3.23.39 \(\int \frac {(A+B x) (d+e x)^4}{(a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=608 \[ \frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (-2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )-x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )+4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )-8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 B e \left (c d^2-5 a e^2\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {e \sqrt {a+b x+c x^2} \left (-4 b^2 c e \left (25 a B e^2+A c d e+2 B c d^2\right )+8 b c^2 \left (5 a A e^3+13 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (5 a e^2+2 c d^2\right )+4 a B e \left (c d^2-2 a e^2\right )\right )-2 b^3 c e^2 (3 A e+7 B d)+15 b^4 B e^3\right )}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {e^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{2 c^{7/2}} \]

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Rubi [A]  time = 0.84, antiderivative size = 608, 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 = {818, 640, 621, 206} \begin {gather*} \frac {e \sqrt {a+b x+c x^2} \left (-4 b^2 c e \left (25 a B e^2+A c d e+2 B c d^2\right )+8 b c^2 \left (5 a A e^3+13 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (5 a e^2+2 c d^2\right )+4 a B e \left (c d^2-2 a e^2\right )\right )-2 b^3 c e^2 (3 A e+7 B d)+15 b^4 B e^3\right )}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {2 (d+e x) \left (-x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )-2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )+4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )-8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 B e \left (c d^2-5 a e^2\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {e^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{2 c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^3*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/
(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (2*(d + e*x)*(b^3*B*e*(c*d^2 - 5*a*e^2) - 8*a*c^2*e*(A*c*d^2 + 8
*a*B*d*e + 3*a*A*e^2) - 2*b^2*c*(2*B*c*d^3 + 5*A*c*d^2*e - 2*a*B*d*e^2 - a*A*e^3) + 4*b*c*(2*A*c*d*(c*d^2 + 3*
a*e^2) + a*B*e*(5*c*d^2 + 7*a*e^2)) - (5*b^4*B*e^3 - 2*b^3*c*e^2*(3*B*d + A*e) - 4*b^2*c*e*(B*c*d^2 + A*c*d*e
+ 8*a*B*e^2) + 8*b*c^2*(B*c*d^3 + 3*A*c*d^2*e + 6*a*B*d*e^2 + 2*a*A*e^3) - 16*c^2*(2*a*B*e*(c*d^2 - a*e^2) + A
*c*d*(c*d^2 + 2*a*e^2)))*x))/(3*c^2*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) + (e*(15*b^4*B*e^3 - 2*b^3*c*e^2*(7
*B*d + 3*A*e) - 4*b^2*c*e*(2*B*c*d^2 + A*c*d*e + 25*a*B*e^2) + 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 13*a*B*d*e^2
+ 5*a*A*e^3) - 16*c^2*(4*a*B*e*(c*d^2 - 2*a*e^2) + A*c*d*(2*c*d^2 + 5*a*e^2)))*Sqrt[a + b*x + c*x^2])/(3*c^3*
(b^2 - 4*a*c)^2) + (e^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/
(2*c^(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 621

