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

Optimal. Leaf size=436 \[ \frac {2 \left (c x \left ((2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {e^3 (B d-A e) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

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Rubi [A]  time = 0.55, antiderivative size = 436, 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, 12, 724, 206} \begin {gather*} \frac {2 \left (c x \left ((2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 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 (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 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 \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 ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {e^3 (B d-A e) \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 )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)*(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)*(a + b*x + c*x^2)^(3/2)) + (2*(4*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)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B*d*e + 3*a*
A*e^2)) + c*(4*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)*(3*b^2*e*(B*d - A*e) - 4*
b*c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B*d*e + 3*a*A*e^2)))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*S
qrt[a + b*x + c*x^2]) - (e^3*(B*d - A*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])])/(c*d^2 - b*d*e + a*e^2)^(5/2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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) \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 ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )-2 c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x) \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 ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 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 (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )+c \left (4 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 (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A 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 \sqrt {a+b x+c x^2}}+\frac {4 \int -\frac {3 \left (b^2-4 a c\right )^2 e^3 (B d-A e)}{4 (d+e x) \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 ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 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 (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )+c \left (4 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 (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A 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 \sqrt {a+b x+c x^2}}-\frac {\left (e^3 (B d-A e)\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{\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 ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 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 (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )+c \left (4 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 (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A 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 \sqrt {a+b x+c x^2}}+\frac {\left (2 e^3 (B d-A e)\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 )^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 ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (4 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 (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A e^2\right )\right )+c \left (4 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 (3 b^2 e (B d-A e)-4 b c d (B d+A e)+4 c \left (2 A c d^2-a B d e+3 a A 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 \sqrt {a+b x+c x^2}}-\frac {e^3 (B d-A e) \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 )}{\left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 1.04, size = 437, normalized size = 1.00 \begin {gather*} \frac {4 \left (c x \left (\frac {1}{2} (2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+2 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\frac {1}{2} \left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+2 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 \sqrt {a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2}+\frac {2 \left (2 A c (a e+c d x)+a B (b e-2 c d+2 c e x)-A b^2 e+A b c (d-e x)-b B c d x\right )}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2} \left (e (b d-a e)-c d^2\right )}+\frac {e^3 (B d-A e) \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 )}{\left (e (a e-b d)+c d^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(2*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)*(3*b^2*e*(B*d - A*e
) - 4*b*c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B*d*e + 3*a*A*e^2)))/2 + c*(2*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)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B*d*e + 3*
a*A*e^2)))/2)*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)]) + (2*(-(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*(-2*c*d + b*e + 2*c*e*x)))/(3*(b^2 - 4*a*c)*(-(c*d^2) +
e*(b*d - a*e))*(a + x*(b + c*x))^(3/2)) + (e^3*(B*d - A*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)])])/(c*d^2 + e*(-(b*d) + a*e))^(5/2)

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IntegrateAlgebraic [F]  time = 180.97, 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)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

$Aborted

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fricas [B]  time = 24.72, size = 6068, normalized size = 13.92

