^5*b*c^4)*d^6*e^3 + (a^2*
b^8 + 4*a^3*b^6*c - 74*a^4*b^4*c^2 + 144*a^5*b^2*c^3 + 96*a^6*c^4)*d^5*e^4 - 4*(a^3*b^7 - 5*a^4*b^5*c - 8*a^5*
b^3*c^2 + 48*a^6*b*c^3)*d^4*e^5 + 2*(3*a^4*b^6 - 22*a^5*b^4*c + 32*a^6*b^2*c^2 + 32*a^7*c^3)*d^3*e^6 - 4*(a^5*
b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*d^2*e^7 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*d*e^8 + ((b^4*c^6 - 8*a*b^2*c
^7 + 16*a^2*c^8)*d^8*e - 4*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^7*e^2 + 2*(3*b^6*c^4 - 22*a*b^4*c^5 + 32*a
^2*b^2*c^6 + 32*a^3*c^7)*d^6*e^3 - 4*(b^7*c^3 - 5*a*b^5*c^4 - 8*a^2*b^3*c^5 + 48*a^3*b*c^6)*d^5*e^4 + (b^8*c^2
 + 4*a*b^6*c^3 - 74*a^2*b^4*c^4 + 144*a^3*b^2*c^5 + 96*a^4*c^6)*d^4*e^5 - 4*(a*b^7*c^2 - 5*a^2*b^5*c^3 - 8*a^3
*b^3*c^4 + 48*a^4*b*c^5)*d^3*e^6 + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a^5*c^5)*d^2*e^7 -
4*(a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 16*a^5*b*c^4)*d*e^8 + (a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*e^9)*x^5 + (
(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^9 - 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^8*e - 2*(b^6*c^4 - 10*a*
b^4*c^5 + 32*a^2*b^2*c^6 - 32*a^3*c^7)*d^7*e^2 + 4*(2*b^7*c^3 - 17*a*b^5*c^4 + 40*a^2*b^3*c^5 - 16*a^3*b*c^6)*
d^6*e^3 - (7*b^8*c^2 - 44*a*b^6*c^3 + 10*a^2*b^4*c^4 + 240*a^3*b^2*c^5 - 96*a^4*c^6)*d^5*e^4 + 2*(b^9*c + 2*a*
b^7*c^2 - 64*a^2*b^5*c^3 + 160*a^3*b^3*c^4)*d^4*e^5 - 2*(4*a*b^8*c - 23*a^2*b^6*c^2 - 10*a^3*b^4*c^3 + 160*a^4
*b^2*c^4 - 32*a^5*c^5)*d^3*e^6 + 4*(3*a^2*b^7*c - 23*a^3*b^5*c^2 + 40*a^4*b^3*c^3 + 16*a^5*b*c^4)*d^2*e^7 - (8
*a^3*b^6*c - 65*a^4*b^4*c^2 + 136*a^5*b^2*c^3 - 16*a^6*c^4)*d*e^8 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^
3)*e^9)*x^4 + (2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^9 - (7*b^6*c^4 - 58*a*b^4*c^5 + 128*a^2*b^2*c^6 - 32
*a^3*c^7)*d^8*e + 8*(b^7*c^3 - 8*a*b^5*c^4 + 16*a^2*b^3*c^5)*d^7*e^2 - 2*(b^8*c^2 - 4*a*b^6*c^3 - 20*a^2*b^4*c
^4 + 96*a^3*b^2*c^5 - 64*a^4*c^6)*d^6*e^3 - 2*(b^9*c - 10*a*b^7*c^2 + 38*a^2*b^5*c^3 - 80*a^3*b^3*c^4 + 96*a^4
*b*c^5)*d^5*e^4 + (b^10 - 2*a*b^8*c - 26*a^2*b^6*c^2 + 60*a^3*b^4*c^3 + 192*a^5*c^5)*d^4*e^5 - 4*(a*b^9 - 6*a^
2*b^7*c + 4*a^3*b^5*c^2 + 64*a^5*b*c^4)*d^3*e^6 + 2*(3*a^2*b^8 - 20*a^3*b^6*c + 20*a^4*b^4*c^2 + 32*a^5*b^2*c^
3 + 64*a^6*c^4)*d^2*e^7 - 2*(2*a^3*b^7 - 13*a^4*b^5*c + 8*a^5*b^3*c^2 + 48*a^6*b*c^3)*d*e^8 + (a^4*b^6 - 6*a^5
*b^4*c + 32*a^7*c^3)*e^9)*x^3 + ((b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*d^9 - 2*(2*b^7*c^3 - 13*a*b^5*c^4 + 8*a^
2*b^3*c^5 + 48*a^3*b*c^6)*d^8*e + 2*(3*b^8*c^2 - 20*a*b^6*c^3 + 20*a^2*b^4*c^4 + 32*a^3*b^2*c^5 + 64*a^4*c^6)*
d^7*e^2 - 4*(b^9*c - 6*a*b^7*c^2 + 4*a^2*b^5*c^3 + 64*a^4*b*c^5)*d^6*e^3 + (b^10 - 2*a*b^8*c - 26*a^2*b^6*c^2
+ 60*a^3*b^4*c^3 + 192*a^5*c^5)*d^5*e^4 - 2*(a*b^9 - 10*a^2*b^7*c + 38*a^3*b^5*c^2 - 80*a^4*b^3*c^3 + 96*a^5*b
*c^4)*d^4*e^5 - 2*(a^2*b^8 - 4*a^3*b^6*c - 20*a^4*b^4*c^2 + 96*a^5*b^2*c^3 - 64*a^6*c^4)*d^3*e^6 + 8*(a^3*b^7
- 8*a^4*b^5*c + 16*a^5*b^3*c^2)*d^2*e^7 - (7*a^4*b^6 - 58*a^5*b^4*c + 128*a^6*b^2*c^2 - 32*a^7*c^3)*d*e^8 + 2*
(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*e^9)*x^2 + (2*(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*d^9 - (8*a*b^6
*c^3 - 65*a^2*b^4*c^4 + 136*a^3*b^2*c^5 - 16*a^4*c^6)*d^8*e + 4*(3*a*b^7*c^2 - 23*a^2*b^5*c^3 + 40*a^3*b^3*c^4
 + 16*a^4*b*c^5)*d^7*e^2 - 2*(4*a*b^8*c - 23*a^2*b^6*c^2 - 10*a^3*b^4*c^3 + 160*a^4*b^2*c^4 - 32*a^5*c^5)*d^6*
e^3 + 2*(a*b^9 + 2*a^2*b^7*c - 64*a^3*b^5*c^2 + 160*a^4*b^3*c^3)*d^5*e^4 - (7*a^2*b^8 - 44*a^3*b^6*c + 10*a^4*
b^4*c^2 + 240*a^5*b^2*c^3 - 96*a^6*c^4)*d^4*e^5 + 4*(2*a^3*b^7 - 17*a^4*b^5*c + 40*a^5*b^3*c^2 - 16*a^6*b*c^3)
*d^3*e^6 - 2*(a^4*b^6 - 10*a^5*b^4*c + 32*a^6*b^2*c^2 - 32*a^7*c^3)*d^2*e^7 - 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^
