3.2487 \(\int \frac{(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=942 \[ \frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^5}{c^{7/2}}+\frac{2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{c x^2+b x+a}}+\frac{2 (d+e x)^2 \left (B e \left (3 c d^2-5 a e^2\right ) b^3-4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2+4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \]

[Out]

(2*(d + e*x)^4*(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))/(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(
d + e*x)^2*(b^3*B*e*(3*c*d^2 - 5*a*e^2) - 4*b^2*c*d*(2*B*c*d^2 + 4*A*c*d*e + a*B
*e^2) - 16*a*c^2*e*(5*a*B*d*e + 2*A*(c*d^2 + a*e^2)) + 4*b*c*(9*a*B*e*(c*d^2 + a
*e^2) + 4*A*c*d*(c*d^2 + 3*a*e^2)) + (2*b^3*B*c*d*e^2 - 5*b^4*B*e^3 + 2*b^2*c*e*
(7*B*c*d^2 + 8*A*c*d*e + 19*a*B*e^2) - 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 7*a*B*
d*e^2 + 2*a*A*e^3) + 8*c^2*(5*a*B*e*(c*d^2 - a*e^2) + 4*A*c*d*(c*d^2 + a*e^2)))*
x))/(15*c^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (2*(4*b^4*B*c^2*d^3*e^2 +
5*b^5*B*e^3*(c*d^2 - 3*a*e^2) + 32*b^2*c^3*d^2*(2*B*c*d^3 + 8*A*c*d^2*e + 17*a*
B*d*e^2 + 16*a*A*e^3) + 64*a*c^3*e*(4*A*(c*d^2 + a*e^2)^2 + 5*a*B*d*e*(c*d^2 + 4
*a*e^2)) - 8*b^3*c*e*(16*A*c^2*d^3*e + B*(11*c^2*d^4 + 7*a*c*d^2*e^2 - 20*a^2*e^
4)) - 16*b*c^2*(8*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 + 5*a^2*e^4) + a*B*e*(18*c^2*d^
4 + 71*a*c*d^2*e^2 + 33*a^2*e^4)) + (10*b^5*B*c*d*e^4 - 15*b^6*B*e^5 + 2*b^4*B*c
*e^3*(3*c*d^2 + 85*a*e^2) + 16*b^3*c^2*d*e^2*(6*B*c*d^2 + 8*A*c*d*e - 7*a*B*e^2)
- 32*c^3*(8*A*c*d*(c*d^2 + a*e^2)^2 + 5*a*B*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 - 3*a^
2*e^4)) - 16*b^2*c^2*e*(16*A*c*d*e*(2*c*d^2 + a*e^2) + B*(15*c^2*d^4 + 29*a*c*d^
2*e^2 + 39*a^2*e^4)) + 32*b*c^3*(4*A*e*(5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4) + B
*(4*c^2*d^5 + 28*a*c*d^3*e^2 + 29*a^2*d*e^4)))*x))/(15*c^3*(b^2 - 4*a*c)^3*Sqrt[
a + b*x + c*x^2]) + (B*e^5*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])
])/c^(7/2)

