3.1496 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^7} \, dx\)

Optimal. Leaf size=167 \[ \frac{6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac{15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac{20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac{15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac{6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac{(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac{b^6 \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]  time = 0.131526, antiderivative size = 167, 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} \[ \frac{6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac{15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac{20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac{15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac{6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac{(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac{b^6 \log (d+e x)}{e^7} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^7,x]

[Out]

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

Mathematica [A]  time = 0.125531, size = 233, normalized size = 1.4 \[ \frac{\frac{(b d-a e) \left (a^2 b^3 e^2 \left (282 d^2 e x+57 d^3+525 d e^2 x^2+400 e^3 x^3\right )+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+2 a^4 b e^4 (11 d+36 e x)+10 a^5 e^5+a b^4 e \left (975 d^2 e^2 x^2+462 d^3 e x+87 d^4+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )\right )}{(d+e x)^6}+60 b^6 \log (d+e x)}{60 e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^7,x]

[Out]

(((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 + 162*d*e*x + 225*e^2*x^2) + a^2
*b^3*e^2*(57*d^3 + 282*d^2*e*x + 525*d*e^2*x^2 + 400*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*x + 975*d^2*e^2*x^
2 + 1000*d*e^3*x^3 + 450*e^4*x^4) + b^5*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*
e^4*x^4 + 360*e^5*x^5)))/(d + e*x)^6 + 60*b^6*Log[d + e*x])/(60*e^7)

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Maple [B]  time = 0.048, size = 513, normalized size = 3.1 \begin{align*}{\frac{{b}^{6}\ln \left ( ex+d \right ) }{{e}^{7}}}-12\,{\frac{{a}^{3}{b}^{3}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{5}}}+12\,{\frac{{a}^{2}{b}^{4}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{5}}}+15\,{\frac{{a}^{3}{b}^{3}d}{{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{45\,{a}^{2}{b}^{4}{d}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}+15\,{\frac{a{b}^{5}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{4}}}+15\,{\frac{a{b}^{5}d}{{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{{a}^{5}bd}{{e}^{2} \left ( ex+d \right ) ^{6}}}-{\frac{5\,{a}^{4}{b}^{2}{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{6}}}+{\frac{10\,{a}^{3}{b}^{3}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{5\,{a}^{2}{b}^{4}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}+{\frac{a{b}^{5}{d}^{5}}{{e}^{6} \left ( ex+d \right ) ^{6}}}-6\,{\frac{a{b}^{5}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{{d}^{6}{b}^{6}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{15\,{a}^{4}{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{b}^{6}{d}^{4}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{6\,{a}^{5}b}{5\,{e}^{2} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{b}^{6}{d}^{5}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+20\,{\frac{{a}^{2}{b}^{4}d}{{e}^{5} \left ( ex+d \right ) ^{3}}}-20\,{\frac{a{b}^{5}{d}^{2}}{{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{6}}{6\,e \left ( ex+d \right ) ^{6}}}-{\frac{15\,{a}^{2}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{b}^{6}{d}^{2}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-6\,{\frac{a{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}+6\,{\frac{{b}^{6}d}{{e}^{7} \left ( ex+d \right ) }}-{\frac{20\,{a}^{3}{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{20\,{b}^{6}{d}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+6\,{\frac{{a}^{4}{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{5}}} \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)^7,x)

[Out]

b^6*ln(e*x+d)/e^7-12*b^3/e^4/(e*x+d)^5*a^3*d^2+12*b^4/e^5/(e*x+d)^5*a^2*d^3+15*b^3/e^4/(e*x+d)^4*a^3*d-45/2*b^
4/e^5/(e*x+d)^4*d^2*a^2+15*b^5/e^6/(e*x+d)^4*a*d^3+15*b^5/e^6/(e*x+d)^2*a*d+1/e^2/(e*x+d)^6*a^5*b*d-5/2/e^3/(e
*x+d)^6*d^2*a^4*b^2+10/3/e^4/(e*x+d)^6*a^3*b^3*d^3-5/2/e^5/(e*x+d)^6*a^2*b^4*d^4+1/e^6/(e*x+d)^6*a*b^5*d^5-6*b
^5/e^6/(e*x+d)^5*a*d^4-1/6/e^7/(e*x+d)^6*d^6*b^6-15/4*b^2/e^3/(e*x+d)^4*a^4-15/4*b^6/e^7/(e*x+d)^4*d^4-6/5*b/e
^2/(e*x+d)^5*a^5+6/5*b^6/e^7/(e*x+d)^5*d^5+20*b^4/e^5/(e*x+d)^3*a^2*d-20*b^5/e^6/(e*x+d)^3*a*d^2-1/6/e/(e*x+d)
^6*a^6-15/2*b^4/e^5/(e*x+d)^2*a^2-15/2*b^6/e^7/(e*x+d)^2*d^2-6*b^5/e^6/(e*x+d)*a+6*b^6/e^7/(e*x+d)*d-20/3*b^3/
e^4/(e*x+d)^3*a^3+20/3*b^6/e^7/(e*x+d)^3*d^3+6*b^2/e^3/(e*x+d)^5*a^4*d

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Maxima [B]  time = 1.27061, size = 562, normalized size = 3.37 \begin{align*} \frac{147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \,{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \,{\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \,{\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \,{\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{b^{6} \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)^7,x, algorithm="maxima")

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d
*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 +
200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d
^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a
^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^
11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + b^6*log(e*x + d)/e^7

