3.1493 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=156 \[ -\frac{3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac{15 b^4 x (b d-a e)^2}{e^6}-\frac{15 b^2 (b d-a e)^4}{e^7 (d+e x)}-\frac{20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}+\frac{3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac{(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac{b^6 (d+e x)^3}{3 e^7} \]

[Out]

(15*b^4*(b*d - a*e)^2*x)/e^6 - (b*d - a*e)^6/(3*e^7*(d + e*x)^3) + (3*b*(b*d - a*e)^5)/(e^7*(d + e*x)^2) - (15
*b^2*(b*d - a*e)^4)/(e^7*(d + e*x)) - (3*b^5*(b*d - a*e)*(d + e*x)^2)/e^7 + (b^6*(d + e*x)^3)/(3*e^7) - (20*b^
3*(b*d - a*e)^3*Log[d + e*x])/e^7

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Rubi [A]  time = 0.160576, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ -\frac{3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac{15 b^4 x (b d-a e)^2}{e^6}-\frac{15 b^2 (b d-a e)^4}{e^7 (d+e x)}-\frac{20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}+\frac{3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac{(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac{b^6 (d+e x)^3}{3 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*b^4*(b*d - a*e)^2*x)/e^6 - (b*d - a*e)^6/(3*e^7*(d + e*x)^3) + (3*b*(b*d - a*e)^5)/(e^7*(d + e*x)^2) - (15
*b^2*(b*d - a*e)^4)/(e^7*(d + e*x)) - (3*b^5*(b*d - a*e)*(d + e*x)^2)/e^7 + (b^6*(d + e*x)^3)/(3*e^7) - (20*b^
3*(b*d - a*e)^3*Log[d + e*x])/e^7

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{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^4} \, dx\\ &=\int \left (\frac{15 b^4 (b d-a e)^2}{e^6}+\frac{(-b d+a e)^6}{e^6 (d+e x)^4}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^3}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^2}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)}-\frac{6 b^5 (b d-a e) (d+e x)}{e^6}+\frac{b^6 (d+e x)^2}{e^6}\right ) \, dx\\ &=\frac{15 b^4 (b d-a e)^2 x}{e^6}-\frac{(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac{3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac{15 b^2 (b d-a e)^4}{e^7 (d+e x)}-\frac{3 b^5 (b d-a e) (d+e x)^2}{e^7}+\frac{b^6 (d+e x)^3}{3 e^7}-\frac{20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.11547, size = 302, normalized size = 1.94 \[ \frac{15 a^2 b^4 e^2 \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )+10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )-15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-3 a^5 b e^5 (d+3 e x)-a^6 e^6+3 a b^5 e \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )-60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left (39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-51 d^5 e x-37 d^6-3 d e^5 x^5+e^6 x^6\right )}{3 e^7 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^6*e^6) - 3*a^5*b*e^5*(d + 3*e*x) - 15*a^4*b^2*e^4*(d^2 + 3*d*e*x + 3*e^2*x^2) + 10*a^3*b^3*d*e^3*(11*d^2
+ 27*d*e*x + 18*e^2*x^2) + 15*a^2*b^4*e^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + 3
*a*b^5*e*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) + b^6*(-37*d^6 - 51
*d^5*e*x + 39*d^4*e^2*x^2 + 73*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 3*d*e^5*x^5 + e^6*x^6) - 60*b^3*(b*d - a*e)^3*(d
+ e*x)^3*Log[d + e*x])/(3*e^7*(d + e*x)^3)

