3.1495 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=155 \[ -\frac{15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac{10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac{5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}-\frac{6 b^5 (b d-a e) \log (d+e x)}{e^7}+\frac{3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac{(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac{b^6 x}{e^6} \]

[Out]

(b^6*x)/e^6 - (b*d - a*e)^6/(5*e^7*(d + e*x)^5) + (3*b*(b*d - a*e)^5)/(2*e^7*(d + e*x)^4) - (5*b^2*(b*d - a*e)
^4)/(e^7*(d + e*x)^3) + (10*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)^2) - (15*b^4*(b*d - a*e)^2)/(e^7*(d + e*x)) - (6
*b^5*(b*d - a*e)*Log[d + e*x])/e^7

________________________________________________________________________________________

Rubi [A]  time = 0.140043, antiderivative size = 155, 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} \[ -\frac{15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac{10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac{5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}-\frac{6 b^5 (b d-a e) \log (d+e x)}{e^7}+\frac{3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac{(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac{b^6 x}{e^6} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^6,x]

[Out]

(b^6*x)/e^6 - (b*d - a*e)^6/(5*e^7*(d + e*x)^5) + (3*b*(b*d - a*e)^5)/(2*e^7*(d + e*x)^4) - (5*b^2*(b*d - a*e)
^4)/(e^7*(d + e*x)^3) + (10*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)^2) - (15*b^4*(b*d - a*e)^2)/(e^7*(d + e*x)) - (6
*b^5*(b*d - a*e)*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)^6} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^6} \, dx\\ &=\int \left (\frac{b^6}{e^6}+\frac{(-b d+a e)^6}{e^6 (d+e x)^6}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^5}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^4}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^3}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^2}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{b^6 x}{e^6}-\frac{(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac{3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac{5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac{10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac{15 b^4 (b d-a e)^2}{e^7 (d+e x)}-\frac{6 b^5 (b d-a e) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.124327, size = 297, normalized size = 1.92 \[ -\frac{30 a^2 b^4 e^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )+10 a^3 b^3 e^3 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a^5 b e^5 (d+5 e x)+2 a^6 e^6-a b^5 d e \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )+60 b^5 (d+e x)^5 (b d-a e) \log (d+e x)+b^6 \left (600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4+375 d^5 e x+87 d^6-50 d e^5 x^5-10 e^6 x^6\right )}{10 e^7 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^6,x]

[Out]

-(2*a^6*e^6 + 3*a^5*b*e^5*(d + 5*e*x) + 5*a^4*b^2*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 10*a^3*b^3*e^3*(d^3 + 5*d
^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 30*a^2*b^4*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*
x^4) - a*b^5*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) + b^6*(87*d^6 + 375*
d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + 60*b^5*(b*d - a*e)
*(d + e*x)^5*Log[d + e*x])/(10*e^7*(d + e*x)^5)

