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

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

[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 = 0.21049, 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, Rules used = {27, 43} \[ -\frac{7 b^6 (d+e x)^3 (b d-a e)}{3 e^8}+\frac{21 b^5 (d+e x)^2 (b d-a e)^2}{2 e^8}-\frac{35 b^4 x (b d-a e)^3}{e^7}+\frac{21 b^2 (b d-a e)^5}{e^8 (d+e x)}+\frac{35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}-\frac{7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac{(b d-a e)^7}{3 e^8 (d+e x)^3}+\frac{b^7 (d+e x)^4}{4 e^8} \]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx &=\int \frac{(a+b x)^7}{(d+e x)^4} \, dx\\ &=\int \left (-\frac{35 b^4 (b d-a e)^3}{e^7}+\frac{(-b d+a e)^7}{e^7 (d+e x)^4}+\frac{7 b (b d-a e)^6}{e^7 (d+e x)^3}-\frac{21 b^2 (b d-a e)^5}{e^7 (d+e x)^2}+\frac{35 b^3 (b d-a e)^4}{e^7 (d+e x)}+\frac{21 b^5 (b d-a e)^2 (d+e x)}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^2}{e^7}+\frac{b^7 (d+e x)^3}{e^7}\right ) \, dx\\ &=-\frac{35 b^4 (b d-a e)^3 x}{e^7}+\frac{(b d-a e)^7}{3 e^8 (d+e x)^3}-\frac{7 b (b d-a e)^6}{2 e^8 (d+e x)^2}+\frac{21 b^2 (b d-a e)^5}{e^8 (d+e x)}+\frac{21 b^5 (b d-a e)^2 (d+e x)^2}{2 e^8}-\frac{7 b^6 (b d-a e) (d+e x)^3}{3 e^8}+\frac{b^7 (d+e x)^4}{4 e^8}+\frac{35 b^3 (b d-a e)^4 \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.105561, size = 199, normalized size = 1.06 \[ \frac{6 b^5 e^2 x^2 \left (21 a^2 e^2-28 a b d e+10 b^2 d^2\right )-12 b^4 e x \left (84 a^2 b d e^2-35 a^3 e^3-70 a b^2 d^2 e+20 b^3 d^3\right )-4 b^6 e^3 x^3 (4 b d-7 a e)+\frac{252 b^2 (b d-a e)^5}{d+e x}+420 b^3 (b d-a e)^4 \log (d+e x)-\frac{42 b (b d-a e)^6}{(d+e x)^2}+\frac{4 (b d-a e)^7}{(d+e x)^3}+3 b^7 e^4 x^4}{12 e^8} \]

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 = 0.015, size = 622, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

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]

1/3/e^8/(e*x+d)^3*b^7*d^7-7/2*b/e^2/(e*x+d)^2*a^6-7/2*b^7/e^8/(e*x+d)^2*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+210*b^5/e^6*ln(e*x+d)*a^2*d^2-140*b^6/e^7*ln(e*x+
d)*a*d^3-140*b^4/e^5*ln(e*x+d)*a^3*d-35/3/e^5/(e*x+d)^3*a^3*b^4*d^4+7/e^6/(e*x+d)^3*a^2*b^5*d^5-7/3/e^7/(e*x+d
)^3*a*b^6*d^6+105*b^3/e^4/(e*x+d)*a^4*d-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^4-14*b^6/e^5*x^2*a*d-84*b^5/e^5*a^2*d*x-105/2*b^5/e^6/(e*x+d)^2*a^2*d^4+21*b^2/e^3/(e*x+d)^2*a^5*d-1
05/2*b^3/e^4/(e*x+d)^2*a^4*d^2+70*b^6/e^6*a*d^2*x+7/3/e^2/(e*x+d)^3*d*a^6*b-7/e^3/(e*x+d)^3*d^2*a^5*b^2+35/3/e
^4/(e*x+d)^3*d^3*a^4*b^3+21*b^6/e^7/(e*x+d)^2*a*d^5+70*b^4/e^5/(e*x+d)^2*a^3*d^3+1/4*b^7/e^4*x^4-1/3/e/(e*x+d)
^3*a^7

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Maxima [B]  time = 1.03992, size = 655, normalized size = 3.5 \begin{align*} \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}} \end{align*}

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, 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 - 10
0*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 [B]  time = 1.9373, size = 1520, normalized size = 8.13 \begin{align*} \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 )}} \end{align*}

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, 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 - 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*(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 [B]  time = 8.2327, size = 468, normalized size = 2.5 \begin{align*} \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}} \end{align*}

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 [B]  time = 1.09648, size = 597, normalized size = 3.19 \begin{align*} 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 )} \log \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}} \end{align*}

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, 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)*log(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 - 4
2*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