7*b*c^2)*d*e^8 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*e^9)*x), -1/6*(3*(8*(B*a^2*b^4*c - 8*B*a^3*b^2*c^2 + 16*
B*a^4*c^3)*d^3*e^3 - (3*B*a^2*b^5 + 160*A*a^4*c^3 + 16*(3*B*a^4*b - 5*A*a^3*b^2)*c^2 - 2*(12*B*a^3*b^3 - 5*A*a
^2*b^4)*c)*d^2*e^4 - (2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(2*B*a^5 - 5*A*a^4*b)*c^2 - 8*(2*B*a^4*b^2 - 5*A*a^3*b^3)
*c)*d*e^5 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d^2*e^4 - (3*B*b^5*c^2 + 160*A*a^2*c^5 + 16*(3*B*a^2
*b - 5*A*a*b^2)*c^4 - 2*(12*B*a*b^3 - 5*A*b^4)*c^3)*d*e^5 - (16*(2*B*a^3 - 5*A*a^2*b)*c^4 - 8*(2*B*a^2*b^2 - 5
*A*a*b^3)*c^3 + (2*B*a*b^4 - 5*A*b^5)*c^2)*e^6)*x^5 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d^3*e^3 +
(13*B*b^5*c^2 - 160*A*a^2*c^5 + 16*(13*B*a^2*b + 5*A*a*b^2)*c^4 - 2*(52*B*a*b^3 + 5*A*b^4)*c^3)*d^2*e^4 - (6*B
*b^6*c + 16*(2*B*a^3 + 15*A*a^2*b)*c^4 + 40*(2*B*a^2*b^2 - 3*A*a*b^3)*c^3 - (46*B*a*b^4 - 15*A*b^5)*c^2)*d*e^5
 - 2*(16*(2*B*a^3*b - 5*A*a^2*b^2)*c^3 - 8*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2 + (2*B*a*b^5 - 5*A*b^6)*c)*e^6)*x^4 +
 (16*(B*b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c^4)*d^3*e^3 + 2*(B*b^6*c - 10*A*b^5*c^2 + 32*(4*B*a^3 - 5*A*a^2*
b)*c^4 - 16*(3*B*a^2*b^2 - 5*A*a*b^3)*c^3)*d^2*e^4 - (3*B*b^7 - 14*B*a*b^5*c + 320*A*a^3*c^4 + 160*(B*a^3*b -
A*a^2*b^2)*c^3 - 4*(8*B*a^2*b^3 - 5*A*a*b^4)*c^2)*d*e^5 - (2*B*a*b^6 - 5*A*b^7 + 32*(2*B*a^4 - 5*A*a^3*b)*c^3
- 6*(2*B*a^2*b^4 - 5*A*a*b^5)*c)*e^6)*x^3 + (8*(B*b^6*c - 6*B*a*b^4*c^2 + 32*B*a^3*c^4)*d^3*e^3 - (3*B*b^7 - 1
60*B*a^3*b*c^3 + 320*A*a^3*c^4 + 4*(32*B*a^2*b^3 - 15*A*a*b^4)*c^2 - 2*(17*B*a*b^5 - 5*A*b^6)*c)*d^2*e^4 - (8*
B*a*b^6 - 5*A*b^7 + 32*(2*B*a^4 + 5*A*a^3*b)*c^3 + 32*(3*B*a^3*b^2 - 5*A*a^2*b^3)*c^2 - 10*(6*B*a^2*b^4 - 5*A*
a*b^5)*c)*d*e^5 - 2*(2*B*a^2*b^5 - 5*A*a*b^6 + 16*(2*B*a^4*b - 5*A*a^3*b^2)*c^2 - 8*(2*B*a^3*b^3 - 5*A*a^2*b^4
)*c)*e^6)*x^2 + (16*(B*a*b^5*c - 8*B*a^2*b^3*c^2 + 16*B*a^3*b*c^3)*d^3*e^3 - 2*(3*B*a*b^6 - 32*(2*B*a^4 - 5*A*
a^3*b)*c^3 + 80*(B*a^3*b^2 - A*a^2*b^3)*c^2 - 2*(14*B*a^2*b^4 - 5*A*a*b^5)*c)*d^2*e^4 - (7*B*a^2*b^5 - 10*A*a*
b^6 + 160*A*a^4*c^3 + 16*(7*B*a^4*b - 15*A*a^3*b^2)*c^2 - 2*(28*B*a^3*b^3 - 45*A*a^2*b^4)*c)*d*e^5 - (2*B*a^3*
b^4 - 5*A*a^2*b^5 + 16*(2*B*a^5 - 5*A*a^4*b)*c^2 - 8*(2*B*a^4*b^2 - 5*A*a^3*b^3)*c)*e^6)*x)*sqrt(-c*d^2 + b*d*
e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c
*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) + 2*(2*(4*(2*
B*a^2 - 3*A*a*b)*c^5 + (2*B*a*b^2 + A*b^3)*c^4)*d^7 - 8*(4*A*a^2*c^5 + (4*B*a^2*b - 11*A*a*b^2)*c^4 + (2*B*a*b
^3 + A*b^4)*c^3)*d^6*e + 4*(4*(10*B*a^3 - A*a^2*b)*c^4 - 2*(2*B*a^2*b^2 + 13*A*a*b^3)*c^3 + 3*(2*B*a*b^4 + A*b
^5)*c^2)*d^5*e^2 - 8*(32*A*a^3*c^4 + 4*(9*B*a^3*b - 8*A*a^2*b^2)*c^3 - (7*B*a^2*b^3 + 3*A*a*b^4)*c^2 + (2*B*a*
b^5 + A*b^6)*c)*d^4*e^3 + (4*B*a*b^6 + 2*A*b^7 + 8*(4*B*a^4 + 45*A*a^3*b)*c^3 + 2*(86*B*a^3*b^2 - 155*A*a^2*b^
3)*c^2 - (31*B*a^2*b^4 - 32*A*a*b^5)*c)*d^3*e^4 + (7*B*a^2*b^5 - 16*A*a*b^6 - 176*A*a^4*c^3 - 16*(B*a^4*b + A*
a^3*b^2)*c^2 - (48*B*a^3*b^3 - 91*A*a^2*b^4)*c)*d^2*e^5 - (11*B*a^3*b^4 - 11*A*a^2*b^5 + 112*(B*a^5 - A*a^4*b)
*c^2 - 4*(20*B*a^4*b^2 - 19*A*a^3*b^3)*c)*d*e^6 + 3*(A*a^3*b^4 - 8*A*a^4*b^2*c + 16*A*a^5*c^2)*e^7 + (16*(B*b*
c^6 - 2*A*c^7)*d^6*e - 8*(7*B*b^2*c^5 - 4*(B*a + 3*A*b)*c^6)*d^5*e^2 + 2*(29*B*b^3*c^4 - 88*A*a*c^6 + 2*(2*B*a
*b - 19*A*b^2)*c^5)*d^4*e^3 - (27*B*b^4*c^3 + 176*(B*a^2 - 2*A*a*b)*c^5 + 8*(B*a*b^2 + A*b^3)*c^4)*d^3*e^4 + (
9*B*b^5*c^2 - 16*A*a^2*c^5 + 16*(17*B*a^2*b - 16*A*a*b^2)*c^4 - (44*B*a*b^3 - 35*A*b^4)*c^3)*d^2*e^5 - (16*(13
*B*a^3 - A*a^2*b)*c^4 - 16*(B*a^2*b^2 + 5*A*a*b^3)*c^3 + 3*(B*a*b^4 + 5*A*b^5)*c^2)*d*e^6 + (128*A*a^3*c^4 + 2
0*(2*B*a^3*b - 5*A*a^2*b^2)*c^3 - 3*(2*B*a^2*b^3 - 5*A*a*b^4)*c^2)*e^7)*x^4 + 2*(8*(B*b*c^6 - 2*A*c^7)*d^7 - 8
