3.2488 \(\int \frac{(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{128 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^4 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{16 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^4)/(5*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(5/2)) - (16*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(d + e*x)^2*(b*d - 2*a*e +
(2*c*d - b*e)*x))/(15*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (128*(b*B*d - 2
*A*c*d + A*b*e - 2*a*B*e)*(c*d^2 - b*d*e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x
))/(15*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.416062, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{128 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^4 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{16 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^4)/(5*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(5/2)) - (16*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(d + e*x)^2*(b*d - 2*a*e +
(2*c*d - b*e)*x))/(15*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (128*(b*B*d - 2
*A*c*d + A*b*e - 2*a*B*e)*(c*d^2 - b*d*e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x
))/(15*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 49.8752, size = 211, normalized size = 1. \[ - \frac{2 \left (d + e x\right )^{4} \left (A b - 2 B a + x \left (2 A c - B b\right )\right )}{5 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}} + \frac{16 \left (d + e x\right )^{2} \left (2 a e - b d + x \left (b e - 2 c d\right )\right ) \left (- 2 A c d - 2 B a e + b \left (A e + B d\right )\right )}{15 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{64 \left (4 a e - 2 b d + x \left (2 b e - 4 c d\right )\right ) \left (a e^{2} - b d e + c d^{2}\right ) \left (- 2 A c d - 2 B a e + b \left (A e + B d\right )\right )}{15 \left (- 4 a c + b^{2}\right )^{3} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**4/(c*x**2+b*x+a)**(7/2),x)

[Out]

-2*(d + e*x)**4*(A*b - 2*B*a + x*(2*A*c - B*b))/(5*(-4*a*c + b**2)*(a + b*x + c*
x**2)**(5/2)) + 16*(d + e*x)**2*(2*a*e - b*d + x*(b*e - 2*c*d))*(-2*A*c*d - 2*B*
a*e + b*(A*e + B*d))/(15*(-4*a*c + b**2)**2*(a + b*x + c*x**2)**(3/2)) - 64*(4*a
*e - 2*b*d + x*(2*b*e - 4*c*d))*(a*e**2 - b*d*e + c*d**2)*(-2*A*c*d - 2*B*a*e +
b*(A*e + B*d))/(15*(-4*a*c + b**2)**3*sqrt(a + b*x + c*x**2))