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

Rule 640

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

Rule 818

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

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac {2 (d+e x)^3 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {(d+e x)^2 \left (\frac {1}{2} \left (-8 A c^2 d^2-2 b B e \left (\frac {b d}{2}-3 a e\right )-4 a c e (4 B d+3 A e)+2 b c d (2 B d+5 A e)\right )+\frac {1}{2} e \left (4 A c^2 d+5 b^2 B e-2 c (b B d+A b e+8 a B e)\right ) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=\frac {2 (d+e x)^3 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (b^3 B e \left (c d^2-5 a e^2\right )-8 a c^2 e \left (A c d^2+8 a B d e+3 a A e^2\right )-2 b^2 c \left (2 B c d^3+5 A c d^2 e-2 a B d e^2-a A e^3\right )+4 b c \left (2 A c d \left (c d^2+3 a e^2\right )+a B e \left (5 c d^2+7 a e^2\right )\right )-\left (5 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-4 b^2 c e \left (B c d^2+A c d e+8 a B e^2\right )+8 b c^2 \left (B c d^3+3 A c d^2 e+6 a B d e^2+2 a A e^3\right )-16 c^2 \left (2 a B e \left (c d^2-a e^2\right )+A c d \left (c d^2+2 a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} e \left (5 b^4 B d e^2+48 a^2 c^2 e^2 (4 B d+A e)-2 b^3 e \left (2 B c d^2+A c d e-5 a B e^2\right )+4 b^2 c \left (2 B c d^3+6 A c d^2 e-11 a B d e^2-a A e^3\right )-8 b c \left (A c d \left (2 c d^2+5 a e^2\right )+a B e \left (4 c d^2+7 a e^2\right )\right )\right )+\frac {1}{4} e \left (15 b^4 B e^3-2 b^3 c e^2 (7 B d+3 A e)-4 b^2 c e \left (2 B c d^2+A c d e+25 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+13 a B d e^2+5 a A e^3\right )-16 c^2 \left (4 a B e \left (c d^2-2 a e^2\right )+A c d \left (2 c d^2+5 a e^2\right )\right )\right ) x}{\sqrt {a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )^2}\\ &=\frac {2 (d+e x)^3 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (b^3 B e \left (c d^2-5 a e^2\right )-8 a c^2 e \left (A c d^2+8 a B d e+3 a A e^2\right )-2 b^2 c \left (2 B c d^3+5 A c d^2 e-2 a B d e^2-a A e^3\right )+4 b c \left (2 A c d \left (c d^2+3 a e^2\right )+a B e \left (5 c d^2+7 a e^2\right )\right )-\left (5 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-4 b^2 c e \left (B c d^2+A c d e+8 a B e^2\right )+8 b c^2 \left (B c d^3+3 A c d^2 e+6 a B d e^2+2 a A e^3\right )-16 c^2 \left (2 a B e \left (c d^2-a e^2\right )+A c d \left (c d^2+2 a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {e \left (15 b^4 B e^3-2 b^3 c e^2 (7 B d+3 A e)-4 b^2 c e \left (2 B c d^2+A c d e+25 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+13 a B d e^2+5 a A e^3\right )-16 c^2 \left (4 a B e \left (c d^2-2 a e^2\right )+A c d \left (2 c d^2+5 a e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {\left (e^3 (8 B c d-5 b B e+2 A c e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c^3}\\ &=\frac {2 (d+e x)^3 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (b^3 B e \left (c d^2-5 a e^2\right )-8 a c^2 e \left (A c d^2+8 a B d e+3 a A e^2\right )-2 b^2 c \left (2 B c d^3+5 A c d^2 e-2 a B d e^2-a A e^3\right )+4 b c \left (2 A c d \left (c d^2+3 a e^2\right )+a B e \left (5 c d^2+7 a e^2\right )\right )-\left (5 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-4 b^2 c e \left (B c d^2+A c d e+8 a B e^2\right )+8 b c^2 \left (B c d^3+3 A c d^2 e+6 a B d e^2+2 a A e^3\right )-16 c^2 \left (2 a B e \left (c d^2-a e^2\right )+A c d \left (c d^2+2 a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {e \left (15 b^4 B e^3-2 b^3 c e^2 (7 B d+3 A e)-4 b^2 c e \left (2 B c d^2+A c d e+25 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+13 a B d e^2+5 a A e^3\right )-16 c^2 \left (4 a B e \left (c d^2-2 a e^2\right )+A c d \left (2 c d^2+5 a e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {\left (e^3 (8 B c d-5 b B e+2 A c e)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{c^3}\\ &=\frac {2 (d+e x)^3 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (d+e x) \left (b^3 B e \left (c d^2-5 a e^2\right )-8 a c^2 e \left (A c d^2+8 a B d e+3 a A e^2\right )-2 b^2 c \left (2 B c d^3+5 A c d^2 e-2 a B d e^2-a A e^3\right )+4 b c \left (2 A c d \left (c d^2+3 a e^2\right )+a B e \left (5 c d^2+7 a e^2\right )\right )-\left (5 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-4 b^2 c e \left (B c d^2+A c d e+8 a B e^2\right )+8 b c^2 \left (B c d^3+3 A c d^2 e+6 a B d e^2+2 a A e^3\right )-16 c^2 \left (2 a B e \left (c d^2-a e^2\right )+A c d \left (c d^2+2 a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {e \left (15 b^4 B e^3-2 b^3 c e^2 (7 B d+3 A e)-4 b^2 c e \left (2 B c d^2+A c d e+25 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+13 a B d e^2+5 a A e^3\right )-16 c^2 \left (4 a B e \left (c d^2-2 a e^2\right )+A c d \left (2 c d^2+5 a e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{3 c^3 \left (b^2-4 a c\right )^2}+\frac {e^3 (8 B c d-5 b B e+2 A c e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 2.40, size = 849, normalized size = 1.40 \begin {gather*} \frac {(8 B c d-5 b B e+2 A c e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) e^3}{2 c^{7/2}}+\frac {B \left (128 a^4 c^2 e^4+4 a^3 c \left (-48 \left (d^2+e x d-e^2 x^2\right ) c^2+2 b e (20 d+39 e x) c-25 b^2 e^2\right ) e^2+b x \left (15 e^4 x b^5+4 c e^3 x (5 e x-6 d) b^4+c^2 e^3 x^2 (3 e x-32 d) b^3-6 c^3 d^2 \left (d^2-4 e x d-2 e^2 x^2\right ) b^2+8 c^4 d^3 x (2 e x-3 d) b-16 c^5 d^4 x^2\right )+a^2 \left (-16 \left (d^4+18 e^2 x^2 d^2+16 e^3 x^3 d-3 e^4 x^4\right ) c^4+32 b e \left (2 d^3-9 e x d^2+8 e^3 x^3\right ) c^3+48 b^2 e^3 x (7 d+e x) c^2-6 b^3 e^3 (4 d+35 e x) c+15 b^4 e^4\right )+2 a \left (15 e^4 x b^5-3 c e^3 x (8 d+15 e x) b^4+2 c^2 e^3 x^2 (36 d-37 e x) b^3-2 c^3 \left (d^4-24 e x d^3+18 e^2 x^2 d^2-56 e^3 x^3 d+6 e^4 x^4\right ) b^2-12 c^4 d^2 x \left (d^2-4 e x d+6 e^2 x^2\right ) b+32 c^5 d^3 e x^3\right )\right )-2 A c \left (3 e^4 x^2 b^5+2 e^4 x \left (2 c x^2+3 a\right ) b^4+\left (3 a^2 e^4-18 a c x^2 e^4+c^2 d \left (d^3+12 e x d^2-18 e^2 x^2 d-4 e^3 x^3\right )\right ) b^3-2 c \left (21 a^2 x e^4+2 a c \left (-2 d^3+18 e x d^2-6 e^2 x^2 d+7 e^3 x^3\right ) e+3 c^2 d^2 x \left (d^2-8 e x d+2 e^2 x^2\right )\right ) b^2-4 c \left (5 a^3 e^4+12 a^2 c d (d-2 e x) e^2+2 c^3 d^3 x^2 (3 d-4 e x)+3 a c^2 d \left (d^3-4 e x d^2+6 e^2 x^2 d-4 e^3 x^3\right )\right ) b+8 c^2 \left (-2 c^3 x^3 d^4-3 a c^2 x \left (d^2+2 e^2 x^2\right ) d^2+a^3 e^3 (8 d+3 e x)+4 a^2 c e \left (d^3+3 e^2 x^2 d+e^3 x^3\right )\right )\right )}{3 c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*A*c*(3*b^5*e^4*x^2 + 2*b^4*e^4*x*(3*a + 2*c*x^2) + b^3*(3*a^2*e^4 - 18*a*c*e^4*x^2 + c^2*d*(d^3 + 12*d^2*e
*x - 18*d*e^2*x^2 - 4*e^3*x^3)) - 4*b*c*(5*a^3*e^4 + 2*c^3*d^3*x^2*(3*d - 4*e*x) + 12*a^2*c*d*e^2*(d - 2*e*x)
+ 3*a*c^2*d*(d^3 - 4*d^2*e*x + 6*d*e^2*x^2 - 4*e^3*x^3)) + 8*c^2*(-2*c^3*d^4*x^3 + a^3*e^3*(8*d + 3*e*x) - 3*a
*c^2*d^2*x*(d^2 + 2*e^2*x^2) + 4*a^2*c*e*(d^3 + 3*d*e^2*x^2 + e^3*x^3)) - 2*b^2*c*(21*a^2*e^4*x + 3*c^2*d^2*x*
(d^2 - 8*d*e*x + 2*e^2*x^2) + 2*a*c*e*(-2*d^3 + 18*d^2*e*x - 6*d*e^2*x^2 + 7*e^3*x^3))) + B*(128*a^4*c^2*e^4 +
b*x*(15*b^5*e^4*x - 16*c^5*d^4*x^2 + 8*b*c^4*d^3*x*(-3*d + 2*e*x) + b^3*c^2*e^3*x^2*(-32*d + 3*e*x) + 4*b^4*c
*e^3*x*(-6*d + 5*e*x) - 6*b^2*c^3*d^2*(d^2 - 4*d*e*x - 2*e^2*x^2)) + 4*a^3*c*e^2*(-25*b^2*e^2 + 2*b*c*e*(20*d
+ 39*e*x) - 48*c^2*(d^2 + d*e*x - e^2*x^2)) + a^2*(15*b^4*e^4 + 48*b^2*c^2*e^3*x*(7*d + e*x) - 6*b^3*c*e^3*(4*
d + 35*e*x) + 32*b*c^3*e*(2*d^3 - 9*d^2*e*x + 8*e^3*x^3) - 16*c^4*(d^4 + 18*d^2*e^2*x^2 + 16*d*e^3*x^3 - 3*e^4
*x^4)) + 2*a*(15*b^5*e^4*x + 32*c^5*d^3*e*x^3 + 2*b^3*c^2*e^3*x^2*(36*d - 37*e*x) - 3*b^4*c*e^3*x*(8*d + 15*e*
x) - 12*b*c^4*d^2*x*(d^2 - 4*d*e*x + 6*e^2*x^2) - 2*b^2*c^3*(d^4 - 24*d^3*e*x + 18*d^2*e^2*x^2 - 56*d*e^3*x^3
+ 6*e^4*x^4))))/(3*c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (e^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[(
b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(2*c^(7/2))