*(2*B*b^2*c^5 - (2*B*a + 3*A*b)*c^6)*d^6*e - (13*B*b^3*c^4 + 88*A*a*c^6 - 2*(14*B*a*b + 17*A*b^2)*c^5)*d^5*e^2
 + (36*B*b^4*c^3 + 4*(2*B*a^2 + 11*A*a*b)*c^5 - (46*B*a*b^2 + 61*A*b^3)*c^4)*d^4*e^3 - 2*(12*B*b^5*c^2 + 64*A*
a^2*c^5 + 2*(32*B*a^2*b - 49*A*a*b^2)*c^4 - (19*B*a*b^3 + 2*A*b^4)*c^3)*d^3*e^4 + (9*B*b^6*c - 16*(2*B*a^3 - 1
1*A*a^2*b)*c^4 + 2*(106*B*a^2*b^2 - 121*A*a*b^3)*c^3 - 6*(8*B*a*b^4 - 5*A*b^5)*c^2)*d^2*e^5 - (56*A*a^3*c^4 +
2*(74*B*a^3*b + 19*A*a^2*b^2)*c^3 - 15*(B*a^2*b^3 + 6*A*a*b^4)*c^2 + 3*(B*a*b^5 + 5*A*b^6)*c)*d*e^6 - 3*(4*(2*
B*a^4 - 13*A*a^3*b)*c^3 - 7*(2*B*a^3*b^2 - 5*A*a^2*b^3)*c^2 + (2*B*a^2*b^4 - 5*A*a*b^5)*c)*e^7)*x^3 + 3*(8*(B*
b^2*c^5 - 2*A*b*c^6)*d^7 - 2*(13*B*b^3*c^4 + 8*A*a*c^6 - 2*(6*B*a*b + 11*A*b^2)*c^5)*d^6*e + 2*(12*B*b^4*c^3 +
 4*(6*B*a^2 - 5*A*a*b)*c^5 - (18*B*a*b^2 + 13*A*b^3)*c^4)*d^5*e^2 - 4*(B*b^5*c^2 + 32*A*a^2*c^5 + 2*(6*B*a^2*b
 - 17*A*a*b^2)*c^4 - 2*(B*a*b^3 - 2*A*b^4)*c^3)*d^4*e^3 - (5*B*b^6*c + 68*A*a*b^3*c^3 + 64*(B*a^3 - 2*A*a^2*b)
*c^4 - 14*(B*a*b^4 + A*b^5)*c^2)*d^3*e^4 + (3*B*b^7 - 48*A*a^3*c^4 + 4*(22*B*a^3*b + 7*A*a^2*b^2)*c^3 + 2*(11*
B*a^2*b^3 - 21*A*a*b^4)*c^2 - (14*B*a*b^5 - 5*A*b^6)*c)*d^2*e^5 - (B*a*b^6 + 5*A*b^7 + 8*(14*B*a^4 + A*a^3*b)*
c^3 + 2*(2*B*a^3*b^2 + 9*A*a^2*b^3)*c^2 - 2*(B*a^2*b^4 + 15*A*a*b^5)*c)*d*e^6 - (2*B*a^2*b^5 - 5*A*a*b^6 - 16*
A*a^3*b^2*c^2 - 64*A*a^4*c^3 - 6*(2*B*a^3*b^3 - 5*A*a^2*b^4)*c)*e^7)*x^2 + 2*(3*(B*b^3*c^4 - 8*A*a*c^6 + 2*(2*
B*a*b - A*b^2)*c^5)*d^7 - (12*B*b^4*c^3 - 4*(2*B*a^2 + 15*A*a*b)*c^5 + (22*B*a*b^2 - 19*A*b^3)*c^4)*d^6*e + 2*
(9*B*b^5*c^2 - 56*A*a^2*c^5 + 4*(13*B*a^2*b - 8*A*a*b^2)*c^4 - 2*(5*B*a*b^3 + 4*A*b^4)*c^3)*d^5*e^2 - 2*(6*B*b
^6*c + 4*(2*B*a^3 - 11*A*a^2*b)*c^4 + 2*(44*B*a^2*b^2 - 23*A*a*b^3)*c^3 - 3*(8*B*a*b^4 - A*b^5)*c^2)*d^4*e^3 +
 (3*B*b^7 - 152*A*a^3*c^4 + 2*(6*B*a^3*b + 61*A*a^2*b^2)*c^3 + (91*B*a^2*b^3 - 102*A*a*b^4)*c^2 - (23*B*a*b^5
- 14*A*b^6)*c)*d^3*e^4 + (5*B*a*b^6 - 5*A*b^7 - 4*(14*B*a^4 - 39*A*a^3*b)*c^3 + (74*B*a^3*b^2 - 125*A*a^2*b^3)
*c^2 - (38*B*a^2*b^4 - 43*A*a*b^5)*c)*d^2*e^5 - (4*B*a^2*b^5 + 5*A*a*b^6 + 64*A*a^4*c^3 + 20*(4*B*a^4*b + A*a^
3*b^2)*c^2 - 2*(15*B*a^3*b^3 + 16*A*a^2*b^4)*c)*d*e^6 - 2*(2*B*a^3*b^4 - 5*A*a^2*b^5 + 16*(B*a^5 - 4*A*a^4*b)*
c^2 - (14*B*a^4*b^2 - 37*A*a^3*b^3)*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4*c
^6)*d^9 - 4*(a^2*b^5*c^3 - 8*a^3*b^3*c^4 + 16*a^4*b*c^5)*d^8*e + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^
2*c^4 + 32*a^5*c^5)*d^7*e^2 - 4*(a^2*b^7*c - 5*a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 48*a^5*b*c^4)*d^6*e^3 + (a^2*b^8
+ 4*a^3*b^6*c - 74*a^4*b^4*c^2 + 144*a^5*b^2*c^3 + 96*a^6*c^4)*d^5*e^4 - 4*(a^3*b^7 - 5*a^4*b^5*c - 8*a^5*b^3*
c^2 + 48*a^6*b*c^3)*d^4*e^5 + 2*(3*a^4*b^6 - 22*a^5*b^4*c + 32*a^6*b^2*c^2 + 32*a^7*c^3)*d^3*e^6 - 4*(a^5*b^5
- 8*a^6*b^3*c + 16*a^7*b*c^2)*d^2*e^7 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*d*e^8 + ((b^4*c^6 - 8*a*b^2*c^7 +
 16*a^2*c^8)*d^8*e - 4*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^7*e^2 + 2*(3*b^6*c^4 - 22*a*b^4*c^5 + 32*a^2*b
^2*c^6 + 32*a^3*c^7)*d^6*e^3 - 4*(b^7*c^3 - 5*a*b^5*c^4 - 8*a^2*b^3*c^5 + 48*a^3*b*c^6)*d^5*e^4 + (b^8*c^2 + 4
*a*b^6*c^3 - 74*a^2*b^4*c^4 + 144*a^3*b^2*c^5 + 96*a^4*c^6)*d^4*e^5 - 4*(a*b^7*c^2 - 5*a^2*b^5*c^3 - 8*a^3*b^3
*c^4 + 48*a^4*b*c^5)*d^3*e^6 + 2*(3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a^5*c^5)*d^2*e^7 - 4*(a
^3*b^5*c^2 - 8*a^4*b^3*c^3 + 16*a^5*b*c^4)*d*e^8 + (a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*e^9)*x^5 + ((b^4
*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^9 - 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^8*e - 2*(b^6*c^4 - 10*a*b^4*
c^5 + 32*a^2*b^2*c^6 - 32*a^3*c^7)*d^7*e^2 + 4*(2*b^7*c^3 - 17*a*b^5*c^4 + 40*a^2*b^3*c^5 - 16*a^3*b*c^6)*d^6*
e^3 - (7*b^8*c^2 - 44*a*b^6*c^3 + 10*a^2*b^4*c^4 + 240*a^3*b^2*c^5 - 96*a^4*c^6)*d^5*e^4 + 2*(b^9*c + 2*a*b^7*