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Maple [B]  time = 0.051, size = 483, normalized size = 3.1 \begin{align*} -30\,{\frac{{a}^{3}{b}^{3}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-5\,{\frac{{a}^{2}{b}^{4}{d}^{4}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+60\,{\frac{{a}^{3}{b}^{3}d}{{e}^{4} \left ( ex+d \right ) }}-90\,{\frac{{a}^{2}{b}^{4}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}+60\,{\frac{a{b}^{5}{d}^{3}}{{e}^{6} \left ( ex+d \right ) }}+20\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{3}}{{e}^{4}}}-20\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{3}}{{e}^{7}}}-15\,{\frac{{a}^{4}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}-15\,{\frac{{b}^{6}{d}^{4}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{{b}^{5}{x}^{2}a}{{e}^{4}}}-2\,{\frac{{b}^{6}{x}^{2}d}{{e}^{5}}}+15\,{\frac{{a}^{2}{b}^{4}x}{{e}^{4}}}+10\,{\frac{{b}^{6}{d}^{2}x}{{e}^{6}}}-{\frac{{d}^{6}{b}^{6}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-3\,{\frac{{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{6}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{2}}}+{\frac{20\,{a}^{3}{d}^{3}{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+2\,{\frac{a{b}^{5}{d}^{5}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+15\,{\frac{{a}^{4}{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-5\,{\frac{{d}^{2}{a}^{4}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-24\,{\frac{a{b}^{5}dx}{{e}^{5}}}+2\,{\frac{d{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{6}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{6}{x}^{3}}{3\,{e}^{4}}}+30\,{\frac{{a}^{2}{b}^{4}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-15\,{\frac{a{b}^{5}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-60\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{2}d}{{e}^{5}}}+60\,{\frac{{b}^{5}\ln \left ( ex+d \right ) a{d}^{2}}{{e}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30*b^3/e^4/(e*x+d)^2*a^3*d^2-5/e^5/(e*x+d)^3*a^2*b^4*d^4+60*b^3/e^4/(e*x+d)*a^3*d-90*b^4/e^5/(e*x+d)*d^2*a^2+
60*b^5/e^6/(e*x+d)*a*d^3+20*b^3/e^4*ln(e*x+d)*a^3-20*b^6/e^7*ln(e*x+d)*d^3-15*b^2/e^3/(e*x+d)*a^4-15*b^6/e^7/(
e*x+d)*d^4+3*b^5/e^4*x^2*a-2*b^6/e^5*x^2*d+15*b^4/e^4*a^2*x+10*b^6/e^6*d^2*x-1/3/e^7/(e*x+d)^3*d^6*b^6-3*b/e^2
/(e*x+d)^2*a^5+3*b^6/e^7/(e*x+d)^2*d^5+20/3/e^4/(e*x+d)^3*d^3*a^3*b^3+2/e^6/(e*x+d)^3*a*b^5*d^5+15*b^2/e^3/(e*
x+d)^2*a^4*d-5/e^3/(e*x+d)^3*d^2*a^4*b^2-24*b^5/e^5*a*d*x+2/e^2/(e*x+d)^3*d*a^5*b-1/3/e/(e*x+d)^3*a^6+1/3*b^6/
e^4*x^3+30*b^4/e^5/(e*x+d)^2*a^2*d^3-15*b^5/e^6/(e*x+d)^2*a*d^4-60*b^4/e^5*ln(e*x+d)*a^2*d+60*b^5/e^6*ln(e*x+d
)*a*d^2

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Maxima [B]  time = 1.05284, size = 505, normalized size = 3.24 \begin{align*} -\frac{37 \, b^{6} d^{6} - 141 \, a b^{5} d^{5} e + 195 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + a^{6} e^{6} + 45 \,{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (9 \, b^{6} d^{5} e - 35 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} - 30 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac{b^{6} e^{2} x^{3} - 3 \,{\left (2 \, b^{6} d e - 3 \, a b^{5} e^{2}\right )} x^{2} + 3 \,{\left (10 \, b^{6} d^{2} - 24 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x}{3 \, e^{6}} - \frac{20 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(37*b^6*d^6 - 141*a*b^5*d^5*e + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 3*a^5*b*
d*e^5 + a^6*e^6 + 45*(b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 +
9*(9*b^6*d^5*e - 35*a*b^5*d^4*e^2 + 50*a^2*b^4*d^3*e^3 - 30*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)
/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7) + 1/3*(b^6*e^2*x^3 - 3*(2*b^6*d*e - 3*a*b^5*e^2)*x^2 + 3*(10
*b^6*d^2 - 24*a*b^5*d*e + 15*a^2*b^4*e^2)*x)/e^6 - 20*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3
)*log(e*x + d)/e^7