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Rubi [A]  time = 2.90199, antiderivative size = 942, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^5}{c^{7/2}}+\frac{2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{c x^2+b x+a}}+\frac{2 (d+e x)^2 \left (B e \left (3 c d^2-5 a e^2\right ) b^3-4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2+4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^4*(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))/(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(
d + e*x)^2*(b^3*B*e*(3*c*d^2 - 5*a*e^2) - 4*b^2*c*d*(2*B*c*d^2 + 4*A*c*d*e + a*B
*e^2) - 16*a*c^2*e*(5*a*B*d*e + 2*A*(c*d^2 + a*e^2)) + 4*b*c*(9*a*B*e*(c*d^2 + a
*e^2) + 4*A*c*d*(c*d^2 + 3*a*e^2)) + (2*b^3*B*c*d*e^2 - 5*b^4*B*e^3 + 2*b^2*c*e*
(7*B*c*d^2 + 8*A*c*d*e + 19*a*B*e^2) - 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 7*a*B*
d*e^2 + 2*a*A*e^3) + 8*c^2*(5*a*B*e*(c*d^2 - a*e^2) + 4*A*c*d*(c*d^2 + a*e^2)))*
x))/(15*c^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (2*(4*b^4*B*c^2*d^3*e^2 +
5*b^5*B*e^3*(c*d^2 - 3*a*e^2) + 32*b^2*c^3*d^2*(2*B*c*d^3 + 8*A*c*d^2*e + 17*a*
B*d*e^2 + 16*a*A*e^3) + 64*a*c^3*e*(4*A*(c*d^2 + a*e^2)^2 + 5*a*B*d*e*(c*d^2 + 4
*a*e^2)) - 8*b^3*c*e*(16*A*c^2*d^3*e + B*(11*c^2*d^4 + 7*a*c*d^2*e^2 - 20*a^2*e^
4)) - 16*b*c^2*(8*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 + 5*a^2*e^4) + a*B*e*(18*c^2*d^
4 + 71*a*c*d^2*e^2 + 33*a^2*e^4)) + (10*b^5*B*c*d*e^4 - 15*b^6*B*e^5 + 2*b^4*B*c
*e^3*(3*c*d^2 + 85*a*e^2) + 16*b^3*c^2*d*e^2*(6*B*c*d^2 + 8*A*c*d*e - 7*a*B*e^2)
- 32*c^3*(8*A*c*d*(c*d^2 + a*e^2)^2 + 5*a*B*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 - 3*a^
2*e^4)) - 16*b^2*c^2*e*(16*A*c*d*e*(2*c*d^2 + a*e^2) + B*(15*c^2*d^4 + 29*a*c*d^
2*e^2 + 39*a^2*e^4)) + 32*b*c^3*(4*A*e*(5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4) + B
*(4*c^2*d^5 + 28*a*c*d^3*e^2 + 29*a^2*d*e^4)))*x))/(15*c^3*(b^2 - 4*a*c)^3*Sqrt[
a + b*x + c*x^2]) + (B*e^5*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])
])/c^(7/2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**5/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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Mathematica [B]  time = 7.19055, size = 2431, normalized size = 2.58 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]