c^2 - 64*a^2*b^5*c^3 + 160*a^3*b^3*c^4)*d^4*e^5 - 2*(4*a*b^8*c - 23*a^2*b^6*c^2 - 10*a^3*b^4*c^3 + 160*a^4*b^2
*c^4 - 32*a^5*c^5)*d^3*e^6 + 4*(3*a^2*b^7*c - 23*a^3*b^5*c^2 + 40*a^4*b^3*c^3 + 16*a^5*b*c^4)*d^2*e^7 - (8*a^3
*b^6*c - 65*a^4*b^4*c^2 + 136*a^5*b^2*c^3 - 16*a^6*c^4)*d*e^8 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*e
^9)*x^4 + (2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^9 - (7*b^6*c^4 - 58*a*b^4*c^5 + 128*a^2*b^2*c^6 - 32*a^3
*c^7)*d^8*e + 8*(b^7*c^3 - 8*a*b^5*c^4 + 16*a^2*b^3*c^5)*d^7*e^2 - 2*(b^8*c^2 - 4*a*b^6*c^3 - 20*a^2*b^4*c^4 +
 96*a^3*b^2*c^5 - 64*a^4*c^6)*d^6*e^3 - 2*(b^9*c - 10*a*b^7*c^2 + 38*a^2*b^5*c^3 - 80*a^3*b^3*c^4 + 96*a^4*b*c
^5)*d^5*e^4 + (b^10 - 2*a*b^8*c - 26*a^2*b^6*c^2 + 60*a^3*b^4*c^3 + 192*a^5*c^5)*d^4*e^5 - 4*(a*b^9 - 6*a^2*b^
7*c + 4*a^3*b^5*c^2 + 64*a^5*b*c^4)*d^3*e^6 + 2*(3*a^2*b^8 - 20*a^3*b^6*c + 20*a^4*b^4*c^2 + 32*a^5*b^2*c^3 +
64*a^6*c^4)*d^2*e^7 - 2*(2*a^3*b^7 - 13*a^4*b^5*c + 8*a^5*b^3*c^2 + 48*a^6*b*c^3)*d*e^8 + (a^4*b^6 - 6*a^5*b^4
*c + 32*a^7*c^3)*e^9)*x^3 + ((b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*d^9 - 2*(2*b^7*c^3 - 13*a*b^5*c^4 + 8*a^2*b^
3*c^5 + 48*a^3*b*c^6)*d^8*e + 2*(3*b^8*c^2 - 20*a*b^6*c^3 + 20*a^2*b^4*c^4 + 32*a^3*b^2*c^5 + 64*a^4*c^6)*d^7*
e^2 - 4*(b^9*c - 6*a*b^7*c^2 + 4*a^2*b^5*c^3 + 64*a^4*b*c^5)*d^6*e^3 + (b^10 - 2*a*b^8*c - 26*a^2*b^6*c^2 + 60
*a^3*b^4*c^3 + 192*a^5*c^5)*d^5*e^4 - 2*(a*b^9 - 10*a^2*b^7*c + 38*a^3*b^5*c^2 - 80*a^4*b^3*c^3 + 96*a^5*b*c^4
)*d^4*e^5 - 2*(a^2*b^8 - 4*a^3*b^6*c - 20*a^4*b^4*c^2 + 96*a^5*b^2*c^3 - 64*a^6*c^4)*d^3*e^6 + 8*(a^3*b^7 - 8*
a^4*b^5*c + 16*a^5*b^3*c^2)*d^2*e^7 - (7*a^4*b^6 - 58*a^5*b^4*c + 128*a^6*b^2*c^2 - 32*a^7*c^3)*d*e^8 + 2*(a^5
*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*e^9)*x^2 + (2*(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*d^9 - (8*a*b^6*c^3
 - 65*a^2*b^4*c^4 + 136*a^3*b^2*c^5 - 16*a^4*c^6)*d^8*e + 4*(3*a*b^7*c^2 - 23*a^2*b^5*c^3 + 40*a^3*b^3*c^4 + 1
6*a^4*b*c^5)*d^7*e^2 - 2*(4*a*b^8*c - 23*a^2*b^6*c^2 - 10*a^3*b^4*c^3 + 160*a^4*b^2*c^4 - 32*a^5*c^5)*d^6*e^3
+ 2*(a*b^9 + 2*a^2*b^7*c - 64*a^3*b^5*c^2 + 160*a^4*b^3*c^3)*d^5*e^4 - (7*a^2*b^8 - 44*a^3*b^6*c + 10*a^4*b^4*
c^2 + 240*a^5*b^2*c^3 - 96*a^6*c^4)*d^4*e^5 + 4*(2*a^3*b^7 - 17*a^4*b^5*c + 40*a^5*b^3*c^2 - 16*a^6*b*c^3)*d^3
*e^6 - 2*(a^4*b^6 - 10*a^5*b^4*c + 32*a^6*b^2*c^2 - 32*a^7*c^3)*d^2*e^7 - 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*
c^2)*d*e^8 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*e^9)*x)]

________________________________________________________________________________________

giac [B]  time = 3.79, size = 8475, normalized size = 11.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/6*((32*sqrt(c*d^2 - b*d*e + a*e^2)*B*b*c^4*d^4*e^2 - 64*sqrt(c*d^2 - b*d*e + a*e^2)*A*c^5*d^4*e^2 - 80*sqrt(
c*d^2 - b*d*e + a*e^2)*B*b^2*c^3*d^3*e^3 + 64*sqrt(c*d^2 - b*d*e + a*e^2)*B*a*c^4*d^3*e^3 + 128*sqrt(c*d^2 - b
*d*e + a*e^2)*A*b*c^4*d^3*e^3 - 24*B*b^4*c^(3/2)*d^2*e^5*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*
sqrt(c))) + 192*B*a*b^2*c^(5/2)*d^2*e^5*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 384*B
*a^2*c^(7/2)*d^2*e^5*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) + 36*sqrt(c*d^2 - b*d*e +
a*e^2)*B*b^3*c^2*d^2*e^4 + 48*sqrt(c*d^2 - b*d*e + a*e^2)*B*a*b*c^3*d^2*e^4 - 24*sqrt(c*d^2 - b*d*e + a*e^2)*A
*b^2*c^3*d^2*e^4 - 288*sqrt(c*d^2 - b*d*e + a*e^2)*A*a*c^4*d^2*e^4 + 9*B*b^5*sqrt(c)*d*e^6*log(abs(-2*c*d + b*
e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 72*B*a*b^3*c^(3/2)*d*e^6*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b
*d*e + a*e^2)*sqrt(c))) + 30*A*b^4*c^(3/2)*d*e^6*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))
