3.1926 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=187 \[ -\frac{7 b^6 (d+e x)^3 (b d-a e)}{3 e^8}+\frac{21 b^5 (d+e x)^2 (b d-a e)^2}{2 e^8}-\frac{35 b^4 x (b d-a e)^3}{e^7}+\frac{35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}+\frac{21 b^2 (b d-a e)^5}{e^8 (d+e x)}-\frac{7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac{(b d-a e)^7}{3 e^8 (d+e x)^3}+\frac{b^7 (d+e x)^4}{4 e^8} \]

[Out]

(-35*b^4*(b*d - a*e)^3*x)/e^7 + (b*d - a*e)^7/(3*e^8*(d + e*x)^3) - (7*b*(b*d -
a*e)^6)/(2*e^8*(d + e*x)^2) + (21*b^2*(b*d - a*e)^5)/(e^8*(d + e*x)) + (21*b^5*(
b*d - a*e)^2*(d + e*x)^2)/(2*e^8) - (7*b^6*(b*d - a*e)*(d + e*x)^3)/(3*e^8) + (b
^7*(d + e*x)^4)/(4*e^8) + (35*b^3*(b*d - a*e)^4*Log[d + e*x])/e^8

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Rubi [A]  time = 0.478421, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{7 b^6 (d+e x)^3 (b d-a e)}{3 e^8}+\frac{21 b^5 (d+e x)^2 (b d-a e)^2}{2 e^8}-\frac{35 b^4 x (b d-a e)^3}{e^7}+\frac{35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}+\frac{21 b^2 (b d-a e)^5}{e^8 (d+e x)}-\frac{7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac{(b d-a e)^7}{3 e^8 (d+e x)^3}+\frac{b^7 (d+e x)^4}{4 e^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^4,x]

[Out]

(-35*b^4*(b*d - a*e)^3*x)/e^7 + (b*d - a*e)^7/(3*e^8*(d + e*x)^3) - (7*b*(b*d -
a*e)^6)/(2*e^8*(d + e*x)^2) + (21*b^2*(b*d - a*e)^5)/(e^8*(d + e*x)) + (21*b^5*(
b*d - a*e)^2*(d + e*x)^2)/(2*e^8) - (7*b^6*(b*d - a*e)*(d + e*x)^3)/(3*e^8) + (b
^7*(d + e*x)^4)/(4*e^8) + (35*b^3*(b*d - a*e)^4*Log[d + e*x])/e^8

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Rubi in Sympy [A]  time = 123.008, size = 172, normalized size = 0.92 \[ \frac{b^{7} \left (d + e x\right )^{4}}{4 e^{8}} + \frac{7 b^{6} \left (d + e x\right )^{3} \left (a e - b d\right )}{3 e^{8}} + \frac{21 b^{5} \left (d + e x\right )^{2} \left (a e - b d\right )^{2}}{2 e^{8}} + \frac{35 b^{4} x \left (a e - b d\right )^{3}}{e^{7}} + \frac{35 b^{3} \left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{8}} - \frac{21 b^{2} \left (a e - b d\right )^{5}}{e^{8} \left (d + e x\right )} - \frac{7 b \left (a e - b d\right )^{6}}{2 e^{8} \left (d + e x\right )^{2}} - \frac{\left (a e - b d\right )^{7}}{3 e^{8} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**4,x)

[Out]

b**7*(d + e*x)**4/(4*e**8) + 7*b**6*(d + e*x)**3*(a*e - b*d)/(3*e**8) + 21*b**5*
(d + e*x)**2*(a*e - b*d)**2/(2*e**8) + 35*b**4*x*(a*e - b*d)**3/e**7 + 35*b**3*(
a*e - b*d)**4*log(d + e*x)/e**8 - 21*b**2*(a*e - b*d)**5/(e**8*(d + e*x)) - 7*b*
(a*e - b*d)**6/(2*e**8*(d + e*x)**2) - (a*e - b*d)**7/(3*e**8*(d + e*x)**3)

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Mathematica [A]  time = 0.181473, size = 199, normalized size = 1.06 \[ \frac{6 b^5 e^2 x^2 \left (21 a^2 e^2-28 a b d e+10 b^2 d^2\right )-12 b^4 e x \left (-35 a^3 e^3+84 a^2 b d e^2-70 a b^2 d^2 e+20 b^3 d^3\right )-4 b^6 e^3 x^3 (4 b d-7 a e)+420 b^3 (b d-a e)^4 \log (d+e x)+\frac{252 b^2 (b d-a e)^5}{d+e x}-\frac{42 b (b d-a e)^6}{(d+e x)^2}+\frac{4 (b d-a e)^7}{(d+e x)^3}+3 b^7 e^4 x^4}{12 e^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^4,x]