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IntegrateAlgebraic [B]  time = 14.80, size = 1243, normalized size = 2.04 \begin {gather*} \frac {15 B e^4 x^2 b^6+20 B c e^4 x^3 b^5-6 A c e^4 x^2 b^5-24 B c d e^3 x^2 b^5+30 a B e^4 x b^5+15 a^2 B e^4 b^4+3 B c^2 e^4 x^4 b^4-8 A c^2 e^4 x^3 b^4-32 B c^2 d e^3 x^3 b^4-90 a B c e^4 x^2 b^4-12 a A c e^4 x b^4-48 a B c d e^3 x b^4-2 A c^3 d^4 b^3-6 a^2 A c e^4 b^3-24 a^2 B c d e^3 b^3-148 a B c^2 e^4 x^3 b^3+8 A c^3 d e^3 x^3 b^3+12 B c^3 d^2 e^2 x^3 b^3+36 a A c^2 e^4 x^2 b^3+144 a B c^2 d e^3 x^2 b^3+36 A c^3 d^2 e^2 x^2 b^3+24 B c^3 d^3 e x^2 b^3-6 B c^3 d^4 x b^3-210 a^2 B c e^4 x b^3-24 A c^3 d^3 e x b^3-4 a B c^3 d^4 b^2-100 a^3 B c e^4 b^2-24 a B c^3 e^4 x^4 b^2+56 a A c^3 e^4 x^3 b^2+224 a B c^3 d e^3 x^3 b^2+24 A c^4 d^2 e^2 x^3 b^2+16 B c^4 d^3 e x^3 b^2-24 B c^4 d^4 x^2 b^2+48 a^2 B c^2 e^4 x^2 b^2-48 a A c^3 d e^3 x^2 b^2-72 a B c^3 d^2 e^2 x^2 b^2-96 A c^4 d^3 e x^2 b^2-16 a A c^3 d^3 e b^2+12 A c^4 d^4 x b^2+84 a^2 A c^2 e^4 x b^2+336 a^2 B c^2 d e^3 x b^2+144 a A c^3 d^2 e^2 x b^2+96 a B c^3 d^3 e x b^2+24 a A c^4 d^4 b+40 a^3 A c^2 e^4 b+160 a^3 B c^2 d e^3 b-16 B c^5 d^4 x^3 b+256 a^2 B c^3 e^4 x^3 b-96 a A c^4 d e^3 x^3 b-144 a B c^4 d^2 e^2 x^3 b-64 A c^5 d^3 e x^3 b+96 a^2 A c^3 d^2 e^2 b+48 A c^5 d^4 x^2 b+144 a A c^4 d^2 e^2 x^2 b+96 a B c^4 d^3 e x^2 b+64 a^2 B c^3 d^3 e b-24 a B c^4 d^4 x b+312 a^3 B c^2 e^4 x b-192 a^2 A c^3 d e^3 x b-288 a^2 B c^3 d^2 e^2 x b-96 a A c^4 d^3 e x b-16 a^2 B c^4 d^4+128 a^4 B c^2 e^4+48 a^2 B c^4 e^4 x^4-128 a^3 A c^3 d e^3+32 A c^6 d^4 x^3-64 a^2 A c^4 e^4 x^3-256 a^2 B c^4 d e^3 x^3+96 a A c^5 d^2 e^2 x^3+64 a B c^5 d^3 e x^3-192 a^3 B c^3 d^2 e^2+192 a^3 B c^3 e^4 x^2-192 a^2 A c^4 d e^3 x^2-288 a^2 B c^4 d^2 e^2 x^2-64 a^2 A c^4 d^3 e+48 a A c^5 d^4 x-48 a^3 A c^3 e^4 x-192 a^3 B c^3 d e^3 x}{3 c^3 \left (4 a c-b^2\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac {\left (5 b B e^4-2 A c e^4-8 B c d e^3\right ) \log \left (2 x c^4-2 \sqrt {c x^2+b x+a} c^{7/2}+b c^3\right )}{2 c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*A*b^3*c^3*d^4 - 4*a*b^2*B*c^3*d^4 + 24*a*A*b*c^4*d^4 - 16*a^2*B*c^4*d^4 - 16*a*A*b^2*c^3*d^3*e + 64*a^2*b*
B*c^3*d^3*e - 64*a^2*A*c^4*d^3*e + 96*a^2*A*b*c^3*d^2*e^2 - 192*a^3*B*c^3*d^2*e^2 - 24*a^2*b^3*B*c*d*e^3 + 160
*a^3*b*B*c^2*d*e^3 - 128*a^3*A*c^3*d*e^3 + 15*a^2*b^4*B*e^4 - 6*a^2*A*b^3*c*e^4 - 100*a^3*b^2*B*c*e^4 + 40*a^3
*A*b*c^2*e^4 + 128*a^4*B*c^2*e^4 - 6*b^3*B*c^3*d^4*x + 12*A*b^2*c^4*d^4*x - 24*a*b*B*c^4*d^4*x + 48*a*A*c^5*d^
4*x - 24*A*b^3*c^3*d^3*e*x + 96*a*b^2*B*c^3*d^3*e*x - 96*a*A*b*c^4*d^3*e*x + 144*a*A*b^2*c^3*d^2*e^2*x - 288*a
^2*b*B*c^3*d^2*e^2*x - 48*a*b^4*B*c*d*e^3*x + 336*a^2*b^2*B*c^2*d*e^3*x - 192*a^2*A*b*c^3*d*e^3*x - 192*a^3*B*
c^3*d*e^3*x + 30*a*b^5*B*e^4*x - 12*a*A*b^4*c*e^4*x - 210*a^2*b^3*B*c*e^4*x + 84*a^2*A*b^2*c^2*e^4*x + 312*a^3
*b*B*c^2*e^4*x - 48*a^3*A*c^3*e^4*x - 24*b^2*B*c^4*d^4*x^2 + 48*A*b*c^5*d^4*x^2 + 24*b^3*B*c^3*d^3*e*x^2 - 96*
A*b^2*c^4*d^3*e*x^2 + 96*a*b*B*c^4*d^3*e*x^2 + 36*A*b^3*c^3*d^2*e^2*x^2 - 72*a*b^2*B*c^3*d^2*e^2*x^2 + 144*a*A
*b*c^4*d^2*e^2*x^2 - 288*a^2*B*c^4*d^2*e^2*x^2 - 24*b^5*B*c*d*e^3*x^2 + 144*a*b^3*B*c^2*d*e^3*x^2 - 48*a*A*b^2
*c^3*d*e^3*x^2 - 192*a^2*A*c^4*d*e^3*x^2 + 15*b^6*B*e^4*x^2 - 6*A*b^5*c*e^4*x^2 - 90*a*b^4*B*c*e^4*x^2 + 36*a*
A*b^3*c^2*e^4*x^2 + 48*a^2*b^2*B*c^2*e^4*x^2 + 192*a^3*B*c^3*e^4*x^2 - 16*b*B*c^5*d^4*x^3 + 32*A*c^6*d^4*x^3 +
16*b^2*B*c^4*d^3*e*x^3 - 64*A*b*c^5*d^3*e*x^3 + 64*a*B*c^5*d^3*e*x^3 + 12*b^3*B*c^3*d^2*e^2*x^3 + 24*A*b^2*c^
4*d^2*e^2*x^3 - 144*a*b*B*c^4*d^2*e^2*x^3 + 96*a*A*c^5*d^2*e^2*x^3 - 32*b^4*B*c^2*d*e^3*x^3 + 8*A*b^3*c^3*d*e^
3*x^3 + 224*a*b^2*B*c^3*d*e^3*x^3 - 96*a*A*b*c^4*d*e^3*x^3 - 256*a^2*B*c^4*d*e^3*x^3 + 20*b^5*B*c*e^4*x^3 - 8*
A*b^4*c^2*e^4*x^3 - 148*a*b^3*B*c^2*e^4*x^3 + 56*a*A*b^2*c^3*e^4*x^3 + 256*a^2*b*B*c^3*e^4*x^3 - 64*a^2*A*c^4*
e^4*x^3 + 3*b^4*B*c^2*e^4*x^4 - 24*a*b^2*B*c^3*e^4*x^4 + 48*a^2*B*c^4*e^4*x^4)/(3*c^3*(-b^2 + 4*a*c)^2*(a + b*
x + c*x^2)^(3/2)) + ((-8*B*c*d*e^3 + 5*b*B*e^4 - 2*A*c*e^4)*Log[b*c^3 + 2*c^4*x - 2*c^(7/2)*Sqrt[a + b*x + c*x
^2]])/(2*c^(7/2))