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Mathematica [B]  time = 6.7878, size = 1686, normalized size = 8.03 \[ \frac{\left (c x^2+b x+a\right )^4 \left (\frac{2 \left (-B e^4 x b^5-a B e^4 b^4+A c e^4 x b^4+4 B c d e^3 x b^4+a A c e^4 b^3+4 a B c d e^3 b^3+5 a B c e^4 x b^3-4 A c^2 d e^3 x b^3-6 B c^2 d^2 e^2 x b^3+4 a^2 B c e^4 b^2-4 a A c^2 d e^3 b^2-6 a B c^2 d^2 e^2 b^2-4 a A c^2 e^4 x b^2-16 a B c^2 d e^3 x b^2+6 A c^3 d^2 e^2 x b^2+4 B c^3 d^3 e x b^2+A c^4 d^4 b-3 a^2 A c^2 e^4 b-12 a^2 B c^2 d e^3 b+6 a A c^3 d^2 e^2 b+4 a B c^3 d^3 e b-B c^4 d^4 x b-5 a^2 B c^2 e^4 x b+12 a A c^3 d e^3 x b+18 a B c^3 d^2 e^2 x b-4 A c^4 d^3 e x b-2 a B c^4 d^4-2 a^3 B c^2 e^4+8 a^2 A c^3 d e^3+12 a^2 B c^3 d^2 e^2-8 a A c^4 d^3 e+2 A c^5 d^4 x+2 a^2 A c^3 e^4 x+8 a^2 B c^3 d e^3 x-12 a A c^4 d^2 e^2 x-8 a B c^4 d^3 e x\right )}{5 c^4 \left (4 a c-b^2\right ) \left (c x^2+b x+a\right )^3}+\frac{2 \left (6 B e^4 b^6-A c e^4 b^5-4 B c d e^3 b^5-3 B c e^4 x b^5-70 a B c e^4 b^4-16 A c^2 d e^3 b^4-24 B c^2 d^2 e^2 b^4-2 A c^2 e^4 x b^4-8 B c^2 d e^3 x b^4+24 a A c^2 e^4 b^3+96 a B c^2 d e^3 b^3+144 A c^3 d^2 e^2 b^3+96 B c^3 d^3 e b^3+40 a B c^2 e^4 x b^3-32 A c^3 d e^3 x b^3-48 B c^3 d^2 e^2 x b^3-64 B c^4 d^4 b^2+240 a^2 B c^2 e^4 b^2-192 a A c^3 d e^3 b^2-288 a B c^3 d^2 e^2 b^2-256 A c^4 d^3 e b^2+48 a A c^3 e^4 x b^2+192 a B c^3 d e^3 x b^2+288 A c^4 d^2 e^2 x b^2+192 B c^4 d^3 e x b^2+128 A c^5 d^4 b+48 a^2 A c^3 e^4 b+192 a^2 B c^3 d e^3 b+192 a A c^4 d^2 e^2 b+128 a B c^4 d^3 e b-128 B c^5 d^4 x b-240 a^2 B c^3 e^4 x b-384 a A c^4 d e^3 x b-576 a B c^4 d^2 e^2 x b-512 A c^5 d^3 e x b-480 a^3 B c^3 e^4+256 A c^6 d^4 x+96 a^2 A c^4 e^4 x+384 a^2 B c^4 d e^3 x+384 a A c^5 d^2 e^2 x+256 a B c^5 d^3 e x\right )}{15 c^3 \left (4 a c-b^2\right )^3 \left (c x^2+b x+a\right )}+\frac{2 \left (-3 B e^4 b^6+3 A c e^4 b^5+12 B c d e^3 b^5+9 B c e^4 x b^5+30 a B c e^4 b^4-12 A c^2 d e^3 b^4-18 B c^2 d^2 e^2 b^4-4 A c^2 e^4 x b^4-16 B c^2 d e^3 x b^4-22 a A c^2 e^4 b^3-88 a B c^2 d e^3 b^3+18 A c^3 d^2 e^2 b^3+12 B c^3 d^3 e b^3-70 a B c^2 e^4 x b^3-4 A c^3 d e^3 x b^3-6 B c^3 d^2 e^2 x b^3-8 B c^4 d^4 b^2-100 a^2 B c^2 e^4 b^2+56 a A c^3 d e^3 b^2+84 a B c^3 d^2 e^2 b^2-32 A c^4 d^3 e b^2+36 a A c^3 e^4 x b^2+144 a B c^3 d e^3 x b^2+36 A c^4 d^2 e^2 x b^2+24 B c^4 d^3 e x b^2+16 A c^5 d^4 b+56 a^2 A c^3 e^4 b+224 a^2 B c^3 d e^3 b+24 a A c^4 d^2 e^2 b+16 a B c^4 d^3 e b-16 B c^5 d^4 x b+120 a^2 B c^3 e^4 x b-48 a A c^4 d e^3 x b-72 a B c^4 d^2 e^2 x b-64 A c^5 d^3 e x b+80 a^3 B c^3 e^4-160 a^2 A c^4 d e^3-240 a^2 B c^4 d^2 e^2+32 A c^6 d^4 x-48 a^2 A c^4 e^4 x-192 a^2 B c^4 d e^3 x+48 a A c^5 d^2 e^2 x+32 a B c^5 d^3 e x\right )}{15 c^4 \left (4 a c-b^2\right )^2 \left (c x^2+b x+a\right )^2}\right )}{(a+x (b+c x))^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(7/2),x]

[Out]