[Out]

(-12*b^4*e*(20*b^3*d^3 - 70*a*b^2*d^2*e + 84*a^2*b*d*e^2 - 35*a^3*e^3)*x + 6*b^5
*e^2*(10*b^2*d^2 - 28*a*b*d*e + 21*a^2*e^2)*x^2 - 4*b^6*e^3*(4*b*d - 7*a*e)*x^3
+ 3*b^7*e^4*x^4 + (4*(b*d - a*e)^7)/(d + e*x)^3 - (42*b*(b*d - a*e)^6)/(d + e*x)
^2 + (252*b^2*(b*d - a*e)^5)/(d + e*x) + 420*b^3*(b*d - a*e)^4*Log[d + e*x])/(12
*e^8)

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Maple [B]  time = 0.02, size = 622, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^4,x)

[Out]

7/e^6/(e*x+d)^3*a^2*b^5*d^5-140*b^4/e^5*ln(e*x+d)*a^3*d+210*b^5/e^6*ln(e*x+d)*a^
2*d^2-140*b^6/e^7*ln(e*x+d)*a*d^3+1/4*b^7/e^4*x^4-1/3/e/(e*x+d)^3*a^7+70*b^6/e^6
*a*d^2*x+21*b^2/e^3/(e*x+d)^2*a^5*d-105/2*b^3/e^4/(e*x+d)^2*a^4*d^2+70*b^4/e^5/(
e*x+d)^2*a^3*d^3-105/2*b^5/e^6/(e*x+d)^2*d^4*a^2+21*b^6/e^7/(e*x+d)^2*d^5*a-14*b
^6/e^5*x^2*a*d-84*b^5/e^5*a^2*d*x-7/3/e^7/(e*x+d)^3*a*b^6*d^6+35*b^3/e^4*ln(e*x+
d)*a^4+35*b^7/e^8*ln(e*x+d)*d^4-21*b^2/e^3/(e*x+d)*a^5+21*b^7/e^8/(e*x+d)*d^5+7/
3*b^6/e^4*x^3*a-4/3*b^7/e^5*x^3*d+21/2*b^5/e^4*x^2*a^2+5*b^7/e^6*x^2*d^2+35*b^4/
e^4*a^3*x-20*b^7/e^7*d^3*x+1/3/e^8/(e*x+d)^3*b^7*d^7-7/2*b/e^2/(e*x+d)^2*a^6-210
*b^4/e^5/(e*x+d)*a^3*d^2+210*b^5/e^6/(e*x+d)*a^2*d^3-105*b^6/e^7/(e*x+d)*a*d^4+3
5/3/e^4/(e*x+d)^3*a^4*b^3*d^3+7/3/e^2/(e*x+d)^3*a^6*b*d-7/e^3/(e*x+d)^3*a^5*b^2*
d^2-7/2*b^7/e^8/(e*x+d)^2*d^6-35/3/e^5/(e*x+d)^3*a^3*b^4*d^4+105*b^3/e^4/(e*x+d)
*a^4*d

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Maxima [A]  time = 0.73351, size = 655, normalized size = 3.5 \[ \frac{107 \, b^{7} d^{7} - 518 \, a b^{6} d^{6} e + 987 \, a^{2} b^{5} d^{5} e^{2} - 910 \, a^{3} b^{4} d^{4} e^{3} + 385 \, a^{4} b^{3} d^{3} e^{4} - 42 \, a^{5} b^{2} d^{2} e^{5} - 7 \, a^{6} b d e^{6} - 2 \, a^{7} e^{7} + 126 \,{\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 21 \,{\left (11 \, b^{7} d^{6} e - 54 \, a b^{6} d^{5} e^{2} + 105 \, a^{2} b^{5} d^{4} e^{3} - 100 \, a^{3} b^{4} d^{3} e^{4} + 45 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x}{6 \,{\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} + \frac{3 \, b^{7} e^{3} x^{4} - 4 \,{\left (4 \, b^{7} d e^{2} - 7 \, a b^{6} e^{3}\right )} x^{3} + 6 \,{\left (10 \, b^{7} d^{2} e - 28 \, a b^{6} d e^{2} + 21 \, a^{2} b^{5} e^{3}\right )} x^{2} - 12 \,{\left (20 \, b^{7} d^{3} - 70 \, a b^{6} d^{2} e + 84 \, a^{2} b^{5} d e^{2} - 35 \, a^{3} b^{4} e^{3}\right )} x}{12 \, e^{7}} + \frac{35 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} \log \left (e x + d\right )}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)/(e*x + d)^4,x, algorithm="maxima")