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Fricas [B]  time = 1.84461, size = 1022, normalized size = 6.12 \begin{align*} \frac{147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \,{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \,{\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \,{\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \,{\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} e^{6} x^{6} + 6 \, b^{6} d e^{5} x^{5} + 15 \, b^{6} d^{2} e^{4} x^{4} + 20 \, b^{6} d^{3} e^{3} x^{3} + 15 \, b^{6} d^{4} e^{2} x^{2} + 6 \, b^{6} d^{5} e x + b^{6} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d
*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 +
200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d
^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a
^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x + 60*(b^6*e^6*x^6 + 6*b^6*d*e^5*x^5 +
15*b^6*d^2*e^4*x^4 + 20*b^6*d^3*e^3*x^3 + 15*b^6*d^4*e^2*x^2 + 6*b^6*d^5*e*x + b^6*d^6)*log(e*x + d))/(e^13*x
^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

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Sympy [B]  time = 96.2197, size = 439, normalized size = 2.63 \begin{align*} \frac{b^{6} \log{\left (d + e x \right )}}{e^{7}} - \frac{10 a^{6} e^{6} + 12 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} + 20 a^{3} b^{3} d^{3} e^{3} + 30 a^{2} b^{4} d^{4} e^{2} + 60 a b^{5} d^{5} e - 147 b^{6} d^{6} + x^{5} \left (360 a b^{5} e^{6} - 360 b^{6} d e^{5}\right ) + x^{4} \left (450 a^{2} b^{4} e^{6} + 900 a b^{5} d e^{5} - 1350 b^{6} d^{2} e^{4}\right ) + x^{3} \left (400 a^{3} b^{3} e^{6} + 600 a^{2} b^{4} d e^{5} + 1200 a b^{5} d^{2} e^{4} - 2200 b^{6} d^{3} e^{3}\right ) + x^{2} \left (225 a^{4} b^{2} e^{6} + 300 a^{3} b^{3} d e^{5} + 450 a^{2} b^{4} d^{2} e^{4} + 900 a b^{5} d^{3} e^{3} - 1875 b^{6} d^{4} e^{2}\right ) + x \left (72 a^{5} b e^{6} + 90 a^{4} b^{2} d e^{5} + 120 a^{3} b^{3} d^{2} e^{4} + 180 a^{2} b^{4} d^{3} e^{3} + 360 a b^{5} d^{4} e^{2} - 822 b^{6} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{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)**7,x)

[Out]

b**6*log(d + e*x)/e**7 - (10*a**6*e**6 + 12*a**5*b*d*e**5 + 15*a**4*b**2*d**2*e**4 + 20*a**3*b**3*d**3*e**3 +
30*a**2*b**4*d**4*e**2 + 60*a*b**5*d**5*e - 147*b**6*d**6 + x**5*(360*a*b**5*e**6 - 360*b**6*d*e**5) + x**4*(4
50*a**2*b**4*e**6 + 900*a*b**5*d*e**5 - 1350*b**6*d**2*e**4) + x**3*(400*a**3*b**3*e**6 + 600*a**2*b**4*d*e**5
+ 1200*a*b**5*d**2*e**4 - 2200*b**6*d**3*e**3) + x**2*(225*a**4*b**2*e**6 + 300*a**3*b**3*d*e**5 + 450*a**2*b
**4*d**2*e**4 + 900*a*b**5*d**3*e**3 - 1875*b**6*d**4*e**2) + x*(72*a**5*b*e**6 + 90*a**4*b**2*d*e**5 + 120*a*
*3*b**3*d**2*e**4 + 180*a**2*b**4*d**3*e**3 + 360*a*b**5*d**4*e**2 - 822*b**6*d**5*e))/(60*d**6*e**7 + 360*d**
5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6)

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Giac [B]  time = 1.15095, size = 458, normalized size = 2.74 \begin{align*} b^{6} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (360 \,{\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{5} + 450 \,{\left (3 \, b^{6} d^{2} e^{3} - 2 \, a b^{5} d e^{4} - a^{2} b^{4} e^{5}\right )} x^{4} + 200 \,{\left (11 \, b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} - 3 \, a^{2} b^{4} d e^{4} - 2 \, a^{3} b^{3} e^{5}\right )} x^{3} + 75 \,{\left (25 \, b^{6} d^{4} e - 12 \, a b^{5} d^{3} e^{2} - 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} - 3 \, a^{4} b^{2} e^{5}\right )} x^{2} + 6 \,{\left (137 \, b^{6} d^{5} - 60 \, a b^{5} d^{4} e - 30 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} - 15 \, a^{4} b^{2} d e^{4} - 12 \, a^{5} b e^{5}\right )} x +{\left (147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{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)^7,x, algorithm="giac")

[Out]

b^6*e^(-7)*log(abs(x*e + d)) + 1/60*(360*(b^6*d*e^4 - a*b^5*e^5)*x^5 + 450*(3*b^6*d^2*e^3 - 2*a*b^5*d*e^4 - a^
2*b^4*e^5)*x^4 + 200*(11*b^6*d^3*e^2 - 6*a*b^5*d^2*e^3 - 3*a^2*b^4*d*e^4 - 2*a^3*b^3*e^5)*x^3 + 75*(25*b^6*d^4
*e - 12*a*b^5*d^3*e^2 - 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 - 3*a^4*b^2*e^5)*x^2 + 6*(137*b^6*d^5 - 60*a*b^5*d
^4*e - 30*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 15*a^4*b^2*d*e^4 - 12*a^5*b*e^5)*x + (147*b^6*d^6 - 60*a*b^5*
d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 10*a^6*e^6)*e^(-1))*e^
(-6)/(x*e + d)^6