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(3*((B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2)*d*e^3 - (A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*c^2)*e^4 +
((B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*d*e^3 - (A*b^4*c^2 - 8*A*a*b^2*c^3 + 16*A*a^2*c^4)*e^4)*x^4 + 2*((
B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^3)*d*e^3 - (A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*e^4)*x^3 + ((B*b^
6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*d*e^3 - (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*e^4)*x^2 + 2*((B*a*b^5 - 8*B*a^2*
b^3*c + 16*B*a^3*b*c^2)*d*e^3 - (A*a*b^5 - 8*A*a^2*b^3*c + 16*A*a^3*b*c^2)*e^4)*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*((4*(2*B*a^2 - 3*A*a*b)*c^4 + (2*B*a*b^2 + A*b^3)*c^3)*d^5 - (
8*A*a^2*c^4 + 2*(8*B*a^2*b - 17*A*a*b^2)*c^3 + 3*(2*B*a*b^3 + A*b^4)*c^2)*d^4*e + (20*(2*B*a^3 - A*a^2*b)*c^3
+ 3*(2*B*a^2*b^2 - 9*A*a*b^3)*c^2 + 3*(2*B*a*b^4 + A*b^5)*c)*d^3*e^2 - (2*B*a*b^5 + A*b^6 - B*a^2*b^3*c + 40*A
*a^3*c^3 + 2*(26*B*a^3*b - 33*A*a^2*b^2)*c^2)*d^2*e^3 + (B*a^2*b^4 + 5*A*a*b^5 + 16*(2*B*a^4 + A*a^3*b)*c^2 +
2*(4*B*a^3*b^2 - 17*A*a^2*b^3)*c)*d*e^4 + (B*a^3*b^3 - 4*A*a^2*b^4 - 32*A*a^4*c^2 - 4*(3*B*a^4*b - 7*A*a^3*b^2
)*c)*e^5 + (8*(B*b*c^5 - 2*A*c^6)*d^5 - 2*(11*B*b^2*c^4 - 4*(B*a + 5*A*b)*c^5)*d^4*e + (17*B*b^3*c^3 - 56*A*a*
c^5 + 2*(6*B*a*b - 13*A*b^2)*c^4)*d^3*e^2 - (3*B*b^4*c^2 + 4*(2*B*a^2 - 21*A*a*b)*c^4 + (26*B*a*b^2 + A*b^3)*c
^3)*d^2*e^3 - (40*A*a^2*c^4 - 2*(14*B*a^2*b - 11*A*a*b^2)*c^3 - 3*(B*a*b^3 + A*b^4)*c^2)*d*e^4 - (3*A*a*b^3*c^
2 + 4*(4*B*a^3 - 5*A*a^2*b)*c^3)*e^5)*x^3 + 3*(4*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - (11*B*b^3*c^3 - 4*(B*a*b + 5*A*
b^2)*c^4)*d^4*e + (9*B*b^4*c^2 + 4*(2*B*a^2 - 7*A*a*b)*c^4 + (2*B*a*b^2 - 13*A*b^3)*c^3)*d^3*e^2 - (2*B*b^5*c
+ 8*A*a^2*c^4 + 2*(6*B*a^2*b - 23*A*a*b^2)*c^3 + (9*B*a*b^3 + A*b^4)*c^2)*d^2*e^3 + (4*(2*B*a^3 - 3*A*a^2*b)*c
^3 + 5*(2*B*a^2*b^2 - 3*A*a*b^3)*c^2 + 2*(B*a*b^4 + A*b^5)*c)*d*e^4 - 2*(A*a*b^4*c + 4*A*a^3*c^3 + (4*B*a^3*b
- 7*A*a^2*b^2)*c^2)*e^5)*x^2 + 3*((B*b^3*c^3 - 8*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^5 - (3*B*b^4*c^2 - 20*A*
a*b*c^4 + (8*B*a*b^2 - 5*A*b^3)*c^3)*d^4*e + (3*B*b^5*c - 24*A*a^2*c^4 + 2*(10*B*a^2*b - 11*A*a*b^2)*c^3 + 3*(
B*a*b^3 - A*b^4)*c^2)*d^3*e^2 - (B*b^6 + A*b^5*c + 4*(2*B*a^3 - 7*A*a^2*b)*c^3 + (22*B*a^2*b^2 - 17*A*a*b^3)*c
^2)*d^2*e^3 + (B*a*b^5 + A*b^6 - 6*A*a*b^4*c - 16*A*a^3*c^3 + 8*(3*B*a^3*b - A*a^2*b^2)*c^2)*d*e^4 - (A*a*b^5
+ 8*B*a^4*c^2 + 2*(B*a^3*b^2 - 3*A*a^2*b^3)*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 +
16*a^4*c^5)*d^6 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^5*e + 3*(a^2*b^6*c - 7*a^3*b^4*c^2 + 8*a^4*
b^2*c^3 + 16*a^5*c^4)*d^4*e^2 - (a^2*b^7 - 2*a^3*b^5*c - 32*a^4*b^3*c^2 + 96*a^5*b*c^3)*d^3*e^3 + 3*(a^3*b^6 -
 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*c^3)*d^2*e^4 - 3*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d*e^5 + (a^5*b^4
 - 8*a^6*b^2*c + 16*a^7*c^2)*e^6 + ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6 - 3*(b^5*c^4 - 8*a*b^3*c^5 + 16*a
^2*b*c^6)*d^5*e + 3*(b^6*c^3 - 7*a*b^4*c^4 + 8*a^2*b^2*c^5 + 16*a^3*c^6)*d^4*e^2 - (b^7*c^2 - 2*a*b^5*c^3 - 32
*a^2*b^3*c^4 + 96*a^3*b*c^5)*d^3*e^3 + 3*(a*b^6*c^2 - 7*a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^2*e^4 - 3*
(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d*e^5 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*e^6)*x^4 + 2*(
(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6 - 3*(b^6*c^3 - 8*a*b^4*c^4 + 16*a^2*b^2*c^5)*d^5*e + 3*(b^7*c^2 - 7
*a*b^5*c^3 + 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^4*e^2 - (b^8*c - 2*a*b^6*c^2 - 32*a^2*b^4*c^3 + 96*a^3*b^2*c^4)*d
^3*e^3 + 3*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^2*e^4 - 3*(a^2*b^6*c - 8*a^3*b^4*c^2 + 1
6*a^4*b^2*c^3)*d*e^5 + (a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*e^6)*x^3 + ((b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*
c^6)*d^6 - 3*(b^7*c^2 - 6*a*b^5*c^3 + 32*a^3*b*c^5)*d^5*e + 3*(b^8*c - 5*a*b^6*c^2 - 6*a^2*b^4*c^3 + 32*a^3*b^
2*c^4 + 32*a^4*c^5)*d^4*e^2 - (b^9 - 36*a^2*b^5*c^2 + 32*a^3*b^3*c^3 + 192*a^4*b*c^4)*d^3*e^3 + 3*(a*b^8 - 5*a
^2*b^6*c - 6*a^3*b^4*c^2 + 32*a^4*b^2*c^3 + 32*a^5*c^4)*d^2*e^4 - 3*(a^2*b^7 - 6*a^3*b^5*c + 32*a^5*b*c^3)*d*e
^5 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*e^6)*x^2 + 2*((a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^6 - 3*(a*
b^6*c^2 - 8*a^2*b^4*c^3 + 16*a^3*b^2*c^4)*d^5*e + 3*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d
^4*e^2 - (a*b^8 - 2*a^2*b^6*c - 32*a^3*b^4*c^2 + 96*a^4*b^2*c^3)*d^3*e^3 + 3*(a^2*b^7 - 7*a^3*b^5*c + 8*a^4*b^
3*c^2 + 16*a^5*b*c^3)*d^2*e^4 - 3*(a^3*b^6 - 8*a^4*b^4*c + 16*a^5*b^2*c^2)*d*e^5 + (a^4*b^5 - 8*a^5*b^3*c + 16
*a^6*b*c^2)*e^6)*x), -1/3*(3*((B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2)*d*e^3 - (A*a^2*b^4 - 8*A*a^3*b^2*c +
16*A*a^4*c^2)*e^4 + ((B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*d*e^3 - (A*b^4*c^2 - 8*A*a*b^2*c^3 + 16*A*a^2*
c^4)*e^4)*x^4 + 2*((B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^3)*d*e^3 - (A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^
3)*e^4)*x^3 + ((B*b^6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*d*e^3 - (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*e^4)*x^2 + 2*
((B*a*b^5 - 8*B*a^2*b^3*c + 16*B*a^3*b*c^2)*d*e^3 - (A*a*b^5 - 8*A*a^2*b^3*c + 16*A*a^3*b*c^2)*e^4)*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*((4*(2*B*a^2 - 3*A*a*b)*c^4 + (2*B*a*b^2 + A*b^3)*c^3)*d^5 - (8*A*a^2*c^4 + 2*(8*B*a^2*b - 17*A*a*b^2)*c^3 +
 3*(2*B*a*b^3 + A*b^4)*c^2)*d^4*e + (20*(2*B*a^3 - A*a^2*b)*c^3 + 3*(2*B*a^2*b^2 - 9*A*a*b^3)*c^2 + 3*(2*B*a*b
^4 + A*b^5)*c)*d^3*e^2 - (2*B*a*b^5 + A*b^6 - B*a^2*b^3*c + 40*A*a^3*c^3 + 2*(26*B*a^3*b - 33*A*a^2*b^2)*c^2)*
d^2*e^3 + (B*a^2*b^4 + 5*A*a*b^5 + 16*(2*B*a^4 + A*a^3*b)*c^2 + 2*(4*B*a^3*b^2 - 17*A*a^2*b^3)*c)*d*e^4 + (B*a
^3*b^3 - 4*A*a^2*b^4 - 32*A*a^4*c^2 - 4*(3*B*a^4*b - 7*A*a^3*b^2)*c)*e^5 + (8*(B*b*c^5 - 2*A*c^6)*d^5 - 2*(11*
B*b^2*c^4 - 4*(B*a + 5*A*b)*c^5)*d^4*e + (17*B*b^3*c^3 - 56*A*a*c^5 + 2*(6*B*a*b - 13*A*b^2)*c^4)*d^3*e^2 - (3
*B*b^4*c^2 + 4*(2*B*a^2 - 21*A*a*b)*c^4 + (26*B*a*b^2 + A*b^3)*c^3)*d^2*e^3 - (40*A*a^2*c^4 - 2*(14*B*a^2*b -
11*A*a*b^2)*c^3 - 3*(B*a*b^3 + A*b^4)*c^2)*d*e^4 - (3*A*a*b^3*c^2 + 4*(4*B*a^3 - 5*A*a^2*b)*c^3)*e^5)*x^3 + 3*
(4*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - (11*B*b^3*c^3 - 4*(B*a*b + 5*A*b^2)*c^4)*d^4*e + (9*B*b^4*c^2 + 4*(2*B*a^2 -
7*A*a*b)*c^4 + (2*B*a*b^2 - 13*A*b^3)*c^3)*d^3*e^2 - (2*B*b^5*c + 8*A*a^2*c^4 + 2*(6*B*a^2*b - 23*A*a*b^2)*c^3
 + (9*B*a*b^3 + A*b^4)*c^2)*d^2*e^3 + (4*(2*B*a^3 - 3*A*a^2*b)*c^3 + 5*(2*B*a^2*b^2 - 3*A*a*b^3)*c^2 + 2*(B*a*
b^4 + A*b^5)*c)*d*e^4 - 2*(A*a*b^4*c + 4*A*a^3*c^3 + (4*B*a^3*b - 7*A*a^2*b^2)*c^2)*e^5)*x^2 + 3*((B*b^3*c^3 -
 8*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^5 - (3*B*b^4*c^2 - 20*A*a*b*c^4 + (8*B*a*b^2 - 5*A*b^3)*c^3)*d^4*e + (
3*B*b^5*c - 24*A*a^2*c^4 + 2*(10*B*a^2*b - 11*A*a*b^2)*c^3 + 3*(B*a*b^3 - A*b^4)*c^2)*d^3*e^2 - (B*b^6 + A*b^5
*c + 4*(2*B*a^3 - 7*A*a^2*b)*c^3 + (22*B*a^2*b^2 - 17*A*a*b^3)*c^2)*d^2*e^3 + (B*a*b^5 + A*b^6 - 6*A*a*b^4*c -
 16*A*a^3*c^3 + 8*(3*B*a^3*b - A*a^2*b^2)*c^2)*d*e^4 - (A*a*b^5 + 8*B*a^4*c^2 + 2*(B*a^3*b^2 - 3*A*a^2*b^3)*c)
*e^5)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^6 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c
^3 + 16*a^4*b*c^4)*d^5*e + 3*(a^2*b^6*c - 7*a^3*b^4*c^2 + 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^4*e^2 - (a^2*b^7 - 2*a
^3*b^5*c - 32*a^4*b^3*c^2 + 96*a^5*b*c^3)*d^3*e^3 + 3*(a^3*b^6 - 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*c^3)*d^2
*e^4 - 3*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d*e^5 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*e^6 + ((b^4*c^5 -
 8*a*b^2*c^6 + 16*a^2*c^7)*d^6 - 3*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^5*e + 3*(b^6*c^3 - 7*a*b^4*c^4 + 8
*a^2*b^2*c^5 + 16*a^3*c^6)*d^4*e^2 - (b^7*c^2 - 2*a*b^5*c^3 - 32*a^2*b^3*c^4 + 96*a^3*b*c^5)*d^3*e^3 + 3*(a*b^
6*c^2 - 7*a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^2*e^4 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d
*e^5 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*e^6)*x^4 + 2*((b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6 - 3
*(b^6*c^3 - 8*a*b^4*c^4 + 16*a^2*b^2*c^5)*d^5*e + 3*(b^7*c^2 - 7*a*b^5*c^3 + 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^4
*e^2 - (b^8*c - 2*a*b^6*c^2 - 32*a^2*b^4*c^3 + 96*a^3*b^2*c^4)*d^3*e^3 + 3*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^
3*c^3 + 16*a^4*b*c^4)*d^2*e^4 - 3*(a^2*b^6*c - 8*a^3*b^4*c^2 + 16*a^4*b^2*c^3)*d*e^5 + (a^3*b^5*c - 8*a^4*b^3*
c^2 + 16*a^5*b*c^3)*e^6)*x^3 + ((b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*d^6 - 3*(b^7*c^2 - 6*a*b^5*c^3 + 32*a^3*b
*c^5)*d^5*e + 3*(b^8*c - 5*a*b^6*c^2 - 6*a^2*b^4*c^3 + 32*a^3*b^2*c^4 + 32*a^4*c^5)*d^4*e^2 - (b^9 - 36*a^2*b^
5*c^2 + 32*a^3*b^3*c^3 + 192*a^4*b*c^4)*d^3*e^3 + 3*(a*b^8 - 5*a^2*b^6*c - 6*a^3*b^4*c^2 + 32*a^4*b^2*c^3 + 32
*a^5*c^4)*d^2*e^4 - 3*(a^2*b^7 - 6*a^3*b^5*c + 32*a^5*b*c^3)*d*e^5 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*e^6)
*x^2 + 2*((a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^6 - 3*(a*b^6*c^2 - 8*a^2*b^4*c^3 + 16*a^3*b^2*c^4)*d^5*
e + 3*(a*b^7*c - 7*a^2*b^5*c^2 + 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^4*e^2 - (a*b^8 - 2*a^2*b^6*c - 32*a^3*b^4*c^2
 + 96*a^4*b^2*c^3)*d^3*e^3 + 3*(a^2*b^7 - 7*a^3*b^5*c + 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*d^2*e^4 - 3*(a^3*b^6 - 8
*a^4*b^4*c + 16*a^5*b^2*c^2)*d*e^5 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*e^6)*x)]