[Out]

1/6*(107*b^7*d^7 - 518*a*b^6*d^6*e + 987*a^2*b^5*d^5*e^2 - 910*a^3*b^4*d^4*e^3 +
385*a^4*b^3*d^3*e^4 - 42*a^5*b^2*d^2*e^5 - 7*a^6*b*d*e^6 - 2*a^7*e^7 + 126*(b^7
*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3
*d*e^6 - a^5*b^2*e^7)*x^2 + 21*(11*b^7*d^6*e - 54*a*b^6*d^5*e^2 + 105*a^2*b^5*d^
4*e^3 - 100*a^3*b^4*d^3*e^4 + 45*a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^6 - a^6*b*e^7)*
x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8) + 1/12*(3*b^7*e^3*x^4 - 4*(
4*b^7*d*e^2 - 7*a*b^6*e^3)*x^3 + 6*(10*b^7*d^2*e - 28*a*b^6*d*e^2 + 21*a^2*b^5*e
^3)*x^2 - 12*(20*b^7*d^3 - 70*a*b^6*d^2*e + 84*a^2*b^5*d*e^2 - 35*a^3*b^4*e^3)*x
)/e^7 + 35*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*
b^3*e^4)*log(e*x + d)/e^8

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Fricas [A]  time = 0.292087, size = 995, normalized size = 5.32 \[ \frac{3 \, b^{7} e^{7} x^{7} + 214 \, b^{7} d^{7} - 1036 \, a b^{6} d^{6} e + 1974 \, a^{2} b^{5} d^{5} e^{2} - 1820 \, a^{3} b^{4} d^{4} e^{3} + 770 \, a^{4} b^{3} d^{3} e^{4} - 84 \, a^{5} b^{2} d^{2} e^{5} - 14 \, a^{6} b d e^{6} - 4 \, a^{7} e^{7} - 7 \,{\left (b^{7} d e^{6} - 4 \, a b^{6} e^{7}\right )} x^{6} + 21 \,{\left (b^{7} d^{2} e^{5} - 4 \, a b^{6} d e^{6} + 6 \, a^{2} b^{5} e^{7}\right )} x^{5} - 105 \,{\left (b^{7} d^{3} e^{4} - 4 \, a b^{6} d^{2} e^{5} + 6 \, a^{2} b^{5} d e^{6} - 4 \, a^{3} b^{4} e^{7}\right )} x^{4} - 2 \,{\left (278 \, b^{7} d^{4} e^{3} - 1022 \, a b^{6} d^{3} e^{4} + 1323 \, a^{2} b^{5} d^{2} e^{5} - 630 \, a^{3} b^{4} d e^{6}\right )} x^{3} - 6 \,{\left (68 \, b^{7} d^{5} e^{2} - 182 \, a b^{6} d^{4} e^{3} + 63 \, a^{2} b^{5} d^{3} e^{4} + 210 \, a^{3} b^{4} d^{2} e^{5} - 210 \, a^{4} b^{3} d e^{6} + 42 \, a^{5} b^{2} e^{7}\right )} x^{2} + 6 \,{\left (37 \, b^{7} d^{6} e - 238 \, a b^{6} d^{5} e^{2} + 567 \, a^{2} b^{5} d^{4} e^{3} - 630 \, a^{3} b^{4} d^{3} e^{4} + 315 \, a^{4} b^{3} d^{2} e^{5} - 42 \, a^{5} b^{2} d e^{6} - 7 \, a^{6} b e^{7}\right )} x + 420 \,{\left (b^{7} d^{7} - 4 \, a b^{6} d^{6} e + 6 \, a^{2} b^{5} d^{5} e^{2} - 4 \, a^{3} b^{4} d^{4} e^{3} + a^{4} b^{3} d^{3} e^{4} +{\left (b^{7} d^{4} e^{3} - 4 \, a b^{6} d^{3} e^{4} + 6 \, a^{2} b^{5} d^{2} e^{5} - 4 \, a^{3} b^{4} d e^{6} + a^{4} b^{3} e^{7}\right )} x^{3} + 3 \,{\left (b^{7} d^{5} e^{2} - 4 \, a b^{6} d^{4} e^{3} + 6 \, a^{2} b^{5} d^{3} e^{4} - 4 \, a^{3} b^{4} d^{2} e^{5} + a^{4} b^{3} d e^{6}\right )} x^{2} + 3 \,{\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 6 \, a^{2} b^{5} d^{4} e^{3} - 4 \, a^{3} b^{4} d^{3} e^{4} + a^{4} b^{3} d^{2} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/12*(3*b^7*e^7*x^7 + 214*b^7*d^7 - 1036*a*b^6*d^6*e + 1974*a^2*b^5*d^5*e^2 - 18
20*a^3*b^4*d^4*e^3 + 770*a^4*b^3*d^3*e^4 - 84*a^5*b^2*d^2*e^5 - 14*a^6*b*d*e^6 -
4*a^7*e^7 - 7*(b^7*d*e^6 - 4*a*b^6*e^7)*x^6 + 21*(b^7*d^2*e^5 - 4*a*b^6*d*e^6 +
6*a^2*b^5*e^7)*x^5 - 105*(b^7*d^3*e^4 - 4*a*b^6*d^2*e^5 + 6*a^2*b^5*d*e^6 - 4*a
^3*b^4*e^7)*x^4 - 2*(278*b^7*d^4*e^3 - 1022*a*b^6*d^3*e^4 + 1323*a^2*b^5*d^2*e^5
- 630*a^3*b^4*d*e^6)*x^3 - 6*(68*b^7*d^5*e^2 - 182*a*b^6*d^4*e^3 + 63*a^2*b^5*d
^3*e^4 + 210*a^3*b^4*d^2*e^5 - 210*a^4*b^3*d*e^6 + 42*a^5*b^2*e^7)*x^2 + 6*(37*b
^7*d^6*e - 238*a*b^6*d^5*e^2 + 567*a^2*b^5*d^4*e^3 - 630*a^3*b^4*d^3*e^4 + 315*a
^4*b^3*d^2*e^5 - 42*a^5*b^2*d*e^6 - 7*a^6*b*e^7)*x + 420*(b^7*d^7 - 4*a*b^6*d^6*
e + 6*a^2*b^5*d^5*e^2 - 4*a^3*b^4*d^4*e^3 + a^4*b^3*d^3*e^4 + (b^7*d^4*e^3 - 4*a
*b^6*d^3*e^4 + 6*a^2*b^5*d^2*e^5 - 4*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + 3*(b^7*d
^5*e^2 - 4*a*b^6*d^4*e^3 + 6*a^2*b^5*d^3*e^4 - 4*a^3*b^4*d^2*e^5 + a^4*b^3*d*e^6
)*x^2 + 3*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 6*a^2*b^5*d^4*e^3 - 4*a^3*b^4*d^3*e^4 +
a^4*b^3*d^2*e^5)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*
e^8)