[Out]

((a + b*x + c*x^2)^4*((2*(A*b*c^5*d^5 - 2*a*B*c^5*d^5 + 5*a*b*B*c^4*d^4*e - 10*a
*A*c^5*d^4*e - 10*a*b^2*B*c^3*d^3*e^2 + 10*a*A*b*c^4*d^3*e^2 + 20*a^2*B*c^4*d^3*
e^2 + 10*a*b^3*B*c^2*d^2*e^3 - 10*a*A*b^2*c^3*d^2*e^3 - 30*a^2*b*B*c^3*d^2*e^3 +
20*a^2*A*c^4*d^2*e^3 - 5*a*b^4*B*c*d*e^4 + 5*a*A*b^3*c^2*d*e^4 + 20*a^2*b^2*B*c
^2*d*e^4 - 15*a^2*A*b*c^3*d*e^4 - 10*a^3*B*c^3*d*e^4 + a*b^5*B*e^5 - a*A*b^4*c*e
^5 - 5*a^2*b^3*B*c*e^5 + 4*a^2*A*b^2*c^2*e^5 + 5*a^3*b*B*c^2*e^5 - 2*a^3*A*c^3*e
^5 - b*B*c^5*d^5*x + 2*A*c^6*d^5*x + 5*b^2*B*c^4*d^4*e*x - 5*A*b*c^5*d^4*e*x - 1
0*a*B*c^5*d^4*e*x - 10*b^3*B*c^3*d^3*e^2*x + 10*A*b^2*c^4*d^3*e^2*x + 30*a*b*B*c
^4*d^3*e^2*x - 20*a*A*c^5*d^3*e^2*x + 10*b^4*B*c^2*d^2*e^3*x - 10*A*b^3*c^3*d^2*
e^3*x - 40*a*b^2*B*c^3*d^2*e^3*x + 30*a*A*b*c^4*d^2*e^3*x + 20*a^2*B*c^4*d^2*e^3
*x - 5*b^5*B*c*d*e^4*x + 5*A*b^4*c^2*d*e^4*x + 25*a*b^3*B*c^2*d*e^4*x - 20*a*A*b
^2*c^3*d*e^4*x - 25*a^2*b*B*c^3*d*e^4*x + 10*a^2*A*c^4*d*e^4*x + b^6*B*e^5*x - A
*b^5*c*e^5*x - 6*a*b^4*B*c*e^5*x + 5*a*A*b^3*c^2*e^5*x + 9*a^2*b^2*B*c^2*e^5*x -
5*a^2*A*b*c^3*e^5*x - 2*a^3*B*c^3*e^5*x))/(5*c^5*(-b^2 + 4*a*c)*(a + b*x + c*x^
2)^3) + (2*(-8*b^2*B*c^5*d^5 + 16*A*b*c^6*d^5 + 15*b^3*B*c^4*d^4*e - 40*A*b^2*c^
5*d^4*e + 20*a*b*B*c^5*d^4*e - 30*b^4*B*c^3*d^3*e^2 + 30*A*b^3*c^4*d^3*e^2 + 140
*a*b^2*B*c^4*d^3*e^2 + 40*a*A*b*c^5*d^3*e^2 - 400*a^2*B*c^5*d^3*e^2 + 30*b^5*B*c
^2*d^2*e^3 - 30*A*b^4*c^3*d^2*e^3 - 220*a*b^3*B*c^3*d^2*e^3 + 140*a*A*b^2*c^4*d^
2*e^3 + 560*a^2*b*B*c^4*d^2*e^3 - 400*a^2*A*c^5*d^2*e^3 - 15*b^6*B*c*d*e^4 + 15*
A*b^5*c^2*d*e^4 + 150*a*b^4*B*c^2*d*e^4 - 110*a*A*b^3*c^3*d*e^4 - 500*a^2*b^2*B*
c^3*d*e^4 + 280*a^2*A*b*c^4*d*e^4 + 400*a^3*B*c^4*d*e^4 + 3*b^7*B*e^5 - 3*A*b^6*
c*e^5 - 38*a*b^5*B*c*e^5 + 30*a*A*b^4*c^2*e^5 + 157*a^2*b^3*B*c^2*e^5 - 100*a^2*
A*b^2*c^3*e^5 - 196*a^3*b*B*c^3*e^5 + 80*a^3*A*c^4*e^5 - 16*b*B*c^6*d^5*x + 32*A
*c^7*d^5*x + 30*b^2*B*c^5*d^4*e*x - 80*A*b*c^6*d^4*e*x + 40*a*B*c^6*d^4*e*x - 10
*b^3*B*c^4*d^3*e^2*x + 60*A*b^2*c^5*d^3*e^2*x - 120*a*b*B*c^5*d^3*e^2*x + 80*a*A
*c^6*d^3*e^2*x - 40*b^4*B*c^3*d^2*e^3*x - 10*A*b^3*c^4*d^2*e^3*x + 360*a*b^2*B*c