) + 144*B*a^2*b*c^(5/2)*d*e^6*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 240*A*a*b^2*c^(
5/2)*d*e^6*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) + 480*A*a^2*c^(7/2)*d*e^6*log(abs(-2
*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 18*sqrt(c*d^2 - b*d*e + a*e^2)*B*b^4*c*d*e^5 + 112*sqrt
(c*d^2 - b*d*e + a*e^2)*B*a*b^2*c^2*d*e^5 - 40*sqrt(c*d^2 - b*d*e + a*e^2)*A*b^3*c^2*d*e^5 - 416*sqrt(c*d^2 -
b*d*e + a*e^2)*B*a^2*c^3*d*e^5 + 288*sqrt(c*d^2 - b*d*e + a*e^2)*A*a*b*c^3*d*e^5 + 6*B*a*b^4*sqrt(c)*e^7*log(a
bs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 15*A*b^5*sqrt(c)*e^7*log(abs(-2*c*d + b*e + 2*sqrt
(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 48*B*a^2*b^2*c^(3/2)*e^7*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e
^2)*sqrt(c))) + 120*A*a*b^3*c^(3/2)*e^7*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) + 96*B*
a^3*c^(5/2)*e^7*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 240*A*a^2*b*c^(5/2)*e^7*log(a
bs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 12*sqrt(c*d^2 - b*d*e + a*e^2)*B*a*b^3*c*e^6 + 30*
sqrt(c*d^2 - b*d*e + a*e^2)*A*b^4*c*e^6 + 80*sqrt(c*d^2 - b*d*e + a*e^2)*B*a^2*b*c^2*e^6 - 200*sqrt(c*d^2 - b*
d*e + a*e^2)*A*a*b^2*c^2*e^6 + 256*sqrt(c*d^2 - b*d*e + a*e^2)*A*a^2*c^3*e^6)*sgn(1/(x*e + d))/(sqrt(c*d^2 - b
*d*e + a*e^2)*b^4*c^(7/2)*d^6 - 8*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^2*c^(9/2)*d^6 + 16*sqrt(c*d^2 - b*d*e + a*e^
2)*a^2*c^(11/2)*d^6 - 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^5*c^(5/2)*d^5*e + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^3*c
^(7/2)*d^5*e - 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b*c^(9/2)*d^5*e + 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^6*c^(3/2)*
d^4*e^2 - 21*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^4*c^(5/2)*d^4*e^2 + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^2*c^(7/2
)*d^4*e^2 + 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*c^(9/2)*d^4*e^2 - sqrt(c*d^2 - b*d*e + a*e^2)*b^7*sqrt(c)*d^3*e
^3 + 2*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^5*c^(3/2)*d^3*e^3 + 32*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^3*c^(5/2)*d^3*
e^3 - 96*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b*c^(7/2)*d^3*e^3 + 3*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^6*sqrt(c)*d^2*e
^4 - 21*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^4*c^(3/2)*d^2*e^4 + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b^2*c^(5/2)*d
^2*e^4 + 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^4*c^(7/2)*d^2*e^4 - 3*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^5*sqrt(c)*d*
e^5 + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b^3*c^(3/2)*d*e^5 - 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^4*b*c^(5/2)*d*e^
5 + sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b^4*sqrt(c)*e^6 - 8*sqrt(c*d^2 - b*d*e + a*e^2)*a^4*b^2*c^(3/2)*e^6 + 16*s
qrt(c*d^2 - b*d*e + a*e^2)*a^5*c^(5/2)*e^6) + 2*((((4*(4*B*b*c^5*d^7*e^16*sgn(1/(x*e + d)) - 8*A*c^6*d^7*e^16*
sgn(1/(x*e + d)) - 16*B*b^2*c^4*d^6*e^17*sgn(1/(x*e + d)) + 8*B*a*c^5*d^6*e^17*sgn(1/(x*e + d)) + 28*A*b*c^5*d
^6*e^17*sgn(1/(x*e + d)) + 21*B*b^3*c^3*d^5*e^18*sgn(1/(x*e + d)) - 30*A*b^2*c^4*d^5*e^18*sgn(1/(x*e + d)) - 4
8*A*a*c^5*d^5*e^18*sgn(1/(x*e + d)) - 18*B*b^4*c^2*d^4*e^19*sgn(1/(x*e + d)) + 34*B*a*b^2*c^3*d^4*e^19*sgn(1/(
x*e + d)) + 5*A*b^3*c^3*d^4*e^19*sgn(1/(x*e + d)) - 128*B*a^2*c^4*d^4*e^19*sgn(1/(x*e + d)) + 120*A*a*b*c^4*d^
4*e^19*sgn(1/(x*e + d)) + 12*B*b^5*c*d^3*e^20*sgn(1/(x*e + d)) - 66*B*a*b^3*c^2*d^3*e^20*sgn(1/(x*e + d)) + 18
*A*b^4*c^2*d^3*e^20*sgn(1/(x*e + d)) + 212*B*a^2*b*c^3*d^3*e^20*sgn(1/(x*e + d)) - 164*A*a*b^2*c^3*d^3*e^20*sg
n(1/(x*e + d)) + 88*A*a^2*c^4*d^3*e^20*sgn(1/(x*e + d)) - 3*B*b^6*d^2*e^21*sgn(1/(x*e + d)) + 18*B*a*b^4*c*d^2