________________________________________________________________________________________

Maple [B]  time = 0.052, size = 508, normalized size = 3.3 \begin{align*}{\frac{6\,d{a}^{5}b}{5\,{e}^{2} \left ( ex+d \right ) ^{5}}}-3\,{\frac{{d}^{2}{a}^{4}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{5}}}+4\,{\frac{{a}^{3}{d}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{5}}}-3\,{\frac{{d}^{4}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{5}}}+30\,{\frac{{a}^{2}{b}^{4}d}{{e}^{5} \left ( ex+d \right ) ^{2}}}-30\,{\frac{a{b}^{5}{d}^{2}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+20\,{\frac{{a}^{3}{b}^{3}d}{{e}^{4} \left ( ex+d \right ) ^{3}}}-30\,{\frac{{b}^{4}{d}^{2}{a}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+20\,{\frac{a{b}^{5}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{15\,{a}^{4}{b}^{2}d}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-15\,{\frac{{a}^{3}{b}^{3}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{4}}}+15\,{\frac{{a}^{2}{b}^{4}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{15\,a{b}^{5}{d}^{4}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}+{\frac{{b}^{6}x}{{e}^{6}}}-5\,{\frac{{b}^{6}{d}^{4}}{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{3\,{a}^{5}b}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}+{\frac{3\,{b}^{6}{d}^{5}}{2\,{e}^{7} \left ( ex+d \right ) ^{4}}}+6\,{\frac{{b}^{5}\ln \left ( ex+d \right ) a}{{e}^{6}}}-6\,{\frac{{b}^{6}\ln \left ( ex+d \right ) d}{{e}^{7}}}-10\,{\frac{{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{b}^{6}{d}^{3}}{{e}^{7} \left ( ex+d \right ) ^{2}}}-15\,{\frac{{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{{b}^{6}{d}^{2}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{{d}^{6}{b}^{6}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{d}^{5}a{b}^{5}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}+30\,{\frac{a{b}^{5}d}{{e}^{6} \left ( ex+d \right ) }}-{\frac{{a}^{6}}{5\,e \left ( ex+d \right ) ^{5}}}-5\,{\frac{{a}^{4}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}} \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)^6,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.09464, size = 536, normalized size = 3.46 \begin{align*} \frac{b^{6} x}{e^{6}} - \frac{87 \, b^{6} d^{6} - 137 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + 2 \, a^{6} e^{6} + 150 \,{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \,{\left (5 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 50 \,{\left (13 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 5 \,{\left (77 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} - \frac{6 \,{\left (b^{6} d - a b^{5} e\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)^6,x, algorithm="maxima")

[Out]

b^6*x/e^6 - 1/10*(87*b^6*d^6 - 137*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 +
3*a^5*b*d*e^5 + 2*a^6*e^6 + 150*(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 100*(5*b^6*d^3*e^3 - 9*a*b^
5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 50*(13*b^6*d^4*e^2 - 22*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 2
*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 5*(77*b^6*d^5*e - 125*a*b^5*d^4*e^2 + 30*a^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*
e^4 + 5*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^
8*x + d^5*e^7) - 6*(b^6*d - a*b^5*e)*log(e*x + d)/e^7

________________________________________________________________________________________

Fricas [B]  time = 1.88706, size = 1095, normalized size = 7.06 \begin{align*} \frac{10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \,{\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \,{\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \,{\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \,{\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - a b^{5} d^{5} e +{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \,{\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

1/10*(10*b^6*e^6*x^6 + 50*b^6*d*e^5*x^5 - 87*b^6*d^6 + 137*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 10*a^3*b^3*d^3*e
^3 - 5*a^4*b^2*d^2*e^4 - 3*a^5*b*d*e^5 - 2*a^6*e^6 - 50*(b^6*d^2*e^4 - 6*a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 - 10
0*(4*b^6*d^3*e^3 - 9*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 - 50*(12*b^6*d^4*e^2 - 22*a*b^5*d^3*e^
3 + 6*a^2*b^4*d^2*e^4 + 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 - 5*(75*b^6*d^5*e - 125*a*b^5*d^4*e^2 + 30*a^2*b^4*
d^3*e^3 + 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - a*b^5*d^5*e + (b^6*d*e^5 - a*b
^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - a*b^5*d*e^5)*x^4 + 10*(b^6*d^3*e^3 - a*b^5*d^2*e^4)*x^3 + 10*(b^6*d^4*e^2 - a*b
^5*d^3*e^3)*x^2 + 5*(b^6*d^5*e - a*b^5*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 +
10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