^4*d^2*e^3*x - 120*a*A*b*c^5*d^2*e^3*x - 480*a^2*B*c^5*d^2*e^3*x + 45*b^5*B*c^2*
d*e^4*x - 20*A*b^4*c^3*d*e^4*x - 350*a*b^3*B*c^3*d*e^4*x + 180*a*A*b^2*c^4*d*e^4
*x + 600*a^2*b*B*c^4*d*e^4*x - 240*a^2*A*c^5*d*e^4*x - 14*b^6*B*c*e^5*x + 9*A*b^
5*c^2*e^5*x + 114*a*b^4*B*c^2*e^5*x - 70*a*A*b^3*c^3*e^5*x - 246*a^2*b^2*B*c^3*e
^5*x + 120*a^2*A*b*c^4*e^5*x + 88*a^3*B*c^4*e^5*x))/(15*c^5*(-b^2 + 4*a*c)^2*(a
+ b*x + c*x^2)^2) + (2*(-64*b^2*B*c^5*d^5 + 128*A*b*c^6*d^5 + 120*b^3*B*c^4*d^4*
e - 320*A*b^2*c^5*d^4*e + 160*a*b*B*c^5*d^4*e - 40*b^4*B*c^3*d^3*e^2 + 240*A*b^3
*c^4*d^3*e^2 - 480*a*b^2*B*c^4*d^3*e^2 + 320*a*A*b*c^5*d^3*e^2 - 10*b^5*B*c^2*d^
2*e^3 - 40*A*b^4*c^3*d^2*e^3 + 240*a*b^3*B*c^3*d^2*e^3 - 480*a*A*b^2*c^4*d^2*e^3
+ 480*a^2*b*B*c^4*d^2*e^3 + 30*b^6*B*c*d*e^4 - 5*A*b^5*c^2*d*e^4 - 350*a*b^4*B*
c^2*d*e^4 + 120*a*A*b^3*c^3*d*e^4 + 1200*a^2*b^2*B*c^3*d*e^4 + 240*a^2*A*b*c^4*d
*e^4 - 2400*a^3*B*c^4*d*e^4 - 11*b^7*B*e^5 + 6*A*b^6*c*e^5 + 141*a*b^5*B*c*e^5 -
70*a*A*b^4*c^2*e^5 - 624*a^2*b^3*B*c^2*e^5 + 240*a^2*A*b^2*c^3*e^5 + 1072*a^3*b
*B*c^3*e^5 - 480*a^3*A*c^4*e^5 - 128*b*B*c^6*d^5*x + 256*A*c^7*d^5*x + 240*b^2*B
*c^5*d^4*e*x - 640*A*b*c^6*d^4*e*x + 320*a*B*c^6*d^4*e*x - 80*b^3*B*c^4*d^3*e^2*
x + 480*A*b^2*c^5*d^3*e^2*x - 960*a*b*B*c^5*d^3*e^2*x + 640*a*A*c^6*d^3*e^2*x -
20*b^4*B*c^3*d^2*e^3*x - 80*A*b^3*c^4*d^2*e^3*x + 480*a*b^2*B*c^4*d^2*e^3*x - 96
0*a*A*b*c^5*d^2*e^3*x + 960*a^2*B*c^5*d^2*e^3*x - 15*b^5*B*c^2*d*e^4*x - 10*A*b^
4*c^3*d*e^4*x + 200*a*b^3*B*c^3*d*e^4*x + 240*a*A*b^2*c^4*d*e^4*x - 1200*a^2*b*B
*c^4*d*e^4*x + 480*a^2*A*c^5*d*e^4*x + 23*b^6*B*c*e^5*x - 3*A*b^5*c^2*e^5*x - 25
8*a*b^4*B*c^2*e^5*x + 40*a*A*b^3*c^3*e^5*x + 912*a^2*b^2*B*c^3*e^5*x - 240*a^2*A
*b*c^4*e^5*x - 736*a^3*B*c^4*e^5*x))/(15*c^4*(-b^2 + 4*a*c)^3*(a + b*x + c*x^2))
))/(a + x*(b + c*x))^(7/2) + (B*e^5*(a + b*x + c*x^2)^(7/2)*Log[b + 2*c*x + 2*Sq
rt[c]*Sqrt[a + b*x + c*x^2]])/(c^(7/2)*(a + x*(b + c*x))^(7/2))

_______________________________________________________________________________________

Maple [B]  time = 0.031, size = 7765, normalized size = 8.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^5/(c*x^2 + b*x + a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 3.65949, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^5/(c*x^2 + b*x + a)^(7/2),x, algorithm="fricas")