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fricas [B]  time = 14.41, size = 3181, normalized size = 5.23

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(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*e^3 - (5*B*a^2*b^5 - 32*A*a^4*c^3 + 16*(5*B*a^4*b
+ A*a^3*b^2)*c^2 - 2*(20*B*a^3*b^3 + A*a^2*b^4)*c)*e^4 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d*e^3
- (5*B*b^5*c^2 - 32*A*a^2*c^5 + 16*(5*B*a^2*b + A*a*b^2)*c^4 - 2*(20*B*a*b^3 + A*b^4)*c^3)*e^4)*x^4 + 2*(8*(B*
b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c^4)*d*e^3 - (5*B*b^6*c - 32*A*a^2*b*c^4 + 16*(5*B*a^2*b^2 + A*a*b^3)*c^3
- 2*(20*B*a*b^4 + A*b^5)*c^2)*e^4)*x^3 + (8*(B*b^6*c - 6*B*a*b^4*c^2 + 32*B*a^3*c^4)*d*e^3 - (5*B*b^7 + 12*A*
a*b^4*c^2 + 160*B*a^3*b*c^3 - 64*A*a^3*c^4 - 2*(15*B*a*b^5 + A*b^6)*c)*e^4)*x^2 + 2*(8*(B*a*b^5*c - 8*B*a^2*b^
3*c^2 + 16*B*a^3*b*c^3)*d*e^3 - (5*B*a*b^6 - 32*A*a^3*b*c^3 + 16*(5*B*a^3*b^2 + A*a^2*b^3)*c^2 - 2*(20*B*a^2*b
^4 + A*a*b^5)*c)*e^4)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c)
- 4*a*c) - 4*(96*(2*B*a^3 - A*a^2*b)*c^4*d^2*e^2 - 3*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*e^4*x^4 + 2*(4
*(2*B*a^2 - 3*A*a*b)*c^5 + (2*B*a*b^2 + A*b^3)*c^4)*d^4 + 16*(4*A*a^2*c^5 - (4*B*a^2*b - A*a*b^2)*c^4)*d^3*e +
8*(3*B*a^2*b^3*c^2 - 20*B*a^3*b*c^3 + 16*A*a^3*c^4)*d*e^3 - (15*B*a^2*b^4*c + 8*(16*B*a^4 + 5*A*a^3*b)*c^3 -
2*(50*B*a^3*b^2 + 3*A*a^2*b^3)*c^2)*e^4 + 4*(4*(B*b*c^6 - 2*A*c^7)*d^4 - 4*(B*b^2*c^5 + 4*(B*a - A*b)*c^6)*d^3
*e - 3*(B*b^3*c^4 + 8*A*a*c^6 - 2*(6*B*a*b - A*b^2)*c^5)*d^2*e^2 + 2*(4*B*b^4*c^3 + 4*(8*B*a^2 + 3*A*a*b)*c^5
- (28*B*a*b^2 + A*b^3)*c^4)*d*e^3 - (5*B*b^5*c^2 - 16*A*a^2*c^5 + 2*(32*B*a^2*b + 7*A*a*b^2)*c^4 - (37*B*a*b^3
+ 2*A*b^4)*c^3)*e^4)*x^3 + 3*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 8*(B*b^3*c^4 + 4*(B*a*b - A*b^2)*c^5)*d^3*e + 1
2*(4*(2*B*a^2 - A*a*b)*c^5 + (2*B*a*b^2 - A*b^3)*c^4)*d^2*e^2 + 8*(B*b^5*c^2 - 6*B*a*b^3*c^3 + 2*A*a*b^2*c^4 +
8*A*a^2*c^5)*d*e^3 - (5*B*b^6*c + 64*B*a^3*c^4 + 4*(4*B*a^2*b^2 + 3*A*a*b^3)*c^3 - 2*(15*B*a*b^4 + A*b^5)*c^2
)*e^4)*x^2 + 6*(24*(2*B*a^2*b - A*a*b^2)*c^4*d^2*e^2 + (B*b^3*c^4 - 8*A*a*c^6 + 2*(2*B*a*b - A*b^2)*c^5)*d^4 +
4*(4*A*a*b*c^5 - (4*B*a*b^2 - A*b^3)*c^4)*d^3*e + 8*(B*a*b^4*c^2 - 7*B*a^2*b^2*c^3 + 4*(B*a^3 + A*a^2*b)*c^4)
*d*e^3 - (5*B*a*b^5*c - 8*A*a^3*c^4 + 2*(26*B*a^3*b + 7*A*a^2*b^2)*c^3 - (35*B*a^2*b^3 + 2*A*a*b^4)*c^2)*e^4)*
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 + (b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*x^4
+ 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7)*x^2 + 2*(a*b^5*c^4 - 8*