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Sympy [A]  time = 25.6242, size = 468, normalized size = 2.5 \[ \frac{b^{7} x^{4}}{4 e^{4}} + \frac{35 b^{3} \left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{8}} - \frac{2 a^{7} e^{7} + 7 a^{6} b d e^{6} + 42 a^{5} b^{2} d^{2} e^{5} - 385 a^{4} b^{3} d^{3} e^{4} + 910 a^{3} b^{4} d^{4} e^{3} - 987 a^{2} b^{5} d^{5} e^{2} + 518 a b^{6} d^{6} e - 107 b^{7} d^{7} + x^{2} \left (126 a^{5} b^{2} e^{7} - 630 a^{4} b^{3} d e^{6} + 1260 a^{3} b^{4} d^{2} e^{5} - 1260 a^{2} b^{5} d^{3} e^{4} + 630 a b^{6} d^{4} e^{3} - 126 b^{7} d^{5} e^{2}\right ) + x \left (21 a^{6} b e^{7} + 126 a^{5} b^{2} d e^{6} - 945 a^{4} b^{3} d^{2} e^{5} + 2100 a^{3} b^{4} d^{3} e^{4} - 2205 a^{2} b^{5} d^{4} e^{3} + 1134 a b^{6} d^{5} e^{2} - 231 b^{7} d^{6} e\right )}{6 d^{3} e^{8} + 18 d^{2} e^{9} x + 18 d e^{10} x^{2} + 6 e^{11} x^{3}} + \frac{x^{3} \left (7 a b^{6} e - 4 b^{7} d\right )}{3 e^{5}} + \frac{x^{2} \left (21 a^{2} b^{5} e^{2} - 28 a b^{6} d e + 10 b^{7} d^{2}\right )}{2 e^{6}} + \frac{x \left (35 a^{3} b^{4} e^{3} - 84 a^{2} b^{5} d e^{2} + 70 a b^{6} d^{2} e - 20 b^{7} d^{3}\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**4,x)