[Out]

[1/30*(4*(640*(2*B*a^5 - A*a^4*b)*c^3*d*e^4 + (48*(2*B*a^3 - 5*A*a^2*b)*c^5 + 8*
(6*B*a^2*b^2 + 5*A*a*b^3)*c^4 - (2*B*a*b^4 + 3*A*b^5)*c^3)*d^5 + 10*(48*A*a^3*c^
5 - 24*(2*B*a^3*b - A*a^2*b^2)*c^4 - (4*B*a^2*b^3 + A*a*b^4)*c^3)*d^4*e + 80*(4*
(2*B*a^4 - 3*A*a^3*b)*c^4 + (6*B*a^3*b^2 - A*a^2*b^3)*c^3)*d^3*e^2 + 160*(4*A*a^
4*c^4 - (8*B*a^4*b - 3*A*a^3*b^2)*c^3)*d^2*e^3 - (15*B*a^3*b^5 - 160*B*a^4*b^3*c
+ 528*B*a^5*b*c^2 - 256*A*a^5*c^3)*e^5 + (128*(B*b*c^7 - 2*A*c^8)*d^5 - 80*(3*B
*b^2*c^6 + 4*(B*a - 2*A*b)*c^7)*d^4*e + 80*(B*b^3*c^5 - 8*A*a*c^7 + 6*(2*B*a*b -
A*b^2)*c^6)*d^3*e^2 + 20*(B*b^4*c^4 - 48*(B*a^2 - A*a*b)*c^6 - 4*(6*B*a*b^2 - A
*b^3)*c^5)*d^2*e^3 + 5*(3*B*b^5*c^3 - 96*A*a^2*c^6 + 48*(5*B*a^2*b - A*a*b^2)*c^
5 - 2*(20*B*a*b^3 - A*b^4)*c^4)*d*e^4 - (23*B*b^6*c^2 - 16*(46*B*a^3 + 15*A*a^2*
b)*c^5 + 8*(114*B*a^2*b^2 + 5*A*a*b^3)*c^4 - 3*(86*B*a*b^4 + A*b^5)*c^3)*e^5)*x^
5 + 5*(64*(B*b^2*c^6 - 2*A*b*c^7)*d^5 - 40*(3*B*b^3*c^5 + 4*(B*a*b - 2*A*b^2)*c^
6)*d^4*e + 40*(B*b^4*c^4 - 8*A*a*b*c^6 + 6*(2*B*a*b^2 - A*b^3)*c^5)*d^3*e^2 + 10
*(B*b^5*c^3 - 48*(B*a^2*b - A*a*b^2)*c^5 - 4*(6*B*a*b^3 - A*b^4)*c^4)*d^2*e^3 +
5*(48*(2*B*a^3 - A*a^2*b)*c^5 + 24*(2*B*a^2*b^2 - A*a*b^3)*c^4 - (2*B*a*b^4 - A*
b^5)*c^3)*d*e^4 - (7*B*b^7*c - 75*B*a*b^5*c^2 - 96*A*a^3*c^5 - 16*(5*B*a^3*b + 3
*A*a^2*b^2)*c^4 + 2*(120*B*a^2*b^3 + A*a*b^4)*c^3)*e^5)*x^4 + 5*(16*(3*B*b^3*c^5
- 8*A*a*c^7 + 2*(2*B*a*b - 3*A*b^2)*c^6)*d^5 - 10*(9*B*b^4*c^4 + 16*(B*a^2 - 2*
A*a*b)*c^6 + 24*(B*a*b^2 - A*b^3)*c^5)*d^4*e + 10*(3*B*b^5*c^3 - 32*A*a^2*c^6 +
48*(B*a^2*b - A*a*b^2)*c^5 + 2*(20*B*a*b^3 - 9*A*b^4)*c^4)*d^3*e^2 + 10*(48*A*a^
2*b*c^5 - 8*(12*B*a^2*b^2 - 5*A*a*b^3)*c^4 - (8*B*a*b^4 - 3*A*b^5)*c^3)*d^2*e^3
+ 40*(12*(2*B*a^3*b - A*a^2*b^2)*c^4 + (2*B*a^2*b^3 - A*a*b^4)*c^3)*d*e^4 - (3*B
*b^8 - 20*B*a*b^6*c - 30*B*a^2*b^4*c^2 - 32*(7*B*a^4 + 6*A*a^3*b)*c^4 + 16*(27*B
*a^3*b^2 - A*a^2*b^3)*c^3)*e^5)*x^3 + 5*(8*(B*b^4*c^4 - 24*A*a*b*c^6 + 2*(6*B*a*
b^2 - A*b^3)*c^5)*d^5 - 5*(3*B*b^5*c^3 + 48*(B*a^2*b - 2*A*a*b^2)*c^5 + 8*(5*B*a