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giac [B]  time = 0.90, size = 11729, normalized size = 26.90

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(B*d*e^3 - A*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))
/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2))
- 2/3*((((8*B*b*c^10*d^15 - 16*A*c^11*d^15 - 62*B*b^2*c^9*d^14*e + 8*B*a*c^10*d^14*e + 120*A*b*c^10*d^14*e + 2
07*B*b^3*c^8*d^13*e^2 + 12*B*a*b*c^9*d^13*e^2 - 386*A*b^2*c^9*d^13*e^2 - 136*A*a*c^10*d^13*e^2 - 388*B*b^4*c^7
*d^12*e^3 - 276*B*a*b^2*c^8*d^12*e^3 + 689*A*b^3*c^8*d^12*e^3 + 32*B*a^2*c^9*d^12*e^3 + 884*A*a*b*c^9*d^12*e^3
 + 445*B*b^5*c^6*d^11*e^4 + 938*B*a*b^3*c^7*d^11*e^4 - 732*A*b^4*c^7*d^11*e^4 + 48*B*a^2*b*c^8*d^11*e^4 - 2412
*A*a*b^2*c^8*d^11*e^4 - 480*A*a^2*c^9*d^11*e^4 - 318*B*b^6*c^5*d^10*e^5 - 1530*B*a*b^4*c^6*d^10*e^5 + 451*A*b^
5*c^6*d^10*e^5 - 810*B*a^2*b^2*c^7*d^10*e^5 + 3542*A*a*b^3*c^7*d^10*e^5 + 24*B*a^3*c^8*d^10*e^5 + 2640*A*a^2*b
*c^8*d^10*e^5 + 137*B*b^7*c^4*d^9*e^6 + 1392*B*a*b^5*c^5*d^9*e^6 - 130*A*b^6*c^5*d^9*e^6 + 2165*B*a^2*b^3*c^6*
d^9*e^6 - 2950*A*a*b^4*c^6*d^9*e^6 + 340*B*a^3*b*c^7*d^9*e^6 - 5910*A*a^2*b^2*c^7*d^9*e^6 - 920*A*a^3*c^8*d^9*
e^6 - 32*B*b^8*c^3*d^8*e^7 - 712*B*a*b^6*c^4*d^8*e^7 - 9*A*b^7*c^4*d^8*e^7 - 2640*B*a^2*b^4*c^5*d^8*e^7 + 1296
*A*a*b^5*c^5*d^8*e^7 - 1720*B*a^3*b^2*c^6*d^8*e^7 + 6795*A*a^2*b^3*c^6*d^8*e^7 - 80*B*a^4*c^7*d^8*e^7 + 4140*A
*a^3*b*c^7*d^8*e^7 + 3*B*b^9*c^2*d^7*e^8 + 186*B*a*b^7*c^3*d^7*e^8 + 16*A*b^8*c^3*d^7*e^8 + 1638*B*a^2*b^5*c^4
*d^7*e^8 - 184*A*a*b^6*c^4*d^7*e^8 + 2940*B*a^3*b^3*c^5*d^7*e^8 - 4080*A*a^2*b^4*c^5*d^7*e^8 + 840*B*a^4*b*c^6
*d^7*e^8 - 7240*A*a^3*b^2*c^6*d^7*e^8 - 1040*A*a^4*c^7*d^7*e^8 - 18*B*a*b^8*c^2*d^6*e^9 - 3*A*b^9*c^2*d^6*e^9
- 478*B*a^2*b^6*c^3*d^6*e^9 - 58*A*a*b^7*c^3*d^6*e^9 - 2260*B*a^3*b^4*c^4*d^6*e^9 + 1050*A*a^2*b^5*c^4*d^6*e^9
 - 2130*B*a^4*b^2*c^5*d^6*e^9 + 6020*A*a^3*b^3*c^5*d^6*e^9 - 200*B*a^5*c^6*d^6*e^9 + 3640*A*a^4*b*c^6*d^6*e^9
+ 45*B*a^2*b^7*c^2*d^5*e^10 + 18*A*a*b^8*c^2*d^5*e^10 + 736*B*a^3*b^5*c^3*d^5*e^10 + 30*A*a^2*b^6*c^3*d^5*e^10
 + 2105*B*a^4*b^3*c^4*d^5*e^10 - 2220*A*a^3*b^4*c^4*d^5*e^10 + 948*B*a^5*b*c^5*d^5*e^10 - 4590*A*a^4*b^2*c^5*d
^5*e^10 - 696*A*a^5*c^6*d^5*e^10 - 60*B*a^3*b^6*c^2*d^4*e^11 - 45*A*a^2*b^7*c^2*d^4*e^11 - 780*B*a^4*b^4*c^3*d
^4*e^11 + 160*A*a^3*b^5*c^3*d^4*e^11 - 1332*B*a^5*b^2*c^4*d^4*e^11 + 2375*A*a^4*b^3*c^4*d^4*e^11 - 192*B*a^6*c
^5*d^4*e^11 + 1740*A*a^5*b*c^5*d^4*e^11 + 45*B*a^4*b^5*c^2*d^3*e^12 + 60*A*a^3*b^6*c^2*d^3*e^12 + 602*B*a^5*b^
3*c^3*d^3*e^12 - 340*A*a^4*b^4*c^3*d^3*e^12 + 512*B*a^6*b*c^4*d^3*e^12 - 1356*A*a^5*b^2*c^4*d^3*e^12 - 256*A*a
^6*c^5*d^3*e^12 - 18*B*a^5*b^4*c^2*d^2*e^13 - 45*A*a^4*b^5*c^2*d^2*e^13 - 326*B*a^6*b^2*c^3*d^2*e^13 + 294*A*a
^5*b^3*c^3*d^2*e^13 - 88*B*a^7*c^4*d^2*e^13 + 384*A*a^6*b*c^4*d^2*e^13 + 3*B*a^6*b^3*c^2*d*e^14 + 18*A*a^5*b^4
*c^2*d*e^14 + 108*B*a^7*b*c^3*d*e^14 - 122*A*a^6*b^2*c^3*d*e^14 - 40*A*a^7*c^4*d*e^14 - 3*A*a^6*b^3*c^2*e^15 -
 16*B*a^8*c^3*e^15 + 20*A*a^7*b*c^3*e^15)*x/(b^4*c^8*d^16 - 8*a*b^2*c^9*d^16 + 16*a^2*c^10*d^16 - 8*b^5*c^7*d^
15*e + 64*a*b^3*c^8*d^15*e - 128*a^2*b*c^9*d^15*e + 28*b^6*c^6*d^14*e^2 - 216*a*b^4*c^7*d^14*e^2 + 384*a^2*b^2
*c^8*d^14*e^2 + 128*a^3*c^9*d^14*e^2 - 56*b^7*c^5*d^13*e^3 + 392*a*b^5*c^6*d^13*e^3 - 448*a^2*b^3*c^7*d^13*e^3
 - 896*a^3*b*c^8*d^13*e^3 + 70*b^8*c^4*d^12*e^4 - 392*a*b^6*c^5*d^12*e^4 - 196*a^2*b^4*c^6*d^12*e^4 + 2464*a^3
*b^2*c^7*d^12*e^4 + 448*a^4*c^8*d^12*e^4 - 56*b^9*c^3*d^11*e^5 + 168*a*b^7*c^4*d^11*e^5 + 1176*a^2*b^5*c^5*d^1
1*e^5 - 3136*a^3*b^3*c^6*d^11*e^5 - 2688*a^4*b*c^7*d^11*e^5 + 28*b^10*c^2*d^10*e^6 + 56*a*b^8*c^3*d^10*e^6 - 1
372*a^2*b^6*c^4*d^10*e^6 + 1176*a^3*b^4*c^5*d^10*e^6 + 6272*a^4*b^2*c^6*d^10*e^6 + 896*a^5*c^7*d^10*e^6 - 8*b^
11*c*d^9*e^7 - 104*a*b^9*c^2*d^9*e^7 + 656*a^2*b^7*c^3*d^9*e^7 + 1512*a^3*b^5*c^4*d^9*e^7 - 6720*a^4*b^3*c^5*d
^9*e^7 - 4480*a^5*b*c^6*d^9*e^7 + b^12*d^8*e^8 + 48*a*b^10*c*d^8*e^8 - 12*a^2*b^8*c^2*d^8*e^8 - 1904*a^3*b^6*c
^3*d^8*e^8 + 2310*a^4*b^4*c^4*d^8*e^8 + 8400*a^5*b^2*c^5*d^8*e^8 + 1120*a^6*c^6*d^8*e^8 - 8*a*b^11*d^7*e^9 - 1