________________________________________________________________________________________

Sympy [B]  time = 36.0141, size = 420, normalized size = 2.71 \begin{align*} \frac{b^{6} x}{e^{6}} + \frac{6 b^{5} \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{2 a^{6} e^{6} + 3 a^{5} b d e^{5} + 5 a^{4} b^{2} d^{2} e^{4} + 10 a^{3} b^{3} d^{3} e^{3} + 30 a^{2} b^{4} d^{4} e^{2} - 137 a b^{5} d^{5} e + 87 b^{6} d^{6} + x^{4} \left (150 a^{2} b^{4} e^{6} - 300 a b^{5} d e^{5} + 150 b^{6} d^{2} e^{4}\right ) + x^{3} \left (100 a^{3} b^{3} e^{6} + 300 a^{2} b^{4} d e^{5} - 900 a b^{5} d^{2} e^{4} + 500 b^{6} d^{3} e^{3}\right ) + x^{2} \left (50 a^{4} b^{2} e^{6} + 100 a^{3} b^{3} d e^{5} + 300 a^{2} b^{4} d^{2} e^{4} - 1100 a b^{5} d^{3} e^{3} + 650 b^{6} d^{4} e^{2}\right ) + x \left (15 a^{5} b e^{6} + 25 a^{4} b^{2} d e^{5} + 50 a^{3} b^{3} d^{2} e^{4} + 150 a^{2} b^{4} d^{3} e^{3} - 625 a b^{5} d^{4} e^{2} + 385 b^{6} d^{5} e\right )}{10 d^{5} e^{7} + 50 d^{4} e^{8} x + 100 d^{3} e^{9} x^{2} + 100 d^{2} e^{10} x^{3} + 50 d e^{11} x^{4} + 10 e^{12} x^{5}} \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)**6,x)

[Out]

b**6*x/e**6 + 6*b**5*(a*e - b*d)*log(d + e*x)/e**7 - (2*a**6*e**6 + 3*a**5*b*d*e**5 + 5*a**4*b**2*d**2*e**4 +
10*a**3*b**3*d**3*e**3 + 30*a**2*b**4*d**4*e**2 - 137*a*b**5*d**5*e + 87*b**6*d**6 + x**4*(150*a**2*b**4*e**6
- 300*a*b**5*d*e**5 + 150*b**6*d**2*e**4) + x**3*(100*a**3*b**3*e**6 + 300*a**2*b**4*d*e**5 - 900*a*b**5*d**2*
e**4 + 500*b**6*d**3*e**3) + x**2*(50*a**4*b**2*e**6 + 100*a**3*b**3*d*e**5 + 300*a**2*b**4*d**2*e**4 - 1100*a
*b**5*d**3*e**3 + 650*b**6*d**4*e**2) + x*(15*a**5*b*e**6 + 25*a**4*b**2*d*e**5 + 50*a**3*b**3*d**2*e**4 + 150
*a**2*b**4*d**3*e**3 - 625*a*b**5*d**4*e**2 + 385*b**6*d**5*e))/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9
*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5)

________________________________________________________________________________________

Giac [B]  time = 1.23668, size = 447, normalized size = 2.88 \begin{align*} b^{6} x e^{\left (-6\right )} - 6 \,{\left (b^{6} d - a b^{5} e\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (87 \, b^{6} d^{6} - 137 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + 2 \, a^{6} e^{6} + 150 \,{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \,{\left (5 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 50 \,{\left (13 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 5 \,{\left (77 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{10 \,{\left (x e + d\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

b^6*x*e^(-6) - 6*(b^6*d - a*b^5*e)*e^(-7)*log(abs(x*e + d)) - 1/10*(87*b^6*d^6 - 137*a*b^5*d^5*e + 30*a^2*b^4*
d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 3*a^5*b*d*e^5 + 2*a^6*e^6 + 150*(b^6*d^2*e^4 - 2*a*b^5*d*e^
5 + a^2*b^4*e^6)*x^4 + 100*(5*b^6*d^3*e^3 - 9*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 50*(13*b^6*
d^4*e^2 - 22*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 5*(77*b^6*d^5*e - 125*a*
b^5*d^4*e^2 + 30*a^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x)*e^(-7)/(x*e + d)^5