((a + b*x + c*x^2)^4*((2*(A*b*c^4*d^4 - 2*a*B*c^4*d^4 + 4*a*b*B*c^3*d^3*e - 8*a*
A*c^4*d^3*e - 6*a*b^2*B*c^2*d^2*e^2 + 6*a*A*b*c^3*d^2*e^2 + 12*a^2*B*c^3*d^2*e^2
+ 4*a*b^3*B*c*d*e^3 - 4*a*A*b^2*c^2*d*e^3 - 12*a^2*b*B*c^2*d*e^3 + 8*a^2*A*c^3*
d*e^3 - a*b^4*B*e^4 + a*A*b^3*c*e^4 + 4*a^2*b^2*B*c*e^4 - 3*a^2*A*b*c^2*e^4 - 2*
a^3*B*c^2*e^4 - b*B*c^4*d^4*x + 2*A*c^5*d^4*x + 4*b^2*B*c^3*d^3*e*x - 4*A*b*c^4*
d^3*e*x - 8*a*B*c^4*d^3*e*x - 6*b^3*B*c^2*d^2*e^2*x + 6*A*b^2*c^3*d^2*e^2*x + 18
*a*b*B*c^3*d^2*e^2*x - 12*a*A*c^4*d^2*e^2*x + 4*b^4*B*c*d*e^3*x - 4*A*b^3*c^2*d*
e^3*x - 16*a*b^2*B*c^2*d*e^3*x + 12*a*A*b*c^3*d*e^3*x + 8*a^2*B*c^3*d*e^3*x - b^
5*B*e^4*x + A*b^4*c*e^4*x + 5*a*b^3*B*c*e^4*x - 4*a*A*b^2*c^2*e^4*x - 5*a^2*b*B*
c^2*e^4*x + 2*a^2*A*c^3*e^4*x))/(5*c^4*(-b^2 + 4*a*c)*(a + b*x + c*x^2)^3) + (2*
(-8*b^2*B*c^4*d^4 + 16*A*b*c^5*d^4 + 12*b^3*B*c^3*d^3*e - 32*A*b^2*c^4*d^3*e + 1
6*a*b*B*c^4*d^3*e - 18*b^4*B*c^2*d^2*e^2 + 18*A*b^3*c^3*d^2*e^2 + 84*a*b^2*B*c^3
*d^2*e^2 + 24*a*A*b*c^4*d^2*e^2 - 240*a^2*B*c^4*d^2*e^2 + 12*b^5*B*c*d*e^3 - 12*
A*b^4*c^2*d*e^3 - 88*a*b^3*B*c^2*d*e^3 + 56*a*A*b^2*c^3*d*e^3 + 224*a^2*b*B*c^3*
d*e^3 - 160*a^2*A*c^4*d*e^3 - 3*b^6*B*e^4 + 3*A*b^5*c*e^4 + 30*a*b^4*B*c*e^4 - 2
2*a*A*b^3*c^2*e^4 - 100*a^2*b^2*B*c^2*e^4 + 56*a^2*A*b*c^3*e^4 + 80*a^3*B*c^3*e^
4 - 16*b*B*c^5*d^4*x + 32*A*c^6*d^4*x + 24*b^2*B*c^4*d^3*e*x - 64*A*b*c^5*d^3*e*
x + 32*a*B*c^5*d^3*e*x - 6*b^3*B*c^3*d^2*e^2*x + 36*A*b^2*c^4*d^2*e^2*x - 72*a*b
*B*c^4*d^2*e^2*x + 48*a*A*c^5*d^2*e^2*x - 16*b^4*B*c^2*d*e^3*x - 4*A*b^3*c^3*d*e
^3*x + 144*a*b^2*B*c^3*d*e^3*x - 48*a*A*b*c^4*d*e^3*x - 192*a^2*B*c^4*d*e^3*x +
9*b^5*B*c*e^4*x - 4*A*b^4*c^2*e^4*x - 70*a*b^3*B*c^2*e^4*x + 36*a*A*b^2*c^3*e^4*
x + 120*a^2*b*B*c^3*e^4*x - 48*a^2*A*c^4*e^4*x))/(15*c^4*(-b^2 + 4*a*c)^2*(a + b
*x + c*x^2)^2) + (2*(-64*b^2*B*c^4*d^4 + 128*A*b*c^5*d^4 + 96*b^3*B*c^3*d^3*e -
256*A*b^2*c^4*d^3*e + 128*a*b*B*c^4*d^3*e - 24*b^4*B*c^2*d^2*e^2 + 144*A*b^3*c^3
*d^2*e^2 - 288*a*b^2*B*c^3*d^2*e^2 + 192*a*A*b*c^4*d^2*e^2 - 4*b^5*B*c*d*e^3 - 1
6*A*b^4*c^2*d*e^3 + 96*a*b^3*B*c^2*d*e^3 - 192*a*A*b^2*c^3*d*e^3 + 192*a^2*b*B*c
^3*d*e^3 + 6*b^6*B*e^4 - A*b^5*c*e^4 - 70*a*b^4*B*c*e^4 + 24*a*A*b^3*c^2*e^4 + 2
40*a^2*b^2*B*c^2*e^4 + 48*a^2*A*b*c^3*e^4 - 480*a^3*B*c^3*e^4 - 128*b*B*c^5*d^4*
x + 256*A*c^6*d^4*x + 192*b^2*B*c^4*d^3*e*x - 512*A*b*c^5*d^3*e*x + 256*a*B*c^5*
d^3*e*x - 48*b^3*B*c^3*d^2*e^2*x + 288*A*b^2*c^4*d^2*e^2*x - 576*a*b*B*c^4*d^2*e
^2*x + 384*a*A*c^5*d^2*e^2*x - 8*b^4*B*c^2*d*e^3*x - 32*A*b^3*c^3*d*e^3*x + 192*
a*b^2*B*c^3*d*e^3*x - 384*a*A*b*c^4*d*e^3*x + 384*a^2*B*c^4*d*e^3*x - 3*b^5*B*c*
e^4*x - 2*A*b^4*c^2*e^4*x + 40*a*b^3*B*c^2*e^4*x + 48*a*A*b^2*c^3*e^4*x - 240*a^
2*b*B*c^3*e^4*x + 96*a^2*A*c^4*e^4*x))/(15*c^3*(-b^2 + 4*a*c)^3*(a + b*x + c*x^2
))))/(a + x*(b + c*x))^(7/2)