*b^3 - A*b^4)*c^4)*d^4*e + 10*(16*(2*B*a^3 - 3*A*a^2*b)*c^5 + 8*(6*B*a^2*b^2 - 5
*A*a*b^3)*c^4 + 3*(6*B*a*b^4 - A*b^5)*c^3)*d^3*e^2 + 20*(16*A*a^3*c^5 - 8*(4*B*a
^3*b - 3*A*a^2*b^2)*c^4 - 3*(8*B*a^2*b^3 - 3*A*a*b^4)*c^3)*d^2*e^3 + 80*(4*(2*B*
a^4 - A*a^3*b)*c^4 + 3*(2*B*a^3*b^2 - A*a^2*b^3)*c^3)*d*e^4 - (9*B*a*b^7 - 93*B*
a^2*b^5*c + 280*B*a^3*b^3*c^2 - 128*A*a^4*c^4 + 48*(B*a^4*b - 2*A*a^3*b^2)*c^3)*
e^5)*x^2 + 5*(320*(2*B*a^4*b - A*a^3*b^2)*c^3*d*e^4 - (B*b^5*c^3 + 96*A*a^2*c^6
- 48*(B*a^2*b - A*a*b^2)*c^5 - 2*(12*B*a*b^3 + A*b^4)*c^4)*d^5 + 5*(48*A*a^2*b*c
^5 - 24*(2*B*a^2*b^2 - A*a*b^3)*c^4 - (4*B*a*b^4 + A*b^5)*c^3)*d^4*e + 40*(4*(2*
B*a^3*b - 3*A*a^2*b^2)*c^4 + (6*B*a^2*b^3 - A*a*b^4)*c^3)*d^3*e^2 + 80*(4*A*a^3*
b*c^4 - (8*B*a^3*b^2 - 3*A*a^2*b^3)*c^3)*d^2*e^3 - (9*B*a^2*b^6 - 98*B*a^3*b^4*c
+ 336*B*a^4*b^2*c^2 - 32*(3*B*a^5 + 4*A*a^4*b)*c^3)*e^5)*x)*sqrt(c*x^2 + b*x +
a)*sqrt(c) + 15*((B*b^6*c^3 - 12*B*a*b^4*c^4 + 48*B*a^2*b^2*c^5 - 64*B*a^3*c^6)*
e^5*x^6 + 3*(B*b^7*c^2 - 12*B*a*b^5*c^3 + 48*B*a^2*b^3*c^4 - 64*B*a^3*b*c^5)*e^5
*x^5 + 3*(B*b^8*c - 11*B*a*b^6*c^2 + 36*B*a^2*b^4*c^3 - 16*B*a^3*b^2*c^4 - 64*B*
a^4*c^5)*e^5*x^4 + (B*b^9 - 6*B*a*b^7*c - 24*B*a^2*b^5*c^2 + 224*B*a^3*b^3*c^3 -
384*B*a^4*b*c^4)*e^5*x^3 + 3*(B*a*b^8 - 11*B*a^2*b^6*c + 36*B*a^3*b^4*c^2 - 16*
B*a^4*b^2*c^3 - 64*B*a^5*c^4)*e^5*x^2 + 3*(B*a^2*b^7 - 12*B*a^3*b^5*c + 48*B*a^4
*b^3*c^2 - 64*B*a^5*b*c^3)*e^5*x + (B*a^3*b^6 - 12*B*a^4*b^4*c + 48*B*a^5*b^2*c^
2 - 64*B*a^6*c^3)*e^5)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2
+ 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/((a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*
c^5 - 64*a^6*c^6 + (b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*x^6 +
3*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x^5 + 3*(b^8*c^4 - 11
*a*b^6*c^5 + 36*a^2*b^4*c^6 - 16*a^3*b^2*c^7 - 64*a^4*c^8)*x^4 + (b^9*c^3 - 6*a*
b^7*c^4 - 24*a^2*b^5*c^5 + 224*a^3*b^3*c^6 - 384*a^4*b*c^7)*x^3 + 3*(a*b^8*c^3 -
11*a^2*b^6*c^4 + 36*a^3*b^4*c^5 - 16*a^4*b^2*c^6 - 64*a^5*c^7)*x^2 + 3*(a^2*b^7
*c^3 - 12*a^3*b^5*c^4 + 48*a^4*b^3*c^5 - 64*a^5*b*c^6)*x)*sqrt(c)), 1/15*(2*(640