*e^21*sgn(1/(x*e + d)) - 18*A*b^5*c*d^2*e^21*sgn(1/(x*e + d)) - 36*B*a^2*b^2*c^2*d^2*e^21*sgn(1/(x*e + d)) + 1
26*A*a*b^3*c^2*d^2*e^21*sgn(1/(x*e + d)) - 120*B*a^3*c^3*d^2*e^21*sgn(1/(x*e + d)) - 132*A*a^2*b*c^3*d^2*e^21*
sgn(1/(x*e + d)) + B*a*b^5*d*e^22*sgn(1/(x*e + d)) + 5*A*b^6*d*e^22*sgn(1/(x*e + d)) - 11*B*a^2*b^3*c*d*e^22*s
gn(1/(x*e + d)) - 24*A*a*b^4*c*d*e^22*sgn(1/(x*e + d)) + 56*B*a^3*b*c^2*d*e^22*sgn(1/(x*e + d)) - 30*A*a^2*b^2
*c^2*d*e^22*sgn(1/(x*e + d)) + 128*A*a^3*c^3*d*e^22*sgn(1/(x*e + d)) + 2*B*a^2*b^4*e^23*sgn(1/(x*e + d)) - 5*A
*a*b^5*e^23*sgn(1/(x*e + d)) - 14*B*a^3*b^2*c*e^23*sgn(1/(x*e + d)) + 37*A*a^2*b^3*c*e^23*sgn(1/(x*e + d)) + 1
6*B*a^4*c^2*e^23*sgn(1/(x*e + d)) - 64*A*a^3*b*c^2*e^23*sgn(1/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2
 - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2 + 16*a^2*c^5*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1
/(x*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d))^2 - 48*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*
d^4*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2*d^4*e^13*sgn(1/(x*e + d))^2 + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(x*e +
d))^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2 + 2*a*b^5*c*d^3*e^14*sgn(1/(x
*e + d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d))^2 - 96*a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d))^2 + 3*a*b^6*d
^2*e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d))^2 + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(x*e + d
))^2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d))^2 - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d))^2 + 24*a^3*b^3*c*d*e^16*sgn(
1/(x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d))^2 + a^3*b^4*e^17*sgn(1/(x*e + d))^2 - 8*a^4*b^2*c*e^17*s
gn(1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/(x*e + d))^2) + 3*(B*b^4*c^2*d^5*e^20*sgn(1/(x*e + d)) - 8*B*a*b^2*c
^3*d^5*e^20*sgn(1/(x*e + d)) + 16*B*a^2*c^4*d^5*e^20*sgn(1/(x*e + d)) - 2*B*b^5*c*d^4*e^21*sgn(1/(x*e + d)) +
16*B*a*b^3*c^2*d^4*e^21*sgn(1/(x*e + d)) - A*b^4*c^2*d^4*e^21*sgn(1/(x*e + d)) - 32*B*a^2*b*c^3*d^4*e^21*sgn(1
/(x*e + d)) + 8*A*a*b^2*c^3*d^4*e^21*sgn(1/(x*e + d)) - 16*A*a^2*c^4*d^4*e^21*sgn(1/(x*e + d)) + B*b^6*d^3*e^2
2*sgn(1/(x*e + d)) - 6*B*a*b^4*c*d^3*e^22*sgn(1/(x*e + d)) + 2*A*b^5*c*d^3*e^22*sgn(1/(x*e + d)) - 16*A*a*b^3*
c^2*d^3*e^22*sgn(1/(x*e + d)) + 32*B*a^3*c^3*d^3*e^22*sgn(1/(x*e + d)) + 32*A*a^2*b*c^3*d^3*e^22*sgn(1/(x*e +
d)) - 2*B*a*b^5*d^2*e^23*sgn(1/(x*e + d)) - A*b^6*d^2*e^23*sgn(1/(x*e + d)) + 16*B*a^2*b^3*c*d^2*e^23*sgn(1/(x
*e + d)) + 6*A*a*b^4*c*d^2*e^23*sgn(1/(x*e + d)) - 32*B*a^3*b*c^2*d^2*e^23*sgn(1/(x*e + d)) - 32*A*a^3*c^3*d^2
*e^23*sgn(1/(x*e + d)) + B*a^2*b^4*d*e^24*sgn(1/(x*e + d)) + 2*A*a*b^5*d*e^24*sgn(1/(x*e + d)) - 8*B*a^3*b^2*c
*d*e^24*sgn(1/(x*e + d)) - 16*A*a^2*b^3*c*d*e^24*sgn(1/(x*e + d)) + 16*B*a^4*c^2*d*e^24*sgn(1/(x*e + d)) + 32*
A*a^3*b*c^2*d*e^24*sgn(1/(x*e + d)) - A*a^2*b^4*e^25*sgn(1/(x*e + d)) + 8*A*a^3*b^2*c*e^25*sgn(1/(x*e + d)) -
16*A*a^4*c^2*e^25*sgn(1/(x*e + d)))*e^(-1)/((b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 8*a*b^2*c^4*d^6*e^11*sgn(1/
(x*e + d))^2 + 16*a^2*c^5*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 24*a*b^3*c^3*d
^5*e^12*sgn(1/(x*e + d))^2 - 48*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 -
21*a*b^4*c^2*d^4*e^13*sgn(1/(x*e + d))^2 + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(x*e + d))^2 + 48*a^3*c^4*d^4*e^13*sg
n(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2 + 2*a*b^5*c*d^3*e^14*sgn(1/(x*e + d))^2 + 32*a^2*b^3*c^2*d^
3*e^14*sgn(1/(x*e + d))^2 - 96*a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d))^2 + 3*a*b^6*d^2*e^15*sgn(1/(x*e + d))^2 - 2
1*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d))^2 + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(x*e + d))^2 + 48*a^4*c^3*d^2*e^15*sgn