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Maple [B]  time = 0.019, size = 1914, normalized size = 9.1 \[ \text{result too large to display} \]

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)^(7/2),x)

[Out]

2/15/(c*x^2+b*x+a)^(5/2)*(96*A*a^2*c^3*e^4*x^5+48*A*a*b^2*c^2*e^4*x^5-384*A*a*b*
c^3*d*e^3*x^5+384*A*a*c^4*d^2*e^2*x^5-2*A*b^4*c*e^4*x^5-32*A*b^3*c^2*d*e^3*x^5+2
88*A*b^2*c^3*d^2*e^2*x^5-512*A*b*c^4*d^3*e*x^5+256*A*c^5*d^4*x^5-240*B*a^2*b*c^2
*e^4*x^5+384*B*a^2*c^3*d*e^3*x^5+40*B*a*b^3*c*e^4*x^5+192*B*a*b^2*c^2*d*e^3*x^5-
576*B*a*b*c^3*d^2*e^2*x^5+256*B*a*c^4*d^3*e*x^5-3*B*b^5*e^4*x^5-8*B*b^4*c*d*e^3*
x^5-48*B*b^3*c^2*d^2*e^2*x^5+192*B*b^2*c^3*d^3*e*x^5-128*B*b*c^4*d^4*x^5+240*A*a
^2*b*c^2*e^4*x^4+120*A*a*b^3*c*e^4*x^4-960*A*a*b^2*c^2*d*e^3*x^4+960*A*a*b*c^3*d
^2*e^2*x^4-5*A*b^5*e^4*x^4-80*A*b^4*c*d*e^3*x^4+720*A*b^3*c^2*d^2*e^2*x^4-1280*A
*b^2*c^3*d^3*e*x^4+640*A*b*c^4*d^4*x^4-480*B*a^3*c^2*e^4*x^4-240*B*a^2*b^2*c*e^4
*x^4+960*B*a^2*b*c^2*d*e^3*x^4+10*B*a*b^4*e^4*x^4+480*B*a*b^3*c*d*e^3*x^4-1440*B
*a*b^2*c^2*d^2*e^2*x^4+640*B*a*b*c^3*d^3*e*x^4-20*B*b^5*d*e^3*x^4-120*B*b^4*c*d^
2*e^2*x^4+480*B*b^3*c^2*d^3*e*x^4-320*B*b^2*c^3*d^4*x^4+480*A*a^2*b^2*c*e^4*x^3-
960*A*a^2*b*c^2*d*e^3*x^3+960*A*a^2*c^3*d^2*e^2*x^3+40*A*a*b^4*e^4*x^3-800*A*a*b
^3*c*d*e^3*x^3+1440*A*a*b^2*c^2*d^2*e^2*x^3-1280*A*a*b*c^3*d^3*e*x^3+640*A*a*c^4
*d^4*x^3-60*A*b^5*d*e^3*x^3+540*A*b^4*c*d^2*e^2*x^3-960*A*b^3*c^2*d^3*e*x^3+480*
A*b^2*c^3*d^4*x^3-960*B*a^3*b*c*e^4*x^3-80*B*a^2*b^3*e^4*x^3+1920*B*a^2*b^2*c*d*
e^3*x^3-1440*B*a^2*b*c^2*d^2*e^2*x^3+640*B*a^2*c^3*d^3*e*x^3+160*B*a*b^4*d*e^3*x
^3-1200*B*a*b^3*c*d^2*e^2*x^3+960*B*a*b^2*c^2*d^3*e*x^3-320*B*a*b*c^3*d^4*x^3-90
*B*b^5*d^2*e^2*x^3+360*B*b^4*c*d^3*e*x^3-240*B*b^3*c^2*d^4*x^3+320*A*a^3*b*c*e^4