a^2*b^3*c^5 + 16*a^3*b*c^6)*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*e^3 - (5*B*a^2*b^5
- 32*A*a^4*c^3 + 16*(5*B*a^4*b + A*a^3*b^2)*c^2 - 2*(20*B*a^3*b^3 + A*a^2*b^4)*c)*e^4 + (8*(B*b^4*c^3 - 8*B*a
*b^2*c^4 + 16*B*a^2*c^5)*d*e^3 - (5*B*b^5*c^2 - 32*A*a^2*c^5 + 16*(5*B*a^2*b + A*a*b^2)*c^4 - 2*(20*B*a*b^3 +
A*b^4)*c^3)*e^4)*x^4 + 2*(8*(B*b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c^4)*d*e^3 - (5*B*b^6*c - 32*A*a^2*b*c^4 +
16*(5*B*a^2*b^2 + A*a*b^3)*c^3 - 2*(20*B*a*b^4 + A*b^5)*c^2)*e^4)*x^3 + (8*(B*b^6*c - 6*B*a*b^4*c^2 + 32*B*a^
3*c^4)*d*e^3 - (5*B*b^7 + 12*A*a*b^4*c^2 + 160*B*a^3*b*c^3 - 64*A*a^3*c^4 - 2*(15*B*a*b^5 + A*b^6)*c)*e^4)*x^2
+ 2*(8*(B*a*b^5*c - 8*B*a^2*b^3*c^2 + 16*B*a^3*b*c^3)*d*e^3 - (5*B*a*b^6 - 32*A*a^3*b*c^3 + 16*(5*B*a^3*b^2 +
A*a^2*b^3)*c^2 - 2*(20*B*a^2*b^4 + A*a*b^5)*c)*e^4)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*
sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(96*(2*B*a^3 - A*a^2*b)*c^4*d^2*e^2 - 3*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*
B*a^2*c^5)*e^4*x^4 + 2*(4*(2*B*a^2 - 3*A*a*b)*c^5 + (2*B*a*b^2 + A*b^3)*c^4)*d^4 + 16*(4*A*a^2*c^5 - (4*B*a^2*
b - A*a*b^2)*c^4)*d^3*e + 8*(3*B*a^2*b^3*c^2 - 20*B*a^3*b*c^3 + 16*A*a^3*c^4)*d*e^3 - (15*B*a^2*b^4*c + 8*(16*
B*a^4 + 5*A*a^3*b)*c^3 - 2*(50*B*a^3*b^2 + 3*A*a^2*b^3)*c^2)*e^4 + 4*(4*(B*b*c^6 - 2*A*c^7)*d^4 - 4*(B*b^2*c^5
+ 4*(B*a - A*b)*c^6)*d^3*e - 3*(B*b^3*c^4 + 8*A*a*c^6 - 2*(6*B*a*b - A*b^2)*c^5)*d^2*e^2 + 2*(4*B*b^4*c^3 + 4
*(8*B*a^2 + 3*A*a*b)*c^5 - (28*B*a*b^2 + A*b^3)*c^4)*d*e^3 - (5*B*b^5*c^2 - 16*A*a^2*c^5 + 2*(32*B*a^2*b + 7*A
*a*b^2)*c^4 - (37*B*a*b^3 + 2*A*b^4)*c^3)*e^4)*x^3 + 3*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 - 8*(B*b^3*c^4 + 4*(B*a*
b - A*b^2)*c^5)*d^3*e + 12*(4*(2*B*a^2 - A*a*b)*c^5 + (2*B*a*b^2 - A*b^3)*c^4)*d^2*e^2 + 8*(B*b^5*c^2 - 6*B*a*
b^3*c^3 + 2*A*a*b^2*c^4 + 8*A*a^2*c^5)*d*e^3 - (5*B*b^6*c + 64*B*a^3*c^4 + 4*(4*B*a^2*b^2 + 3*A*a*b^3)*c^3 - 2
*(15*B*a*b^4 + A*b^5)*c^2)*e^4)*x^2 + 6*(24*(2*B*a^2*b - A*a*b^2)*c^4*d^2*e^2 + (B*b^3*c^4 - 8*A*a*c^6 + 2*(2*
B*a*b - A*b^2)*c^5)*d^4 + 4*(4*A*a*b*c^5 - (4*B*a*b^2 - A*b^3)*c^4)*d^3*e + 8*(B*a*b^4*c^2 - 7*B*a^2*b^2*c^3 +
4*(B*a^3 + A*a^2*b)*c^4)*d*e^3 - (5*B*a*b^5*c - 8*A*a^3*c^4 + 2*(26*B*a^3*b + 7*A*a^2*b^2)*c^3 - (35*B*a^2*b^
3 + 2*A*a*b^4)*c^2)*e^4)*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 + (b^4*c^6 - 8*a*
b^2*c^7 + 16*a^2*c^8)*x^4 + 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32*a^3*c^7
)*x^2 + 2*(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*x)]