*(2*B*a^5 - A*a^4*b)*c^3*d*e^4 + (48*(2*B*a^3 - 5*A*a^2*b)*c^5 + 8*(6*B*a^2*b^2
+ 5*A*a*b^3)*c^4 - (2*B*a*b^4 + 3*A*b^5)*c^3)*d^5 + 10*(48*A*a^3*c^5 - 24*(2*B*a
^3*b - A*a^2*b^2)*c^4 - (4*B*a^2*b^3 + A*a*b^4)*c^3)*d^4*e + 80*(4*(2*B*a^4 - 3*
A*a^3*b)*c^4 + (6*B*a^3*b^2 - A*a^2*b^3)*c^3)*d^3*e^2 + 160*(4*A*a^4*c^4 - (8*B*
a^4*b - 3*A*a^3*b^2)*c^3)*d^2*e^3 - (15*B*a^3*b^5 - 160*B*a^4*b^3*c + 528*B*a^5*
b*c^2 - 256*A*a^5*c^3)*e^5 + (128*(B*b*c^7 - 2*A*c^8)*d^5 - 80*(3*B*b^2*c^6 + 4*
(B*a - 2*A*b)*c^7)*d^4*e + 80*(B*b^3*c^5 - 8*A*a*c^7 + 6*(2*B*a*b - A*b^2)*c^6)*
d^3*e^2 + 20*(B*b^4*c^4 - 48*(B*a^2 - A*a*b)*c^6 - 4*(6*B*a*b^2 - A*b^3)*c^5)*d^
2*e^3 + 5*(3*B*b^5*c^3 - 96*A*a^2*c^6 + 48*(5*B*a^2*b - A*a*b^2)*c^5 - 2*(20*B*a
*b^3 - A*b^4)*c^4)*d*e^4 - (23*B*b^6*c^2 - 16*(46*B*a^3 + 15*A*a^2*b)*c^5 + 8*(1
14*B*a^2*b^2 + 5*A*a*b^3)*c^4 - 3*(86*B*a*b^4 + A*b^5)*c^3)*e^5)*x^5 + 5*(64*(B*
b^2*c^6 - 2*A*b*c^7)*d^5 - 40*(3*B*b^3*c^5 + 4*(B*a*b - 2*A*b^2)*c^6)*d^4*e + 40
*(B*b^4*c^4 - 8*A*a*b*c^6 + 6*(2*B*a*b^2 - A*b^3)*c^5)*d^3*e^2 + 10*(B*b^5*c^3 -
48*(B*a^2*b - A*a*b^2)*c^5 - 4*(6*B*a*b^3 - A*b^4)*c^4)*d^2*e^3 + 5*(48*(2*B*a^
3 - A*a^2*b)*c^5 + 24*(2*B*a^2*b^2 - A*a*b^3)*c^4 - (2*B*a*b^4 - A*b^5)*c^3)*d*e
^4 - (7*B*b^7*c - 75*B*a*b^5*c^2 - 96*A*a^3*c^5 - 16*(5*B*a^3*b + 3*A*a^2*b^2)*c
^4 + 2*(120*B*a^2*b^3 + A*a*b^4)*c^3)*e^5)*x^4 + 5*(16*(3*B*b^3*c^5 - 8*A*a*c^7
+ 2*(2*B*a*b - 3*A*b^2)*c^6)*d^5 - 10*(9*B*b^4*c^4 + 16*(B*a^2 - 2*A*a*b)*c^6 +
24*(B*a*b^2 - A*b^3)*c^5)*d^4*e + 10*(3*B*b^5*c^3 - 32*A*a^2*c^6 + 48*(B*a^2*b -
A*a*b^2)*c^5 + 2*(20*B*a*b^3 - 9*A*b^4)*c^4)*d^3*e^2 + 10*(48*A*a^2*b*c^5 - 8*(
12*B*a^2*b^2 - 5*A*a*b^3)*c^4 - (8*B*a*b^4 - 3*A*b^5)*c^3)*d^2*e^3 + 40*(12*(2*B
*a^3*b - A*a^2*b^2)*c^4 + (2*B*a^2*b^3 - A*a*b^4)*c^3)*d*e^4 - (3*B*b^8 - 20*B*a
*b^6*c - 30*B*a^2*b^4*c^2 - 32*(7*B*a^4 + 6*A*a^3*b)*c^4 + 16*(27*B*a^3*b^2 - A*
a^2*b^3)*c^3)*e^5)*x^3 + 5*(8*(B*b^4*c^4 - 24*A*a*b*c^6 + 2*(6*B*a*b^2 - A*b^3)*
c^5)*d^5 - 5*(3*B*b^5*c^3 + 48*(B*a^2*b - 2*A*a*b^2)*c^5 + 8*(5*B*a*b^3 - A*b^4)
*c^4)*d^4*e + 10*(16*(2*B*a^3 - 3*A*a^2*b)*c^5 + 8*(6*B*a^2*b^2 - 5*A*a*b^3)*c^4
+ 3*(6*B*a*b^4 - A*b^5)*c^3)*d^3*e^2 + 20*(16*A*a^3*c^5 - 8*(4*B*a^3*b - 3*A*a^