(1/(x*e + d))^2 - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d))^2 + 24*a^3*b^3*c*d*e^16*sgn(1/(x*e + d))^2 - 48*a^4*b*c^2*
d*e^16*sgn(1/(x*e + d))^2 + a^3*b^4*e^17*sgn(1/(x*e + d))^2 - 8*a^4*b^2*c*e^17*sgn(1/(x*e + d))^2 + 16*a^5*c^2
*e^17*sgn(1/(x*e + d))^2)*(x*e + d)))*e^(-1)/(x*e + d) - 3*(16*B*b*c^5*d^6*e^15*sgn(1/(x*e + d)) - 32*A*c^6*d^
6*e^15*sgn(1/(x*e + d)) - 56*B*b^2*c^4*d^5*e^16*sgn(1/(x*e + d)) + 32*B*a*c^5*d^5*e^16*sgn(1/(x*e + d)) + 96*A
*b*c^5*d^5*e^16*sgn(1/(x*e + d)) + 60*B*b^3*c^3*d^4*e^17*sgn(1/(x*e + d)) - 80*A*b^2*c^4*d^4*e^17*sgn(1/(x*e +
 d)) - 160*A*a*c^5*d^4*e^17*sgn(1/(x*e + d)) - 42*B*b^4*c^2*d^3*e^18*sgn(1/(x*e + d)) + 96*B*a*b^2*c^3*d^3*e^1
8*sgn(1/(x*e + d)) - 352*B*a^2*c^4*d^3*e^18*sgn(1/(x*e + d)) + 320*A*a*b*c^4*d^3*e^18*sgn(1/(x*e + d)) + 20*B*
b^5*c*d^2*e^19*sgn(1/(x*e + d)) - 120*B*a*b^3*c^2*d^2*e^19*sgn(1/(x*e + d)) + 46*A*b^4*c^2*d^2*e^19*sgn(1/(x*e
 + d)) + 400*B*a^2*b*c^3*d^2*e^19*sgn(1/(x*e + d)) - 368*A*a*b^2*c^3*d^2*e^19*sgn(1/(x*e + d)) + 256*A*a^2*c^4
*d^2*e^19*sgn(1/(x*e + d)) - 3*B*b^6*d*e^20*sgn(1/(x*e + d)) + 26*B*a*b^4*c*d*e^20*sgn(1/(x*e + d)) - 30*A*b^5
*c*d*e^20*sgn(1/(x*e + d)) - 88*B*a^2*b^2*c^2*d*e^20*sgn(1/(x*e + d)) + 208*A*a*b^3*c^2*d*e^20*sgn(1/(x*e + d)
) - 64*B*a^3*c^3*d*e^20*sgn(1/(x*e + d)) - 256*A*a^2*b*c^3*d*e^20*sgn(1/(x*e + d)) - 2*B*a*b^5*e^21*sgn(1/(x*e
 + d)) + 5*A*b^6*e^21*sgn(1/(x*e + d)) + 12*B*a^2*b^3*c*e^21*sgn(1/(x*e + d)) - 30*A*a*b^4*c*e^21*sgn(1/(x*e +
 d)) + 16*A*a^2*b^2*c^2*e^21*sgn(1/(x*e + d)) + 64*A*a^3*c^3*e^21*sgn(1/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x
*e + d))^2 - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2 + 16*a^2*c^5*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*
e^12*sgn(1/(x*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d))^2 - 48*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d))^2
+ 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2*d^4*e^13*sgn(1/(x*e + d))^2 + 24*a^2*b^2*c^3*d^4*e^13*sgn
(1/(x*e + d))^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2 + 2*a*b^5*c*d^3*e^1
4*sgn(1/(x*e + d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d))^2 - 96*a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d))^2 +
 3*a*b^6*d^2*e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d))^2 + 24*a^3*b^2*c^2*d^2*e^15*sgn(
1/(x*e + d))^2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d))^2 - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d))^2 + 24*a^3*b^3*c*d
*e^16*sgn(1/(x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d))^2 + a^3*b^4*e^17*sgn(1/(x*e + d))^2 - 8*a^4*b^
2*c*e^17*sgn(1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/(x*e + d))^2))*e^(-1)/(x*e + d) + 6*(8*B*b*c^5*d^5*e^14*sg
n(1/(x*e + d)) - 16*A*c^6*d^5*e^14*sgn(1/(x*e + d)) - 24*B*b^2*c^4*d^4*e^15*sgn(1/(x*e + d)) + 16*B*a*c^5*d^4*
e^15*sgn(1/(x*e + d)) + 40*A*b*c^5*d^4*e^15*sgn(1/(x*e + d)) + 19*B*b^3*c^3*d^3*e^16*sgn(1/(x*e + d)) + 4*B*a*
b*c^4*d^3*e^16*sgn(1/(x*e + d)) - 22*A*b^2*c^4*d^3*e^16*sgn(1/(x*e + d)) - 72*A*a*c^5*d^3*e^16*sgn(1/(x*e + d)
) - 11*B*b^4*c^2*d^2*e^17*sgn(1/(x*e + d)) + 38*B*a*b^2*c^3*d^2*e^17*sgn(1/(x*e + d)) - 7*A*b^3*c^3*d^2*e^17*s
gn(1/(x*e + d)) - 136*B*a^2*c^4*d^2*e^17*sgn(1/(x*e + d)) + 108*A*a*b*c^4*d^2*e^17*sgn(1/(x*e + d)) + 3*B*b^5*
c*d*e^18*sgn(1/(x*e + d)) - 23*B*a*b^3*c^2*d*e^18*sgn(1/(x*e + d)) + 15*A*b^4*c^2*d*e^18*sgn(1/(x*e + d)) + 84
*B*a^2*b*c^3*d*e^18*sgn(1/(x*e + d)) - 106*A*a*b^2*c^3*d*e^18*sgn(1/(x*e + d)) + 104*A*a^2*c^4*d*e^18*sgn(1/(x
*e + d)) + 2*B*a*b^4*c*e^19*sgn(1/(x*e + d)) - 5*A*b^5*c*e^19*sgn(1/(x*e + d)) - 14*B*a^2*b^2*c^2*e^19*sgn(1/(
x*e + d)) + 35*A*a*b^3*c^2*e^19*sgn(1/(x*e + d)) + 8*B*a^3*c^3*e^19*sgn(1/(x*e + d)) - 52*A*a^2*b*c^3*e^19*sgn