*x^2-640*A*a^3*c^2*d*e^3*x^2+240*A*a^2*b^3*e^4*x^2-960*A*a^2*b^2*c*d*e^3*x^2+144
0*A*a^2*b*c^2*d^2*e^2*x^2-360*A*a*b^4*d*e^3*x^2+1200*A*a*b^3*c*d^2*e^2*x^2-1920*
A*a*b^2*c^2*d^3*e*x^2+960*A*a*b*c^3*d^4*x^2+90*A*b^5*d^2*e^2*x^2-160*A*b^4*c*d^3
*e*x^2+80*A*b^3*c^2*d^4*x^2-640*B*a^4*c*e^4*x^2-480*B*a^3*b^2*e^4*x^2+1280*B*a^3
*b*c*d*e^3*x^2-960*B*a^3*c^2*d^2*e^2*x^2+960*B*a^2*b^3*d*e^3*x^2-1440*B*a^2*b^2*
c*d^2*e^2*x^2+960*B*a^2*b*c^2*d^3*e*x^2-540*B*a*b^4*d^2*e^2*x^2+800*B*a*b^3*c*d^
3*e*x^2-480*B*a*b^2*c^2*d^4*x^2+60*B*b^5*d^3*e*x^2-40*B*b^4*c*d^4*x^2+320*A*a^3*
b^2*e^4*x-640*A*a^3*b*c*d*e^3*x-480*A*a^2*b^3*d*e^3*x+1440*A*a^2*b^2*c*d^2*e^2*x
-960*A*a^2*b*c^2*d^3*e*x+480*A*a^2*c^3*d^4*x+120*A*a*b^4*d^2*e^2*x-480*A*a*b^3*c
*d^3*e*x+240*A*a*b^2*c^2*d^4*x+20*A*b^5*d^3*e*x-10*A*b^4*c*d^4*x-640*B*a^4*b*e^4
*x+1280*B*a^3*b^2*d*e^3*x-960*B*a^3*b*c*d^2*e^2*x-720*B*a^2*b^3*d^2*e^2*x+960*B*
a^2*b^2*c*d^3*e*x-240*B*a^2*b*c^2*d^4*x+80*B*a*b^4*d^3*e*x-120*B*a*b^3*c*d^4*x+5
*B*b^5*d^4*x+128*A*a^4*b*e^4-256*A*a^4*c*d*e^3-192*A*a^3*b^2*d*e^3+576*A*a^3*b*c
*d^2*e^2-384*A*a^3*c^2*d^3*e+48*A*a^2*b^3*d^2*e^2-192*A*a^2*b^2*c*d^3*e+240*A*a^
2*b*c^2*d^4+8*A*a*b^4*d^3*e-40*A*a*b^3*c*d^4+3*A*b^5*d^4-256*B*a^5*e^4+512*B*a^4
*b*d*e^3-384*B*a^4*c*d^2*e^2-288*B*a^3*b^2*d^2*e^2+384*B*a^3*b*c*d^3*e-96*B*a^3*
c^2*d^4+32*B*a^2*b^3*d^3*e-48*B*a^2*b^2*c*d^4+2*B*a*b^4*d^4)/(64*a^3*c^3-48*a^2*
b^2*c^2+12*a*b^4*c-b^6)

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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)^4/(c*x^2 + b*x + a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.92043, size = 2198, normalized size = 10.47 \[ \text{result too large to display} \]

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)^(7/2),x, algorithm="fricas")

[Out]