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Fricas [B]  time = 1.76712, size = 1161, normalized size = 7.44 \begin{align*} \frac{b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \,{\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} +{\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \,{\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \,{\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} +{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^6*e^6*x^6 - 37*b^6*d^6 + 141*a*b^5*d^5*e - 195*a^2*b^4*d^4*e^2 + 110*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e
^4 - 3*a^5*b*d*e^5 - a^6*e^6 - 3*(b^6*d*e^5 - 3*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 3*a*b^5*d*e^5 + 3*a^2*b^4*e
^6)*x^4 + (73*b^6*d^3*e^3 - 189*a*b^5*d^2*e^4 + 135*a^2*b^4*d*e^5)*x^3 + 3*(13*b^6*d^4*e^2 - 9*a*b^5*d^3*e^3 -
45*a^2*b^4*d^2*e^4 + 60*a^3*b^3*d*e^5 - 15*a^4*b^2*e^6)*x^2 - 3*(17*b^6*d^5*e - 81*a*b^5*d^4*e^2 + 135*a^2*b^
4*d^3*e^3 - 90*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - 3*a*b^5*d^5*e + 3*a^2*b^4*d
^4*e^2 - a^3*b^3*d^3*e^3 + (b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 3*(b^6*d^4*e^
2 - 3*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 - a^3*b^3*d*e^5)*x^2 + 3*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*
e^3 - a^3*b^3*d^2*e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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Sympy [B]  time = 6.69639, size = 364, normalized size = 2.33 \begin{align*} \frac{b^{6} x^{3}}{3 e^{4}} + \frac{20 b^{3} \left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} + 3 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 110 a^{3} b^{3} d^{3} e^{3} + 195 a^{2} b^{4} d^{4} e^{2} - 141 a b^{5} d^{5} e + 37 b^{6} d^{6} + x^{2} \left (45 a^{4} b^{2} e^{6} - 180 a^{3} b^{3} d e^{5} + 270 a^{2} b^{4} d^{2} e^{4} - 180 a b^{5} d^{3} e^{3} + 45 b^{6} d^{4} e^{2}\right ) + x \left (9 a^{5} b e^{6} + 45 a^{4} b^{2} d e^{5} - 270 a^{3} b^{3} d^{2} e^{4} + 450 a^{2} b^{4} d^{3} e^{3} - 315 a b^{5} d^{4} e^{2} + 81 b^{6} d^{5} e\right )}{3 d^{3} e^{7} + 9 d^{2} e^{8} x + 9 d e^{9} x^{2} + 3 e^{10} x^{3}} + \frac{x^{2} \left (3 a b^{5} e - 2 b^{6} d\right )}{e^{5}} + \frac{x \left (15 a^{2} b^{4} e^{2} - 24 a b^{5} d e + 10 b^{6} d^{2}\right )}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**6*x**3/(3*e**4) + 20*b**3*(a*e - b*d)**3*log(d + e*x)/e**7 - (a**6*e**6 + 3*a**5*b*d*e**5 + 15*a**4*b**2*d*
*2*e**4 - 110*a**3*b**3*d**3*e**3 + 195*a**2*b**4*d**4*e**2 - 141*a*b**5*d**5*e + 37*b**6*d**6 + x**2*(45*a**4
*b**2*e**6 - 180*a**3*b**3*d*e**5 + 270*a**2*b**4*d**2*e**4 - 180*a*b**5*d**3*e**3 + 45*b**6*d**4*e**2) + x*(9
*a**5*b*e**6 + 45*a**4*b**2*d*e**5 - 270*a**3*b**3*d**2*e**4 + 450*a**2*b**4*d**3*e**3 - 315*a*b**5*d**4*e**2
+ 81*b**6*d**5*e))/(3*d**3*e**7 + 9*d**2*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) + x**2*(3*a*b**5*e - 2*b**6*d)
/e**5 + x*(15*a**2*b**4*e**2 - 24*a*b**5*d*e + 10*b**6*d**2)/e**6

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Giac [B]  time = 1.11391, size = 452, normalized size = 2.9 \begin{align*} -20 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (b^{6} x^{3} e^{8} - 6 \, b^{6} d x^{2} e^{7} + 30 \, b^{6} d^{2} x e^{6} + 9 \, a b^{5} x^{2} e^{8} - 72 \, a b^{5} d x e^{7} + 45 \, a^{2} b^{4} x e^{8}\right )} e^{\left (-12\right )} - \frac{{\left (37 \, b^{6} d^{6} - 141 \, a b^{5} d^{5} e + 195 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + a^{6} e^{6} + 45 \,{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (9 \, b^{6} d^{5} e - 35 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} - 30 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*e^(-7)*log(abs(x*e + d)) + 1/3*(b^6*x^3*e^8 - 6*
b^6*d*x^2*e^7 + 30*b^6*d^2*x*e^6 + 9*a*b^5*x^2*e^8 - 72*a*b^5*d*x*e^7 + 45*a^2*b^4*x*e^8)*e^(-12) - 1/3*(37*b^
6*d^6 - 141*a*b^5*d^5*e + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 3*a^5*b*d*e^5 + a^6
*e^6 + 45*(b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 9*(9*b^6*d
^5*e - 35*a*b^5*d^4*e^2 + 50*a^2*b^4*d^3*e^3 - 30*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*e^(-7)/(x*
e + d)^3