(1/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2 + 16*a^2*c^5*d^6
*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d))^2 -
48*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2*d^4*e^13*sgn(1/(
x*e + d))^2 + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(x*e + d))^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^
14*sgn(1/(x*e + d))^2 + 2*a*b^5*c*d^3*e^14*sgn(1/(x*e + d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d))^2 - 9
6*a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d))^2 + 3*a*b^6*d^2*e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x
*e + d))^2 + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(x*e + d))^2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d))^2 - 3*a^2*b^5*d
*e^16*sgn(1/(x*e + d))^2 + 24*a^3*b^3*c*d*e^16*sgn(1/(x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d))^2 + a
^3*b^4*e^17*sgn(1/(x*e + d))^2 - 8*a^4*b^2*c*e^17*sgn(1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/(x*e + d))^2))*e^
(-1)/(x*e + d) - (16*B*b*c^5*d^4*e^13*sgn(1/(x*e + d)) - 32*A*c^6*d^4*e^13*sgn(1/(x*e + d)) - 40*B*b^2*c^4*d^3
*e^14*sgn(1/(x*e + d)) + 32*B*a*c^5*d^3*e^14*sgn(1/(x*e + d)) + 64*A*b*c^5*d^3*e^14*sgn(1/(x*e + d)) + 18*B*b^
3*c^3*d^2*e^15*sgn(1/(x*e + d)) + 24*B*a*b*c^4*d^2*e^15*sgn(1/(x*e + d)) - 12*A*b^2*c^4*d^2*e^15*sgn(1/(x*e +
d)) - 144*A*a*c^5*d^2*e^15*sgn(1/(x*e + d)) - 9*B*b^4*c^2*d*e^16*sgn(1/(x*e + d)) + 56*B*a*b^2*c^3*d*e^16*sgn(
1/(x*e + d)) - 20*A*b^3*c^3*d*e^16*sgn(1/(x*e + d)) - 208*B*a^2*c^4*d*e^16*sgn(1/(x*e + d)) + 144*A*a*b*c^4*d*
e^16*sgn(1/(x*e + d)) - 6*B*a*b^3*c^2*e^17*sgn(1/(x*e + d)) + 15*A*b^4*c^2*e^17*sgn(1/(x*e + d)) + 40*B*a^2*b*
c^3*e^17*sgn(1/(x*e + d)) - 100*A*a*b^2*c^3*e^17*sgn(1/(x*e + d)) + 128*A*a^2*c^4*e^17*sgn(1/(x*e + d)))/(b^4*
c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d))^2 + 16*a^2*c^5*d^6*e^11*sgn(1/(x*e + d
))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d))^2 - 48*a^2*b*c^4*d^5*e^1
2*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - 21*a*b^4*c^2*d^4*e^13*sgn(1/(x*e + d))^2 + 24*a^2
*b^2*c^3*d^4*e^13*sgn(1/(x*e + d))^2 + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^
2 + 2*a*b^5*c*d^3*e^14*sgn(1/(x*e + d))^2 + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d))^2 - 96*a^3*b*c^3*d^3*e^14
*sgn(1/(x*e + d))^2 + 3*a*b^6*d^2*e^15*sgn(1/(x*e + d))^2 - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d))^2 + 24*a^3*
b^2*c^2*d^2*e^15*sgn(1/(x*e + d))^2 + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d))^2 - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d
))^2 + 24*a^3*b^3*c*d*e^16*sgn(1/(x*e + d))^2 - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d))^2 + a^3*b^4*e^17*sgn(1/(x
*e + d))^2 - 8*a^4*b^2*c*e^17*sgn(1/(x*e + d))^2 + 16*a^5*c^2*e^17*sgn(1/(x*e + d))^2))/(c - 2*c*d/(x*e + d) +
 c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2)^(3/2) + 3*(8*B*c*d^2*e^6 - 3*B*b*d
*e^7 - 10*A*c*d*e^7 - 2*B*a*e^8 + 5*A*b*e^8)*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*(sqrt(c - 2*
c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 -
b*d*e^3 + a*e^4)*e^(-1)/(x*e + d))))/((c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d
^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7)*sqrt(c*d^2 - b*d*e + a
*e^2)*sgn(1/(x*e + d))))*e^(-2)

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maple [B]  time = 0.08, size = 6675, normalized size = 8.95 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x)

[Out]

result too large to display

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?`
 for more details)Is a*e^2-b*d*e                            +c*d^2    positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x)

[Out]

int((A + B*x)/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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