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giac [B]  time = 0.40, size = 1185, normalized size = 1.95 \begin {gather*} \frac {{\left ({\left ({\left (\frac {3 \, {\left (B b^{4} c^{2} e^{4} - 8 \, B a b^{2} c^{3} e^{4} + 16 \, B a^{2} c^{4} e^{4}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} - \frac {4 \, {\left (4 \, B b c^{5} d^{4} - 8 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e - 16 \, B a c^{5} d^{3} e + 16 \, A b c^{5} d^{3} e - 3 \, B b^{3} c^{3} d^{2} e^{2} + 36 \, B a b c^{4} d^{2} e^{2} - 6 \, A b^{2} c^{4} d^{2} e^{2} - 24 \, A a c^{5} d^{2} e^{2} + 8 \, B b^{4} c^{2} d e^{3} - 56 \, B a b^{2} c^{3} d e^{3} - 2 \, A b^{3} c^{3} d e^{3} + 64 \, B a^{2} c^{4} d e^{3} + 24 \, A a b c^{4} d e^{3} - 5 \, B b^{5} c e^{4} + 37 \, B a b^{3} c^{2} e^{4} + 2 \, A b^{4} c^{2} e^{4} - 64 \, B a^{2} b c^{3} e^{4} - 14 \, A a b^{2} c^{3} e^{4} + 16 \, A a^{2} c^{4} e^{4}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {3 \, {\left (8 \, B b^{2} c^{4} d^{4} - 16 \, A b c^{5} d^{4} - 8 \, B b^{3} c^{3} d^{3} e - 32 \, B a b c^{4} d^{3} e + 32 \, A b^{2} c^{4} d^{3} e + 24 \, B a b^{2} c^{3} d^{2} e^{2} - 12 \, A b^{3} c^{3} d^{2} e^{2} + 96 \, B a^{2} c^{4} d^{2} e^{2} - 48 \, A a b c^{4} d^{2} e^{2} + 8 \, B b^{5} c d e^{3} - 48 \, B a b^{3} c^{2} d e^{3} + 16 \, A a b^{2} c^{3} d e^{3} + 64 \, A a^{2} c^{4} d e^{3} - 5 \, B b^{6} e^{4} + 30 \, B a b^{4} c e^{4} + 2 \, A b^{5} c e^{4} - 16 \, B a^{2} b^{2} c^{2} e^{4} - 12 \, A a b^{3} c^{2} e^{4} - 64 \, B a^{3} c^{3} e^{4}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {6 \, {\left (B b^{3} c^{3} d^{4} + 4 \, B a b c^{4} d^{4} - 2 \, A b^{2} c^{4} d^{4} - 8 \, A a c^{5} d^{4} - 16 \, B a b^{2} c^{3} d^{3} e + 4 \, A b^{3} c^{3} d^{3} e + 16 \, A a b c^{4} d^{3} e + 48 \, B a^{2} b c^{3} d^{2} e^{2} - 24 \, A a b^{2} c^{3} d^{2} e^{2} + 8 \, B a b^{4} c d e^{3} - 56 \, B a^{2} b^{2} c^{2} d e^{3} + 32 \, B a^{3} c^{3} d e^{3} + 32 \, A a^{2} b c^{3} d e^{3} - 5 \, B a b^{5} e^{4} + 35 \, B a^{2} b^{3} c e^{4} + 2 \, A a b^{4} c e^{4} - 52 \, B a^{3} b c^{2} e^{4} - 14 \, A a^{2} b^{2} c^{2} e^{4} + 8 \, A a^{3} c^{3} e^{4}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac {4 \, B a b^{2} c^{3} d^{4} + 2 \, A b^{3} c^{3} d^{4} + 16 \, B a^{2} c^{4} d^{4} - 24 \, A a b c^{4} d^{4} - 64 \, B a^{2} b c^{3} d^{3} e + 16 \, A a b^{2} c^{3} d^{3} e + 64 \, A a^{2} c^{4} d^{3} e + 192 \, B a^{3} c^{3} d^{2} e^{2} - 96 \, A a^{2} b c^{3} d^{2} e^{2} + 24 \, B a^{2} b^{3} c d e^{3} - 160 \, B a^{3} b c^{2} d e^{3} + 128 \, A a^{3} c^{3} d e^{3} - 15 \, B a^{2} b^{4} e^{4} + 100 \, B a^{3} b^{2} c e^{4} + 6 \, A a^{2} b^{3} c e^{4} - 128 \, B a^{4} c^{2} e^{4} - 40 \, A a^{3} b c^{2} e^{4}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} - \frac {{\left (8 \, B c d e^{3} - 5 \, B b e^{4} + 2 \, A c e^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/3*((((3*(B*b^4*c^2*e^4 - 8*B*a*b^2*c^3*e^4 + 16*B*a^2*c^4*e^4)*x/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5) - 4*(4
*B*b*c^5*d^4 - 8*A*c^6*d^4 - 4*B*b^2*c^4*d^3*e - 16*B*a*c^5*d^3*e + 16*A*b*c^5*d^3*e - 3*B*b^3*c^3*d^2*e^2 + 3
6*B*a*b*c^4*d^2*e^2 - 6*A*b^2*c^4*d^2*e^2 - 24*A*a*c^5*d^2*e^2 + 8*B*b^4*c^2*d*e^3 - 56*B*a*b^2*c^3*d*e^3 - 2*
A*b^3*c^3*d*e^3 + 64*B*a^2*c^4*d*e^3 + 24*A*a*b*c^4*d*e^3 - 5*B*b^5*c*e^4 + 37*B*a*b^3*c^2*e^4 + 2*A*b^4*c^2*e
^4 - 64*B*a^2*b*c^3*e^4 - 14*A*a*b^2*c^3*e^4 + 16*A*a^2*c^4*e^4)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 3*(
8*B*b^2*c^4*d^4 - 16*A*b*c^5*d^4 - 8*B*b^3*c^3*d^3*e - 32*B*a*b*c^4*d^3*e + 32*A*b^2*c^4*d^3*e + 24*B*a*b^2*c^
3*d^2*e^2 - 12*A*b^3*c^3*d^2*e^2 + 96*B*a^2*c^4*d^2*e^2 - 48*A*a*b*c^4*d^2*e^2 + 8*B*b^5*c*d*e^3 - 48*B*a*b^3*
c^2*d*e^3 + 16*A*a*b^2*c^3*d*e^3 + 64*A*a^2*c^4*d*e^3 - 5*B*b^6*e^4 + 30*B*a*b^4*c*e^4 + 2*A*b^5*c*e^4 - 16*B*
a^2*b^2*c^2*e^4 - 12*A*a*b^3*c^2*e^4 - 64*B*a^3*c^3*e^4)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 6*(B*b^3*c^
3*d^4 + 4*B*a*b*c^4*d^4 - 2*A*b^2*c^4*d^4 - 8*A*a*c^5*d^4 - 16*B*a*b^2*c^3*d^3*e + 4*A*b^3*c^3*d^3*e + 16*A*a*
b*c^4*d^3*e + 48*B*a^2*b*c^3*d^2*e^2 - 24*A*a*b^2*c^3*d^2*e^2 + 8*B*a*b^4*c*d*e^3 - 56*B*a^2*b^2*c^2*d*e^3 + 3
2*B*a^3*c^3*d*e^3 + 32*A*a^2*b*c^3*d*e^3 - 5*B*a*b^5*e^4 + 35*B*a^2*b^3*c*e^4 + 2*A*a*b^4*c*e^4 - 52*B*a^3*b*c
^2*e^4 - 14*A*a^2*b^2*c^2*e^4 + 8*A*a^3*c^3*e^4)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - (4*B*a*b^2*c^3*d^4
+ 2*A*b^3*c^3*d^4 + 16*B*a^2*c^4*d^4 - 24*A*a*b*c^4*d^4 - 64*B*a^2*b*c^3*d^3*e + 16*A*a*b^2*c^3*d^3*e + 64*A*a
^2*c^4*d^3*e + 192*B*a^3*c^3*d^2*e^2 - 96*A*a^2*b*c^3*d^2*e^2 + 24*B*a^2*b^3*c*d*e^3 - 160*B*a^3*b*c^2*d*e^3 +
128*A*a^3*c^3*d*e^3 - 15*B*a^2*b^4*e^4 + 100*B*a^3*b^2*c*e^4 + 6*A*a^2*b^3*c*e^4 - 128*B*a^4*c^2*e^4 - 40*A*a
^3*b*c^2*e^4)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))/(c*x^2 + b*x + a)^(3/2) - 1/2*(8*B*c*d*e^3 - 5*B*b*e^4 + 2
*A*c*e^4)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)