04*a^2*b^9*c*d^7*e^9 + 656*a^3*b^7*c^2*d^7*e^9 + 1512*a^4*b^5*c^3*d^7*e^9 - 6720*a^5*b^3*c^4*d^7*e^9 - 4480*a^
6*b*c^5*d^7*e^9 + 28*a^2*b^10*d^6*e^10 + 56*a^3*b^8*c*d^6*e^10 - 1372*a^4*b^6*c^2*d^6*e^10 + 1176*a^5*b^4*c^3*
d^6*e^10 + 6272*a^6*b^2*c^4*d^6*e^10 + 896*a^7*c^5*d^6*e^10 - 56*a^3*b^9*d^5*e^11 + 168*a^4*b^7*c*d^5*e^11 + 1
176*a^5*b^5*c^2*d^5*e^11 - 3136*a^6*b^3*c^3*d^5*e^11 - 2688*a^7*b*c^4*d^5*e^11 + 70*a^4*b^8*d^4*e^12 - 392*a^5
*b^6*c*d^4*e^12 - 196*a^6*b^4*c^2*d^4*e^12 + 2464*a^7*b^2*c^3*d^4*e^12 + 448*a^8*c^4*d^4*e^12 - 56*a^5*b^7*d^3
*e^13 + 392*a^6*b^5*c*d^3*e^13 - 448*a^7*b^3*c^2*d^3*e^13 - 896*a^8*b*c^3*d^3*e^13 + 28*a^6*b^6*d^2*e^14 - 216
*a^7*b^4*c*d^2*e^14 + 384*a^8*b^2*c^2*d^2*e^14 + 128*a^9*c^3*d^2*e^14 - 8*a^7*b^5*d*e^15 + 64*a^8*b^3*c*d*e^15
 - 128*a^9*b*c^2*d*e^15 + a^8*b^4*e^16 - 8*a^9*b^2*c*e^16 + 16*a^10*c^2*e^16) + 3*(4*B*b^2*c^9*d^15 - 8*A*b*c^
10*d^15 - 31*B*b^3*c^8*d^14*e + 4*B*a*b*c^9*d^14*e + 60*A*b^2*c^9*d^14*e + 104*B*b^4*c^7*d^13*e^2 + 2*B*a*b^2*
c^8*d^13*e^2 - 193*A*b^3*c^8*d^13*e^2 + 8*B*a^2*c^9*d^13*e^2 - 68*A*a*b*c^9*d^13*e^2 - 197*B*b^5*c^6*d^12*e^3
- 114*B*a*b^3*c^7*d^12*e^3 + 344*A*b^4*c^7*d^12*e^3 - 32*B*a^2*b*c^8*d^12*e^3 + 446*A*a*b^2*c^8*d^12*e^3 - 8*A
*a^2*c^9*d^12*e^3 + 230*B*b^6*c^5*d^11*e^4 + 412*B*a*b^4*c^6*d^11*e^4 - 363*A*b^5*c^6*d^11*e^4 + 120*B*a^2*b^2
*c^7*d^11*e^4 - 1230*A*a*b^3*c^7*d^11*e^4 + 48*B*a^3*c^8*d^11*e^4 - 192*A*a^2*b*c^8*d^11*e^4 - 169*B*b^7*c^4*d
^10*e^5 - 700*B*a*b^5*c^5*d^10*e^5 + 218*A*b^6*c^5*d^10*e^5 - 445*B*a^2*b^3*c^6*d^10*e^5 + 1828*A*a*b^4*c^6*d^
10*e^5 - 228*B*a^3*b*c^7*d^10*e^5 + 1224*A*a^2*b^2*c^7*d^10*e^5 - 48*A*a^3*c^8*d^10*e^5 + 76*B*b^8*c^3*d^9*e^6
 + 666*B*a*b^6*c^4*d^9*e^6 - 55*A*b^7*c^4*d^9*e^6 + 970*B*a^2*b^4*c^5*d^9*e^6 - 1540*A*a*b^5*c^5*d^9*e^6 + 590
*B*a^3*b^2*c^6*d^9*e^6 - 2915*A*a^2*b^3*c^6*d^9*e^6 + 120*B*a^4*c^7*d^9*e^6 - 220*A*a^3*b*c^7*d^9*e^6 - 19*B*b
^9*c^2*d^8*e^7 - 362*B*a*b^7*c^3*d^8*e^7 - 12*A*b^8*c^3*d^8*e^7 - 1158*B*a^2*b^5*c^4*d^8*e^7 + 678*A*a*b^6*c^4
*d^8*e^7 - 1100*B*a^3*b^3*c^5*d^8*e^7 + 3510*A*a^2*b^4*c^5*d^8*e^7 - 520*B*a^4*b*c^6*d^8*e^7 + 1650*A*a^3*b^2*
c^6*d^8*e^7 - 120*A*a^4*c^7*d^8*e^7 + 2*B*b^10*c*d^7*e^8 + 104*B*a*b^8*c^2*d^7*e^8 + 11*A*b^9*c^2*d^7*e^8 + 75
2*B*a^2*b^6*c^3*d^7*e^8 - 86*A*a*b^7*c^3*d^7*e^8 + 1360*B*a^3*b^4*c^4*d^7*e^8 - 2202*A*a^2*b^5*c^4*d^7*e^8 + 1
060*B*a^4*b^2*c^5*d^7*e^8 - 3380*A*a^3*b^3*c^5*d^7*e^8 + 160*B*a^5*c^6*d^7*e^8 - 40*A*a^4*b*c^6*d^7*e^8 - 12*B
*a*b^9*c*d^6*e^9 - 2*A*b^10*c*d^6*e^9 - 245*B*a^2*b^7*c^2*d^6*e^9 - 40*A*a*b^8*c^2*d^6*e^9 - 968*B*a^3*b^5*c^3
*d^6*e^9 + 592*A*a^2*b^6*c^3*d^6*e^9 - 1305*B*a^4*b^3*c^4*d^6*e^9 + 3120*A*a^3*b^4*c^4*d^6*e^9 - 580*B*a^5*b*c
^5*d^6*e^9 + 1180*A*a^4*b^2*c^5*d^6*e^9 - 160*A*a^5*c^6*d^6*e^9 + 30*B*a^2*b^8*c*d^5*e^10 + 12*A*a*b^9*c*d^5*e
^10 + 338*B*a^3*b^6*c^2*d^5*e^10 + 21*A*a^2*b^7*c^2*d^5*e^10 + 940*B*a^4*b^4*c^3*d^5*e^10 - 1272*A*a^3*b^5*c^3
*d^5*e^10 + 894*B*a^5*b^2*c^4*d^5*e^10 - 2055*A*a^4*b^3*c^4*d^5*e^10 + 120*B*a^6*c^5*d^5*e^10 + 132*A*a^5*b*c^
5*d^5*e^10 - 40*B*a^3*b^7*c*d^4*e^11 - 30*A*a^2*b^8*c*d^4*e^11 - 325*B*a^4*b^5*c^2*d^4*e^11 + 110*A*a^3*b^6*c^
2*d^4*e^11 - 706*B*a^5*b^3*c^3*d^4*e^11 + 1300*A*a^4*b^4*c^3*d^4*e^11 - 336*B*a^6*b*c^4*d^4*e^11 + 450*A*a^5*b
^2*c^4*d^4*e^11 - 120*A*a^6*c^5*d^4*e^11 + 30*B*a^4*b^6*c*d^3*e^12 + 40*A*a^3*b^7*c*d^3*e^12 + 244*B*a^5*b^4*c
^2*d^3*e^12 - 235*A*a^4*b^5*c^2*d^3*e^12 + 352*B*a^6*b^2*c^3*d^3*e^12 - 638*A*a^5*b^3*c^3*d^3*e^12 + 48*B*a^7*
c^4*d^3*e^12 + 112*A*a^6*b*c^4*d^3*e^12 - 12*B*a^5*b^5*c*d^2*e^13 - 30*A*a^4*b^6*c*d^2*e^13 - 139*B*a^6*b^3*c^
2*d^2*e^13 + 204*A*a^5*b^4*c^2*d^2*e^13 - 92*B*a^7*b*c^3*d^2*e^13 + 96*A*a^6*b^2*c^3*d^2*e^13 - 48*A*a^7*c^4*d
^2*e^13 + 2*B*a^6*b^4*c*d*e^14 + 12*A*a^5*b^5*c*d*e^14 + 50*B*a^7*b^2*c^2*d*e^14 - 85*A*a^6*b^3*c^2*d*e^14 + 8
*B*a^8*c^3*d*e^14 + 28*A*a^7*b*c^3*d*e^14 - 2*A*a^6*b^4*c*e^15 - 8*B*a^8*b*c^2*e^15 + 14*A*a^7*b^2*c^2*e^15 -
8*A*a^8*c^3*e^15)/(b^4*c^8*d^16 - 8*a*b^2*c^9*d^16 + 16*a^2*c^10*d^16 - 8*b^5*c^7*d^15*e + 64*a*b^3*c^8*d^15*e
 - 128*a^2*b*c^9*d^15*e + 28*b^6*c^6*d^14*e^2 - 216*a*b^4*c^7*d^14*e^2 + 384*a^2*b^2*c^8*d^14*e^2 + 128*a^3*c^