2/15*((128*(B*b*c^4 - 2*A*c^5)*d^4 - 64*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4)*d^3*
e + 48*(B*b^3*c^2 - 8*A*a*c^4 + 6*(2*B*a*b - A*b^2)*c^3)*d^2*e^2 + 8*(B*b^4*c -
48*(B*a^2 - A*a*b)*c^3 - 4*(6*B*a*b^2 - A*b^3)*c^2)*d*e^3 + (3*B*b^5 - 96*A*a^2*
c^3 + 48*(5*B*a^2*b - A*a*b^2)*c^2 - 2*(20*B*a*b^3 - A*b^4)*c)*e^4)*x^5 - (2*B*a
*b^4 + 3*A*b^5 - 48*(2*B*a^3 - 5*A*a^2*b)*c^2 - 8*(6*B*a^2*b^2 + 5*A*a*b^3)*c)*d
^4 - 8*(4*B*a^2*b^3 + A*a*b^4 - 48*A*a^3*c^2 + 24*(2*B*a^3*b - A*a^2*b^2)*c)*d^3
*e + 48*(6*B*a^3*b^2 - A*a^2*b^3 + 4*(2*B*a^4 - 3*A*a^3*b)*c)*d^2*e^2 - 64*(8*B*
a^4*b - 3*A*a^3*b^2 - 4*A*a^4*c)*d*e^3 + 128*(2*B*a^5 - A*a^4*b)*e^4 + 5*(64*(B*
b^2*c^3 - 2*A*b*c^4)*d^4 - 32*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^3)*d^3*e + 24
*(B*b^4*c - 8*A*a*b*c^3 + 6*(2*B*a*b^2 - A*b^3)*c^2)*d^2*e^2 + 4*(B*b^5 - 48*(B*
a^2*b - A*a*b^2)*c^2 - 4*(6*B*a*b^3 - A*b^4)*c)*d*e^3 - (2*B*a*b^4 - A*b^5 - 48*
(2*B*a^3 - A*a^2*b)*c^2 - 24*(2*B*a^2*b^2 - A*a*b^3)*c)*e^4)*x^4 + 10*(8*(3*B*b^
3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b - 3*A*b^2)*c^3)*d^4 - 4*(9*B*b^4*c + 16*(B*a^2 -
2*A*a*b)*c^3 + 24*(B*a*b^2 - A*b^3)*c^2)*d^3*e + 3*(3*B*b^5 - 32*A*a^2*c^3 + 48*
(B*a^2*b - A*a*b^2)*c^2 + 2*(20*B*a*b^3 - 9*A*b^4)*c)*d^2*e^2 - 2*(8*B*a*b^4 - 3
*A*b^5 - 48*A*a^2*b*c^2 + 8*(12*B*a^2*b^2 - 5*A*a*b^3)*c)*d*e^3 + 4*(2*B*a^2*b^3
- A*a*b^4 + 12*(2*B*a^3*b - A*a^2*b^2)*c)*e^4)*x^3 + 10*(4*(B*b^4*c - 24*A*a*b*
c^3 + 2*(6*B*a*b^2 - A*b^3)*c^2)*d^4 - 2*(3*B*b^5 + 48*(B*a^2*b - 2*A*a*b^2)*c^2
+ 8*(5*B*a*b^3 - A*b^4)*c)*d^3*e + 3*(18*B*a*b^4 - 3*A*b^5 + 16*(2*B*a^3 - 3*A*
a^2*b)*c^2 + 8*(6*B*a^2*b^2 - 5*A*a*b^3)*c)*d^2*e^2 - 4*(24*B*a^2*b^3 - 9*A*a*b^
4 - 16*A*a^3*c^2 + 8*(4*B*a^3*b - 3*A*a^2*b^2)*c)*d*e^3 + 8*(6*B*a^3*b^2 - 3*A*a
^2*b^3 + 4*(2*B*a^4 - A*a^3*b)*c)*e^4)*x^2 - 5*((B*b^5 + 96*A*a^2*c^3 - 48*(B*a^
2*b - A*a*b^2)*c^2 - 2*(12*B*a*b^3 + A*b^4)*c)*d^4 + 4*(4*B*a*b^4 + A*b^5 - 48*A
*a^2*b*c^2 + 24*(2*B*a^2*b^2 - A*a*b^3)*c)*d^3*e - 24*(6*B*a^2*b^3 - A*a*b^4 + 4
*(2*B*a^3*b - 3*A*a^2*b^2)*c)*d^2*e^2 + 32*(8*B*a^3*b^2 - 3*A*a^2*b^3 - 4*A*a^3*
b*c)*d*e^3 - 64*(2*B*a^4*b - A*a^3*b^2)*e^4)*x)*sqrt(c*x^2 + b*x + a)/(a^3*b^6 -
12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b
^2*c^5 - 64*a^3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b
*c^5)*x^5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c
^5)*x^4 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x
^3 + 3*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2
+ 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)

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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)**4/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27913, 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)^4/(c*x^2 + b*x + a)^(7/2),x, algorithm="giac")

[Out]

Done