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maple [B]  time = 0.09, size = 3912, normalized size = 6.43 \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)^4/(c*x^2+b*x+a)^(5/2),x)

[Out]

-1/6*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d*e^3+2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d*e^3+1
6*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d*e^3-4*b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d*e^3-4/3*b^
4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d*e^3-2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d*e^3-6*b/c*a/
(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^2*e^2-1/3/c/(c*x^2+b*x+a)^(3/2)*B*d^4+1/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))*A*e^4+8/3*B*e^4*a^2/c^3/(c*x^2+b*x+a)^(3/2)+5/96*B*e^4*b^4/c^5/(c*x^2+b*x+a)^(3/2)-5/4*B*e
^4*b^2/c^4/(c*x^2+b*x+a)^(1/2)+B*e^4*x^4/c/(c*x^2+b*x+a)^(3/2)-5/2*B*e^4*b/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))-1/c^2*x/(c*x^2+b*x+a)^(1/2)*A*e^4-8/3*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d^4+4*B*e^4*a^2
/c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+8*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d*e^3-48*b*a/(4*a*c-b
^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^2*e^2+b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d^2*e^2+2/3*b^2/c/(4*a*c-b^2)/
(c*x^2+b*x+a)^(3/2)*x*B*d^3*e+2*a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*A*d^2*e^2+4/c^2*b^2/(4*a*c-b^2)/(c*x^2+b
*x+a)^(1/2)*x*B*d*e^3-16*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d*e^3-24*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x
+a)^(1/2)*B*d^2*e^2+1/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*e^4+b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^
(3/2)*B*d*e^3-38/3*B*e^4*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+32*B*e^4*a^2/c*b/(4*a*c-b^2)^2/(c*x^2+b
*x+a)^(1/2)*x-3*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2-32*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x
*A*d*e^3+1/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^2*e^2+8/3*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x
*A*d*e^3+4*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*e^4+4*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^2
*e^2+1/3*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d*e^3-19/12*B*e^4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/
2)*x+64/3*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^3*e-64/3*b*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d^3*e
+1/4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*e^4+4/3*b/c^3*a/(c*x^2+b*x+a)^(3/2)*B*d*e^3+1/2*b^2/c^3*x/(c*
x^2+b*x+a)^(3/2)*B*d*e^3-4/3/c/(c*x^2+b*x+a)^(3/2)*A*d^3*e+4/3*a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*B*d^3*e+3
2*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d^2*e^2+4/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d*
e^3-5/2*B*e^4*b^3/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+5/48*B*e^4*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+5
/6*B*e^4*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-19/24*B*e^4*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+B*e
^4*a/c^3*b*x/(c*x^2+b*x+a)^(3/2)+2*B*e^4*a^2/c^3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-19/3*B*e^4*b^4/c^3*a/(4*a
*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+1/4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2+2/c^3*b^3/(4*a*c-b^2)/(c*x
^2+b*x+a)^(1/2)*B*d*e^3-2/3*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d*e^3+1/2/c^3*b/(c*x^2+b*x+a)^(1/2)*A*
e^4+16*B*e^4*a^2/c^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-3*x/c/(c*x^2+b*x+a)^(3/2)*A*d^2*e^2+2/3*A*d^4/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)*b-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)*A*e^4-1/48*b^3/c^4/(c*x^2+b*x+a)^(3/2)*A*e^4+5/6*B*
e^4*b/c^2*x^3/(c*x^2+b*x+a)^(3/2)+4*B*e^4*a/c^2*x^2/(c*x^2+b*x+a)^(3/2)+1/2*b/c^2/(c*x^2+b*x+a)^(3/2)*A*d^2*e^
2+1/3*b/c^2/(c*x^2+b*x+a)^(3/2)*B*d^3*e-B*e^4*b^2/c^4*a/(c*x^2+b*x+a)^(3/2)+5/2*B*e^4*b/c^3*x/(c*x^2+b*x+a)^(1
/2)-5/4*B*e^4*b^4/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-5/4*B*e^4*b^2/c^3*x^2/(c*x^2+b*x+a)^(3/2)+5/12*B*e^4*b^6
/c^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-5/16*B*e^4*b^3/c^4*x/(c*x^2+b*x+a)^(3/2)+5/96*B*e^4*b^6/c^5/(4*a*c-b^2)
/(c*x^2+b*x+a)^(3/2)-1/24*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*e^4+16/3*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)
^(1/2)*x*B*d^3*e+4*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d^2*e^2+8/3*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/
2)*B*d^3*e+4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d^2*e^2+8/3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^3*e+16*
a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*A*d^2*e^2+32/3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*B*d^3*e-16/3*b*c/(4
*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^4+1/3*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^3*e+8*b^2/(4*a*c-b^2)^
2/(c*x^2+b*x+a)^(1/2)*x*A*d^2*e^2+2*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d^2*e^2-3/2*b/c^2*x/(c*x^2+b*x
+a)^(3/2)*B*d^2*e^2+1/6*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d*e^3+2*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a
)^(1/2)*A*e^4-b/c^2*x/(c*x^2+b*x+a)^(3/2)*A*d*e^3-8/3*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d^3*e-4/3*b^2/c/(4
*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d^3*e+2*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)*B*d*e^3+1/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b
*x+a)^(3/2)*A*d^2*e^2+4/3*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d*e^3+1/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a
)^(1/2)*x*A*e^4-1/3*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*e^4-1/12*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(
3/2)*B*d*e^3-1/48*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*e^4-1/6*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*
A*e^4+1/3*b/c^3*a/(c*x^2+b*x+a)^(3/2)*A*e^4-4/c^2*x/(c*x^2+b*x+a)^(1/2)*B*d*e^3+2/c^3*b/(c*x^2+b*x+a)^(1/2)*B*
d*e^3-4*x^2/c/(c*x^2+b*x+a)^(3/2)*A*d*e^3-6*x^2/c/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2+1/6*b^2/c^3/(c*x^2+b*x+a)^(3/2
)*A*d*e^3-4/3*x^3/c/(c*x^2+b*x+a)^(3/2)*B*d*e^3+1/4*b^2/c^3/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2-8/3*a/c^2/(c*x^2+b*x
+a)^(3/2)*A*d*e^3+4/3*A*d^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+32/3*A*d^4*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/
2)*x+16/3*A*d^4*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-2/3*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^4-1/3*b^2/c/
(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^4-32/3*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d^3*e-4*a/c^2/(c*x^2+b*x+a)
^(3/2)*B*d^2*e^2+1/2*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)*A*e^4+1/8*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)*A*e^4-1/12*b^3/c^4/
(c*x^2+b*x+a)^(3/2)*B*d*e^3+1/2/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*A*e^4-2*x/c/(c*x^2+b*x+a)^(3/2)*B*d^3*
e

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^4}{{\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)^4)/(a + b*x + c*x^2)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^4)/(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)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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