9*d^14*e^2 - 56*b^7*c^5*d^13*e^3 + 392*a*b^5*c^6*d^13*e^3 - 448*a^2*b^3*c^7*d^13*e^3 - 896*a^3*b*c^8*d^13*e^3
+ 70*b^8*c^4*d^12*e^4 - 392*a*b^6*c^5*d^12*e^4 - 196*a^2*b^4*c^6*d^12*e^4 + 2464*a^3*b^2*c^7*d^12*e^4 + 448*a^
4*c^8*d^12*e^4 - 56*b^9*c^3*d^11*e^5 + 168*a*b^7*c^4*d^11*e^5 + 1176*a^2*b^5*c^5*d^11*e^5 - 3136*a^3*b^3*c^6*d
^11*e^5 - 2688*a^4*b*c^7*d^11*e^5 + 28*b^10*c^2*d^10*e^6 + 56*a*b^8*c^3*d^10*e^6 - 1372*a^2*b^6*c^4*d^10*e^6 +
 1176*a^3*b^4*c^5*d^10*e^6 + 6272*a^4*b^2*c^6*d^10*e^6 + 896*a^5*c^7*d^10*e^6 - 8*b^11*c*d^9*e^7 - 104*a*b^9*c
^2*d^9*e^7 + 656*a^2*b^7*c^3*d^9*e^7 + 1512*a^3*b^5*c^4*d^9*e^7 - 6720*a^4*b^3*c^5*d^9*e^7 - 4480*a^5*b*c^6*d^
9*e^7 + b^12*d^8*e^8 + 48*a*b^10*c*d^8*e^8 - 12*a^2*b^8*c^2*d^8*e^8 - 1904*a^3*b^6*c^3*d^8*e^8 + 2310*a^4*b^4*
c^4*d^8*e^8 + 8400*a^5*b^2*c^5*d^8*e^8 + 1120*a^6*c^6*d^8*e^8 - 8*a*b^11*d^7*e^9 - 104*a^2*b^9*c*d^7*e^9 + 656
*a^3*b^7*c^2*d^7*e^9 + 1512*a^4*b^5*c^3*d^7*e^9 - 6720*a^5*b^3*c^4*d^7*e^9 - 4480*a^6*b*c^5*d^7*e^9 + 28*a^2*b
^10*d^6*e^10 + 56*a^3*b^8*c*d^6*e^10 - 1372*a^4*b^6*c^2*d^6*e^10 + 1176*a^5*b^4*c^3*d^6*e^10 + 6272*a^6*b^2*c^
4*d^6*e^10 + 896*a^7*c^5*d^6*e^10 - 56*a^3*b^9*d^5*e^11 + 168*a^4*b^7*c*d^5*e^11 + 1176*a^5*b^5*c^2*d^5*e^11 -
 3136*a^6*b^3*c^3*d^5*e^11 - 2688*a^7*b*c^4*d^5*e^11 + 70*a^4*b^8*d^4*e^12 - 392*a^5*b^6*c*d^4*e^12 - 196*a^6*
b^4*c^2*d^4*e^12 + 2464*a^7*b^2*c^3*d^4*e^12 + 448*a^8*c^4*d^4*e^12 - 56*a^5*b^7*d^3*e^13 + 392*a^6*b^5*c*d^3*
e^13 - 448*a^7*b^3*c^2*d^3*e^13 - 896*a^8*b*c^3*d^3*e^13 + 28*a^6*b^6*d^2*e^14 - 216*a^7*b^4*c*d^2*e^14 + 384*
a^8*b^2*c^2*d^2*e^14 + 128*a^9*c^3*d^2*e^14 - 8*a^7*b^5*d*e^15 + 64*a^8*b^3*c*d*e^15 - 128*a^9*b*c^2*d*e^15 +
a^8*b^4*e^16 - 8*a^9*b^2*c*e^16 + 16*a^10*c^2*e^16))*x + 3*(B*b^3*c^8*d^15 + 4*B*a*b*c^9*d^15 - 2*A*b^2*c^9*d^
15 - 8*A*a*c^10*d^15 - 8*B*b^4*c^7*d^14*e - 28*B*a*b^2*c^8*d^14*e + 15*A*b^3*c^8*d^14*e + 60*A*a*b*c^9*d^14*e
+ 28*B*b^5*c^6*d^13*e^2 + 88*B*a*b^3*c^7*d^13*e^2 - 48*A*b^4*c^7*d^13*e^2 + 40*B*a^2*b*c^8*d^13*e^2 - 212*A*a*
b^2*c^8*d^13*e^2 - 64*A*a^2*c^9*d^13*e^2 - 56*B*b^6*c^5*d^12*e^3 - 170*B*a*b^4*c^6*d^12*e^3 + 84*A*b^5*c^6*d^1
2*e^3 - 242*B*a^2*b^2*c^7*d^12*e^3 + 472*A*a*b^3*c^7*d^12*e^3 - 8*B*a^3*c^8*d^12*e^3 + 408*A*a^2*b*c^8*d^12*e^
3 + 70*B*b^7*c^4*d^11*e^4 + 236*B*a*b^5*c^5*d^11*e^4 - 84*A*b^6*c^5*d^11*e^4 + 615*B*a^2*b^3*c^6*d^11*e^4 - 72
6*A*a*b^4*c^6*d^11*e^4 + 204*B*a^3*b*c^7*d^11*e^4 - 1158*A*a^2*b^2*c^7*d^11*e^4 - 216*A*a^3*c^8*d^11*e^4 - 56*
B*b^8*c^3*d^10*e^5 - 254*B*a*b^6*c^4*d^10*e^5 + 42*A*b^7*c^4*d^10*e^5 - 860*B*a^2*b^4*c^5*d^10*e^5 + 772*A*a*b
^5*c^5*d^10*e^5 - 912*B*a^3*b^2*c^6*d^10*e^5 + 1961*A*a^2*b^3*c^6*d^10*e^5 - 48*B*a^4*c^7*d^10*e^5 + 1140*A*a^
3*b*c^7*d^10*e^5 + 28*B*b^9*c^2*d^9*e^6 + 208*B*a*b^7*c^3*d^9*e^6 + 745*B*a^2*b^5*c^4*d^9*e^6 - 530*A*a*b^6*c^
4*d^9*e^6 + 1770*B*a^3*b^3*c^5*d^9*e^6 - 2220*A*a^2*b^4*c^5*d^9*e^6 + 560*B*a^4*b*c^6*d^9*e^6 - 2560*A*a^3*b^2
*c^6*d^9*e^6 - 400*A*a^4*c^7*d^9*e^6 - 8*B*b^10*c*d^8*e^7 - 118*B*a*b^8*c^2*d^8*e^7 - 12*A*b^9*c^2*d^8*e^7 - 4
50*B*a^2*b^6*c^3*d^8*e^7 + 192*A*a*b^7*c^3*d^8*e^7 - 1780*B*a^3*b^4*c^4*d^8*e^7 + 1719*A*a^2*b^5*c^4*d^8*e^7 -
 1790*B*a^4*b^2*c^5*d^8*e^7 + 3270*A*a^3*b^3*c^5*d^8*e^7 - 120*B*a^5*c^6*d^8*e^7 + 1680*A*a^4*b*c^6*d^8*e^7 +
B*b^11*d^7*e^8 + 40*B*a*b^9*c*d^7*e^8 + 6*A*b^10*c*d^7*e^8 + 217*B*a^2*b^7*c^2*d^7*e^8 - 2*A*a*b^8*c^2*d^7*e^8
 + 968*B*a^3*b^5*c^3*d^7*e^8 - 838*A*a^2*b^6*c^3*d^7*e^8 + 2495*B*a^4*b^3*c^4*d^7*e^8 - 2700*A*a^3*b^4*c^4*d^7
*e^8 + 860*B*a^5*b*c^5*d^7*e^8 - 2830*A*a^4*b^2*c^5*d^7*e^8 - 440*A*a^5*c^6*d^7*e^8 - 6*B*a*b^10*d^6*e^9 - A*b
^11*d^6*e^9 - 80*B*a^2*b^8*c*d^6*e^9 - 24*A*a*b^9*c*d^6*e^9 - 304*B*a^3*b^6*c^2*d^6*e^9 + 183*A*a^2*b^7*c^2*d^
6*e^9 - 1640*B*a^4*b^4*c^3*d^6*e^9 + 1496*A*a^3*b^5*c^3*d^6*e^9 - 1900*B*a^5*b^2*c^4*d^6*e^9 + 2545*A*a^4*b^3*
c^4*d^6*e^9 - 160*B*a^6*c^5*d^6*e^9 + 1380*A*a^5*b*c^5*d^6*e^9 + 15*B*a^2*b^9*d^5*e^10 + 6*A*a*b^10*d^5*e^10 +
 82*B*a^3*b^7*c*d^5*e^10 + 24*A*a^2*b^8*c*d^5*e^10 + 458*B*a^4*b^5*c^2*d^5*e^10 - 480*A*a^3*b^6*c^2*d^5*e^10 +
 1716*B*a^5*b^3*c^3*d^5*e^10 - 1440*A*a^4*b^4*c^3*d^5*e^10 + 744*B*a^6*b*c^4*d^5*e^10 - 1572*A*a^5*b^2*c^4*d^5