2*b^2)*c^4 - 3*(8*B*a^2*b^3 - 3*A*a*b^4)*c^3)*d^2*e^3 + 80*(4*(2*B*a^4 - A*a^3*b
)*c^4 + 3*(2*B*a^3*b^2 - A*a^2*b^3)*c^3)*d*e^4 - (9*B*a*b^7 - 93*B*a^2*b^5*c + 2
80*B*a^3*b^3*c^2 - 128*A*a^4*c^4 + 48*(B*a^4*b - 2*A*a^3*b^2)*c^3)*e^5)*x^2 + 5*
(320*(2*B*a^4*b - A*a^3*b^2)*c^3*d*e^4 - (B*b^5*c^3 + 96*A*a^2*c^6 - 48*(B*a^2*b
- A*a*b^2)*c^5 - 2*(12*B*a*b^3 + A*b^4)*c^4)*d^5 + 5*(48*A*a^2*b*c^5 - 24*(2*B*
a^2*b^2 - A*a*b^3)*c^4 - (4*B*a*b^4 + A*b^5)*c^3)*d^4*e + 40*(4*(2*B*a^3*b - 3*A
*a^2*b^2)*c^4 + (6*B*a^2*b^3 - A*a*b^4)*c^3)*d^3*e^2 + 80*(4*A*a^3*b*c^4 - (8*B*
a^3*b^2 - 3*A*a^2*b^3)*c^3)*d^2*e^3 - (9*B*a^2*b^6 - 98*B*a^3*b^4*c + 336*B*a^4*
b^2*c^2 - 32*(3*B*a^5 + 4*A*a^4*b)*c^3)*e^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) +
15*((B*b^6*c^3 - 12*B*a*b^4*c^4 + 48*B*a^2*b^2*c^5 - 64*B*a^3*c^6)*e^5*x^6 + 3*
(B*b^7*c^2 - 12*B*a*b^5*c^3 + 48*B*a^2*b^3*c^4 - 64*B*a^3*b*c^5)*e^5*x^5 + 3*(B*
b^8*c - 11*B*a*b^6*c^2 + 36*B*a^2*b^4*c^3 - 16*B*a^3*b^2*c^4 - 64*B*a^4*c^5)*e^5
*x^4 + (B*b^9 - 6*B*a*b^7*c - 24*B*a^2*b^5*c^2 + 224*B*a^3*b^3*c^3 - 384*B*a^4*b
*c^4)*e^5*x^3 + 3*(B*a*b^8 - 11*B*a^2*b^6*c + 36*B*a^3*b^4*c^2 - 16*B*a^4*b^2*c^
3 - 64*B*a^5*c^4)*e^5*x^2 + 3*(B*a^2*b^7 - 12*B*a^3*b^5*c + 48*B*a^4*b^3*c^2 - 6
4*B*a^5*b*c^3)*e^5*x + (B*a^3*b^6 - 12*B*a^4*b^4*c + 48*B*a^5*b^2*c^2 - 64*B*a^6
*c^3)*e^5)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a^3*b^6
*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 - 64*a^6*c^6 + (b^6*c^6 - 12*a*b^4*c^7 +
48*a^2*b^2*c^8 - 64*a^3*c^9)*x^6 + 3*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 -
64*a^3*b*c^8)*x^5 + 3*(b^8*c^4 - 11*a*b^6*c^5 + 36*a^2*b^4*c^6 - 16*a^3*b^2*c^7
- 64*a^4*c^8)*x^4 + (b^9*c^3 - 6*a*b^7*c^4 - 24*a^2*b^5*c^5 + 224*a^3*b^3*c^6 -
384*a^4*b*c^7)*x^3 + 3*(a*b^8*c^3 - 11*a^2*b^6*c^4 + 36*a^3*b^4*c^5 - 16*a^4*b^2
*c^6 - 64*a^5*c^7)*x^2 + 3*(a^2*b^7*c^3 - 12*a^3*b^5*c^4 + 48*a^4*b^3*c^5 - 64*a
^5*b*c^6)*x)*sqrt(-c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**5/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293106, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^5/(c*x^2 + b*x + a)^(7/2),x, algorithm="giac")

[Out]

Done