[Out]

b**7*x**4/(4*e**4) + 35*b**3*(a*e - b*d)**4*log(d + e*x)/e**8 - (2*a**7*e**7 + 7
*a**6*b*d*e**6 + 42*a**5*b**2*d**2*e**5 - 385*a**4*b**3*d**3*e**4 + 910*a**3*b**
4*d**4*e**3 - 987*a**2*b**5*d**5*e**2 + 518*a*b**6*d**6*e - 107*b**7*d**7 + x**2
*(126*a**5*b**2*e**7 - 630*a**4*b**3*d*e**6 + 1260*a**3*b**4*d**2*e**5 - 1260*a*
*2*b**5*d**3*e**4 + 630*a*b**6*d**4*e**3 - 126*b**7*d**5*e**2) + x*(21*a**6*b*e*
*7 + 126*a**5*b**2*d*e**6 - 945*a**4*b**3*d**2*e**5 + 2100*a**3*b**4*d**3*e**4 -
2205*a**2*b**5*d**4*e**3 + 1134*a*b**6*d**5*e**2 - 231*b**7*d**6*e))/(6*d**3*e*
*8 + 18*d**2*e**9*x + 18*d*e**10*x**2 + 6*e**11*x**3) + x**3*(7*a*b**6*e - 4*b**
7*d)/(3*e**5) + x**2*(21*a**2*b**5*e**2 - 28*a*b**6*d*e + 10*b**7*d**2)/(2*e**6)
+ x*(35*a**3*b**4*e**3 - 84*a**2*b**5*d*e**2 + 70*a*b**6*d**2*e - 20*b**7*d**3)
/e**7

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GIAC/XCAS [A]  time = 0.30196, size = 597, normalized size = 3.19 \[ 35 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} e^{\left (-8\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, b^{7} x^{4} e^{12} - 16 \, b^{7} d x^{3} e^{11} + 60 \, b^{7} d^{2} x^{2} e^{10} - 240 \, b^{7} d^{3} x e^{9} + 28 \, a b^{6} x^{3} e^{12} - 168 \, a b^{6} d x^{2} e^{11} + 840 \, a b^{6} d^{2} x e^{10} + 126 \, a^{2} b^{5} x^{2} e^{12} - 1008 \, a^{2} b^{5} d x e^{11} + 420 \, a^{3} b^{4} x e^{12}\right )} e^{\left (-16\right )} + \frac{{\left (107 \, b^{7} d^{7} - 518 \, a b^{6} d^{6} e + 987 \, a^{2} b^{5} d^{5} e^{2} - 910 \, a^{3} b^{4} d^{4} e^{3} + 385 \, a^{4} b^{3} d^{3} e^{4} - 42 \, a^{5} b^{2} d^{2} e^{5} - 7 \, a^{6} b d e^{6} - 2 \, a^{7} e^{7} + 126 \,{\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{2} + 21 \,{\left (11 \, b^{7} d^{6} e - 54 \, a b^{6} d^{5} e^{2} + 105 \, a^{2} b^{5} d^{4} e^{3} - 100 \, a^{3} b^{4} d^{3} e^{4} + 45 \, a^{4} b^{3} d^{2} e^{5} - 6 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)/(e*x + d)^4,x, algorithm="giac")

[Out]

35*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)
*e^(-8)*ln(abs(x*e + d)) + 1/12*(3*b^7*x^4*e^12 - 16*b^7*d*x^3*e^11 + 60*b^7*d^2
*x^2*e^10 - 240*b^7*d^3*x*e^9 + 28*a*b^6*x^3*e^12 - 168*a*b^6*d*x^2*e^11 + 840*a
*b^6*d^2*x*e^10 + 126*a^2*b^5*x^2*e^12 - 1008*a^2*b^5*d*x*e^11 + 420*a^3*b^4*x*e
^12)*e^(-16) + 1/6*(107*b^7*d^7 - 518*a*b^6*d^6*e + 987*a^2*b^5*d^5*e^2 - 910*a^
3*b^4*d^4*e^3 + 385*a^4*b^3*d^3*e^4 - 42*a^5*b^2*d^2*e^5 - 7*a^6*b*d*e^6 - 2*a^7
*e^7 + 126*(b^7*d^5*e^2 - 5*a*b^6*d^4*e^3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*
e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^2 + 21*(11*b^7*d^6*e - 54*a*b^6*d^5*e^2 +
105*a^2*b^5*d^4*e^3 - 100*a^3*b^4*d^3*e^4 + 45*a^4*b^3*d^2*e^5 - 6*a^5*b^2*d*e^
6 - a^6*b*e^7)*x)*e^(-8)/(x*e + d)^3