*e^10 - 288*A*a^6*c^5*d^5*e^10 - 20*B*a^3*b^8*d^4*e^11 - 15*A*a^2*b^9*d^4*e^11 - 50*B*a^4*b^6*c*d^4*e^11 + 30*
A*a^3*b^7*c*d^4*e^11 - 578*B*a^5*b^4*c^2*d^4*e^11 + 550*A*a^4*b^5*c^2*d^4*e^11 - 1038*B*a^6*b^2*c^3*d^4*e^11 +
 860*A*a^5*b^3*c^3*d^4*e^11 - 120*B*a^7*c^4*d^4*e^11 + 600*A*a^6*b*c^4*d^4*e^11 + 15*B*a^4*b^7*d^3*e^12 + 20*A
*a^3*b^8*d^3*e^12 + 28*B*a^5*b^5*c*d^3*e^12 - 90*A*a^4*b^6*c*d^3*e^12 + 473*B*a^6*b^3*c^2*d^3*e^12 - 318*A*a^5
*b^4*c^2*d^3*e^12 + 340*B*a^7*b*c^3*d^3*e^12 - 362*A*a^6*b^2*c^3*d^3*e^12 - 104*A*a^7*c^4*d^3*e^12 - 6*B*a^5*b
^6*d^2*e^13 - 15*A*a^4*b^7*d^2*e^13 - 20*B*a^6*b^4*c*d^2*e^13 + 84*A*a^5*b^5*c*d^2*e^13 - 232*B*a^7*b^2*c^2*d^
2*e^13 + 87*A*a^6*b^3*c^2*d^2*e^13 - 48*B*a^8*c^3*d^2*e^13 + 108*A*a^7*b*c^3*d^2*e^13 + B*a^6*b^5*d*e^14 + 6*A
*a^5*b^6*d*e^14 + 10*B*a^7*b^3*c*d*e^14 - 36*A*a^6*b^4*c*d*e^14 + 64*B*a^8*b*c^2*d*e^14 - 8*A*a^7*b^2*c^2*d*e^
14 - 16*A*a^8*c^3*d*e^14 - A*a^6*b^5*e^15 - 2*B*a^8*b^2*c*e^15 + 6*A*a^7*b^3*c*e^15 - 8*B*a^9*c^2*e^15)/(b^4*c
^8*d^16 - 8*a*b^2*c^9*d^16 + 16*a^2*c^10*d^16 - 8*b^5*c^7*d^15*e + 64*a*b^3*c^8*d^15*e - 128*a^2*b*c^9*d^15*e
+ 28*b^6*c^6*d^14*e^2 - 216*a*b^4*c^7*d^14*e^2 + 384*a^2*b^2*c^8*d^14*e^2 + 128*a^3*c^9*d^14*e^2 - 56*b^7*c^5*
d^13*e^3 + 392*a*b^5*c^6*d^13*e^3 - 448*a^2*b^3*c^7*d^13*e^3 - 896*a^3*b*c^8*d^13*e^3 + 70*b^8*c^4*d^12*e^4 -
392*a*b^6*c^5*d^12*e^4 - 196*a^2*b^4*c^6*d^12*e^4 + 2464*a^3*b^2*c^7*d^12*e^4 + 448*a^4*c^8*d^12*e^4 - 56*b^9*
c^3*d^11*e^5 + 168*a*b^7*c^4*d^11*e^5 + 1176*a^2*b^5*c^5*d^11*e^5 - 3136*a^3*b^3*c^6*d^11*e^5 - 2688*a^4*b*c^7
*d^11*e^5 + 28*b^10*c^2*d^10*e^6 + 56*a*b^8*c^3*d^10*e^6 - 1372*a^2*b^6*c^4*d^10*e^6 + 1176*a^3*b^4*c^5*d^10*e
^6 + 6272*a^4*b^2*c^6*d^10*e^6 + 896*a^5*c^7*d^10*e^6 - 8*b^11*c*d^9*e^7 - 104*a*b^9*c^2*d^9*e^7 + 656*a^2*b^7
*c^3*d^9*e^7 + 1512*a^3*b^5*c^4*d^9*e^7 - 6720*a^4*b^3*c^5*d^9*e^7 - 4480*a^5*b*c^6*d^9*e^7 + b^12*d^8*e^8 + 4
8*a*b^10*c*d^8*e^8 - 12*a^2*b^8*c^2*d^8*e^8 - 1904*a^3*b^6*c^3*d^8*e^8 + 2310*a^4*b^4*c^4*d^8*e^8 + 8400*a^5*b
^2*c^5*d^8*e^8 + 1120*a^6*c^6*d^8*e^8 - 8*a*b^11*d^7*e^9 - 104*a^2*b^9*c*d^7*e^9 + 656*a^3*b^7*c^2*d^7*e^9 + 1
512*a^4*b^5*c^3*d^7*e^9 - 6720*a^5*b^3*c^4*d^7*e^9 - 4480*a^6*b*c^5*d^7*e^9 + 28*a^2*b^10*d^6*e^10 + 56*a^3*b^
8*c*d^6*e^10 - 1372*a^4*b^6*c^2*d^6*e^10 + 1176*a^5*b^4*c^3*d^6*e^10 + 6272*a^6*b^2*c^4*d^6*e^10 + 896*a^7*c^5
*d^6*e^10 - 56*a^3*b^9*d^5*e^11 + 168*a^4*b^7*c*d^5*e^11 + 1176*a^5*b^5*c^2*d^5*e^11 - 3136*a^6*b^3*c^3*d^5*e^
11 - 2688*a^7*b*c^4*d^5*e^11 + 70*a^4*b^8*d^4*e^12 - 392*a^5*b^6*c*d^4*e^12 - 196*a^6*b^4*c^2*d^4*e^12 + 2464*
a^7*b^2*c^3*d^4*e^12 + 448*a^8*c^4*d^4*e^12 - 56*a^5*b^7*d^3*e^13 + 392*a^6*b^5*c*d^3*e^13 - 448*a^7*b^3*c^2*d
^3*e^13 - 896*a^8*b*c^3*d^3*e^13 + 28*a^6*b^6*d^2*e^14 - 216*a^7*b^4*c*d^2*e^14 + 384*a^8*b^2*c^2*d^2*e^14 + 1
28*a^9*c^3*d^2*e^14 - 8*a^7*b^5*d*e^15 + 64*a^8*b^3*c*d*e^15 - 128*a^9*b*c^2*d*e^15 + a^8*b^4*e^16 - 8*a^9*b^2
*c*e^16 + 16*a^10*c^2*e^16))*x + (2*B*a*b^2*c^8*d^15 + A*b^3*c^8*d^15 + 8*B*a^2*c^9*d^15 - 12*A*a*b*c^9*d^15 -
 16*B*a*b^3*c^7*d^14*e - 8*A*b^4*c^7*d^14*e - 56*B*a^2*b*c^8*d^14*e + 94*A*a*b^2*c^8*d^14*e - 8*A*a^2*c^9*d^14
*e + 56*B*a*b^4*c^6*d^13*e^2 + 28*A*b^5*c^6*d^13*e^2 + 176*B*a^2*b^2*c^7*d^13*e^2 - 312*A*a*b^3*c^7*d^13*e^2 +
 80*B*a^3*c^8*d^13*e^2 - 40*A*a^2*b*c^8*d^13*e^2 - 112*B*a*b^5*c^5*d^12*e^3 - 56*A*b^6*c^5*d^12*e^3 - 339*B*a^
2*b^3*c^6*d^12*e^3 + 560*A*a*b^4*c^6*d^12*e^3 - 492*B*a^3*b*c^7*d^12*e^3 + 496*A*a^2*b^2*c^7*d^12*e^3 - 80*A*a
^3*c^8*d^12*e^3 + 140*B*a*b^6*c^4*d^11*e^4 + 70*A*b^7*c^4*d^11*e^4 + 466*B*a^2*b^4*c^5*d^11*e^4 - 560*A*a*b^5*
c^5*d^11*e^4 + 1278*B*a^3*b^2*c^6*d^11*e^4 - 1649*A*a^2*b^3*c^6*d^11*e^4 + 312*B*a^4*c^7*d^11*e^4 + 156*A*a^3*
b*c^7*d^11*e^4 - 112*B*a*b^7*c^3*d^10*e^5 - 56*A*b^8*c^3*d^10*e^5 - 493*B*a^2*b^5*c^4*d^10*e^5 + 252*A*a*b^6*c
^4*d^10*e^5 - 1834*B*a^3*b^3*c^5*d^10*e^5 + 2726*A*a^2*b^4*c^5*d^10*e^5 - 1632*B*a^4*b*c^6*d^10*e^5 + 738*A*a^
3*b^2*c^6*d^10*e^5 - 312*A*a^4*c^7*d^10*e^5 + 56*B*a*b^8*c^2*d^9*e^6 + 28*A*b^9*c^2*d^9*e^6 + 396*B*a^2*b^6*c^
3*d^9*e^6 + 56*A*a*b^7*c^3*d^9*e^6 + 1620*B*a^3*b^4*c^4*d^9*e^6 - 2447*A*a^2*b^5*c^4*d^9*e^6 + 3460*B*a^4*b^2*
c^5*d^9*e^6 - 3150*A*a^3*b^3*c^5*d^9*e^6 + 640*B*a^5*c^6*d^9*e^6 + 960*A*a^4*b*c^6*d^9*e^6 - 16*B*a*b^9*c*d^8*
e^7 - 8*A*b^10*c*d^8*e^7 - 221*B*a^2*b^7*c^2*d^8*e^7 - 128*A*a*b^8*c^2*d^8*e^7 - 960*B*a^3*b^5*c^3*d^8*e^7 + 1
060*A*a^2*b^6*c^3*d^8*e^7 - 3785*B*a^4*b^3*c^4*d^8*e^7 + 4820*A*a^3*b^4*c^4*d^8*e^7 - 2740*B*a^5*b*c^5*d^8*e^7
 - 100*A*a^4*b^2*c^5*d^8*e^7 - 640*A*a^5*c^6*d^8*e^7 + 2*B*a*b^10*d^7*e^8 + A*b^11*d^7*e^8 + 74*B*a^2*b^8*c*d^
7*e^8 + 60*A*a*b^9*c*d^7*e^8 + 422*B*a^3*b^6*c^2*d^7*e^8 - 31*A*a^2*b^7*c^2*d^7*e^8 + 2260*B*a^4*b^4*c^3*d^7*e
^8 - 3520*A*a^3*b^5*c^3*d^7*e^8 + 4510*B*a^5*b^2*c^4*d^7*e^8 - 2865*A*a^4*b^3*c^4*d^7*e^8 + 760*B*a^6*c^5*d^7*
e^8 + 1900*A*a^5*b*c^5*d^7*e^8 - 11*B*a^2*b^9*d^6*e^9 - 10*A*a*b^10*d^6*e^9 - 138*B*a^3*b^7*c*d^6*e^9 - 146*A*
a^2*b^8*c*d^6*e^9 - 742*B*a^4*b^5*c^2*d^6*e^9 + 1034*A*a^3*b^6*c^2*d^6*e^9 - 3500*B*a^5*b^3*c^3*d^6*e^9 + 4180
*A*a^4*b^4*c^3*d^6*e^9 - 2520*B*a^6*b*c^4*d^6*e^9 - 1150*A*a^5*b^2*c^4*d^6*e^9 - 760*A*a^6*c^5*d^6*e^9 + 24*B*
a^3*b^8*d^5*e^10 + 39*A*a^2*b^9*d^5*e^10 + 152*B*a^4*b^6*c*d^5*e^10 + 82*A*a^3*b^7*c*d^5*e^10 + 1240*B*a^5*b^4
*c^2*d^5*e^10 - 2198*A*a^4*b^5*c^2*d^5*e^10 + 2952*B*a^6*b^2*c^3*d^5*e^10 - 1484*A*a^5*b^3*c^3*d^5*e^10 + 528*
B*a^7*c^4*d^5*e^10 + 1848*A*a^6*b*c^4*d^5*e^10 - 25*B*a^4*b^7*d^4*e^11 - 80*A*a^3*b^8*d^4*e^11 - 160*B*a^5*b^5
*c*d^4*e^11 + 240*A*a^4*b^6*c*d^4*e^11 - 1381*B*a^6*b^3*c^2*d^4*e^11 + 1952*A*a^5*b^4*c^2*d^4*e^11 - 1236*B*a^
7*b*c^3*d^4*e^11 - 936*A*a^6*b^2*c^3*d^4*e^11 - 528*A*a^7*c^4*d^4*e^11 + 10*B*a^5*b^6*d^3*e^12 + 95*A*a^4*b^7*
d^3*e^12 + 186*B*a^6*b^4*c*d^3*e^12 - 512*A*a^5*b^5*c*d^3*e^12 + 866*B*a^7*b^2*c^2*d^3*e^12 - 607*A*a^6*b^3*c^
2*d^3*e^12 + 200*B*a^8*c^3*d^3*e^12 + 900*A*a^7*b*c^3*d^3*e^12 + 3*B*a^6*b^5*d^2*e^13 - 66*A*a^5*b^6*d^2*e^13
- 154*B*a^7*b^3*c*d^2*e^13 + 430*A*a^6*b^4*c*d^2*e^13 - 272*B*a^8*b*c^2*d^2*e^13 - 194*A*a^7*b^2*c^2*d^2*e^13
- 200*A*a^8*c^3*d^2*e^13 - 4*B*a^7*b^4*d*e^14 + 25*A*a^6*b^5*d*e^14 + 68*B*a^8*b^2*c*d*e^14 - 174*A*a^7*b^3*c*
d*e^14 + 32*B*a^9*c^2*d*e^14 + 176*A*a^8*b*c^2*d*e^14 + B*a^8*b^3*e^15 - 4*A*a^7*b^4*e^15 - 12*B*a^9*b*c*e^15
+ 28*A*a^8*b^2*c*e^15 - 32*A*a^9*c^2*e^15)/(b^4*c^8*d^16 - 8*a*b^2*c^9*d^16 + 16*a^2*c^10*d^16 - 8*b^5*c^7*d^1
5*e + 64*a*b^3*c^8*d^15*e - 128*a^2*b*c^9*d^15*e + 28*b^6*c^6*d^14*e^2 - 216*a*b^4*c^7*d^14*e^2 + 384*a^2*b^2*
c^8*d^14*e^2 + 128*a^3*c^9*d^14*e^2 - 56*b^7*c^5*d^13*e^3 + 392*a*b^5*c^6*d^13*e^3 - 448*a^2*b^3*c^7*d^13*e^3
- 896*a^3*b*c^8*d^13*e^3 + 70*b^8*c^4*d^12*e^4 - 392*a*b^6*c^5*d^12*e^4 - 196*a^2*b^4*c^6*d^12*e^4 + 2464*a^3*
b^2*c^7*d^12*e^4 + 448*a^4*c^8*d^12*e^4 - 56*b^9*c^3*d^11*e^5 + 168*a*b^7*c^4*d^11*e^5 + 1176*a^2*b^5*c^5*d^11
*e^5 - 3136*a^3*b^3*c^6*d^11*e^5 - 2688*a^4*b*c^7*d^11*e^5 + 28*b^10*c^2*d^10*e^6 + 56*a*b^8*c^3*d^10*e^6 - 13
72*a^2*b^6*c^4*d^10*e^6 + 1176*a^3*b^4*c^5*d^10*e^6 + 6272*a^4*b^2*c^6*d^10*e^6 + 896*a^5*c^7*d^10*e^6 - 8*b^1
1*c*d^9*e^7 - 104*a*b^9*c^2*d^9*e^7 + 656*a^2*b^7*c^3*d^9*e^7 + 1512*a^3*b^5*c^4*d^9*e^7 - 6720*a^4*b^3*c^5*d^
9*e^7 - 4480*a^5*b*c^6*d^9*e^7 + b^12*d^8*e^8 + 48*a*b^10*c*d^8*e^8 - 12*a^2*b^8*c^2*d^8*e^8 - 1904*a^3*b^6*c^
3*d^8*e^8 + 2310*a^4*b^4*c^4*d^8*e^8 + 8400*a^5*b^2*c^5*d^8*e^8 + 1120*a^6*c^6*d^8*e^8 - 8*a*b^11*d^7*e^9 - 10
4*a^2*b^9*c*d^7*e^9 + 656*a^3*b^7*c^2*d^7*e^9 + 1512*a^4*b^5*c^3*d^7*e^9 - 6720*a^5*b^3*c^4*d^7*e^9 - 4480*a^6
*b*c^5*d^7*e^9 + 28*a^2*b^10*d^6*e^10 + 56*a^3*b^8*c*d^6*e^10 - 1372*a^4*b^6*c^2*d^6*e^10 + 1176*a^5*b^4*c^3*d
^6*e^10 + 6272*a^6*b^2*c^4*d^6*e^10 + 896*a^7*c^5*d^6*e^10 - 56*a^3*b^9*d^5*e^11 + 168*a^4*b^7*c*d^5*e^11 + 11
76*a^5*b^5*c^2*d^5*e^11 - 3136*a^6*b^3*c^3*d^5*e^11 - 2688*a^7*b*c^4*d^5*e^11 + 70*a^4*b^8*d^4*e^12 - 392*a^5*
b^6*c*d^4*e^12 - 196*a^6*b^4*c^2*d^4*e^12 + 2464*a^7*b^2*c^3*d^4*e^12 + 448*a^8*c^4*d^4*e^12 - 56*a^5*b^7*d^3*
e^13 + 392*a^6*b^5*c*d^3*e^13 - 448*a^7*b^3*c^2*d^3*e^13 - 896*a^8*b*c^3*d^3*e^13 + 28*a^6*b^6*d^2*e^14 - 216*
a^7*b^4*c*d^2*e^14 + 384*a^8*b^2*c^2*d^2*e^14 + 128*a^9*c^3*d^2*e^14 - 8*a^7*b^5*d*e^15 + 64*a^8*b^3*c*d*e^15
- 128*a^9*b*c^2*d*e^15 + a^8*b^4*e^16 - 8*a^9*b^2*c*e^16 + 16*a^10*c^2*e^16))/(c*x^2 + b*x + a)^(3/2)

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maple [B]  time = 0.04, size = 2996, normalized size = 6.87 \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)/(c*x^2+b*x+a)^(5/2),x)

[Out]

-2/3*e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*x*b
*A-4/3/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2
*x*d^2*B-2/3/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/
2)*b*c*d^2*B-16/3*e/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*x*b*A-32/3/e/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2)*x*d^2*B-16/3/e/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d^2*B-2*e^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c*A+4*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(
x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^2*d*A-4*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)
/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^2*d^2*B+2*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*
e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d*A-2*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(
b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d^2*B+16/3/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2)^2/((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*B*d+2/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*x*b*B*d-1/3*e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/(
(x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2*A-e^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/(
(x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*A+e^2/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*B*d+1/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)
/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2*B*d+4/3*B/e/(4*a*c-b^2)/(c*x^2+b*x+a)^(
3/2)*c*x+32/3*B/e*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+16/3*B/e*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b+1/3*e
/(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*A+e^3/(a*e^2-b*d*e+c*d^
2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*A-1/3/(a*e^2-b*d*e+c*d^2)/((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*B*d+2*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c*B*d+4/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*
c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2*x*d*A+2/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c*d*A+32/3/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^2)^2/((
x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d*A+8/3/(a*e^2-b*d*e+c*d^2)*c/(4*a*c-b^2)^2/
((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*B*d+16/3/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-
b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d*A-8/3*e/(a*e^2-b*d*e+c*d^2)*c/(4*
a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*A+e^2/(a*e^2-b*d*e+c*d^2)^2/(
4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*B*d-e^2/(a*e^2-b*d*e+c*d^2)^2
/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*B*d-e^3/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*A+2/3*B/e/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/
2)*b

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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)/(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((b/e-(2*c*d)/e^2)^2>0)', see `
assume?` for more details)Is (b/e-(2*c*d)/e^2)^2    -(4*c       *((-(b*d)/e)        +(c*d^2)/e^2+a))     /e^2
zero or nonzero?

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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 )\,{\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)*(a + b*x + c*x^2)^(5/2)),x)

[Out]

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

[Out]

Timed out

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