### 3.1492 $$\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^3} \, dx$$

Optimal. Leaf size=158 $-\frac{2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac{15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac{20 b^3 x (b d-a e)^3}{e^6}+\frac{15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac{6 b (b d-a e)^5}{e^7 (d+e x)}-\frac{(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac{b^6 (d+e x)^4}{4 e^7}$

[Out]

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

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Rubi [A]  time = 0.179208, antiderivative size = 158, 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} $-\frac{2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac{15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac{20 b^3 x (b d-a e)^3}{e^6}+\frac{15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac{6 b (b d-a e)^5}{e^7 (d+e x)}-\frac{(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac{b^6 (d+e x)^4}{4 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.108777, size = 303, normalized size = 1.92 $\frac{30 a^2 b^4 e^2 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )+40 a^3 b^3 e^3 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )+30 a^4 b^2 d e^4 (3 d+4 e x)-12 a^5 b e^5 (d+2 e x)-2 a^6 e^6+4 a b^5 e \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )+60 b^2 (d+e x)^2 (b d-a e)^4 \log (d+e x)+b^6 \left (-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-16 d^5 e x+22 d^6-2 d e^5 x^5+e^6 x^6\right )}{4 e^7 (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*a^6*e^6 - 12*a^5*b*e^5*(d + 2*e*x) + 30*a^4*b^2*d*e^4*(3*d + 4*e*x) + 40*a^3*b^3*e^3*(-5*d^3 - 4*d^2*e*x +
4*d*e^2*x^2 + 2*e^3*x^3) + 30*a^2*b^4*e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 4*a*
b^5*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + b^6*(22*d^6 - 16*d^5
*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) + 60*b^2*(b*d - a*e)^4*(d + e*
x)^2*Log[d + e*x])/(4*e^7*(d + e*x)^2)

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Maple [B]  time = 0.052, size = 464, normalized size = 2.9 \begin{align*}{\frac{{b}^{6}{x}^{4}}{4\,{e}^{3}}}-{\frac{{a}^{6}}{2\,e \left ( ex+d \right ) ^{2}}}-9\,{\frac{{b}^{5}{x}^{2}ad}{{e}^{4}}}-45\,{\frac{{a}^{2}{b}^{4}dx}{{e}^{4}}}-{\frac{15\,{d}^{2}{a}^{4}{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{a}^{3}{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{a}^{2}{b}^{4}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+3\,{\frac{a{b}^{5}{d}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-60\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{3}d}{{e}^{4}}}+90\,{\frac{{b}^{4}\ln \left ( ex+d \right ){d}^{2}{a}^{2}}{{e}^{5}}}-60\,{\frac{{b}^{5}\ln \left ( ex+d \right ) a{d}^{3}}{{e}^{6}}}+30\,{\frac{{a}^{4}{b}^{2}d}{{e}^{3} \left ( ex+d \right ) }}-60\,{\frac{{a}^{3}{b}^{3}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+60\,{\frac{{a}^{2}{b}^{4}{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-30\,{\frac{a{b}^{5}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+36\,{\frac{a{b}^{5}{d}^{2}x}{{e}^{5}}}+3\,{\frac{d{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{b}^{5}{x}^{3}a}{{e}^{3}}}-{\frac{{b}^{6}{x}^{3}d}{{e}^{4}}}+{\frac{15\,{b}^{4}{x}^{2}{a}^{2}}{2\,{e}^{3}}}+3\,{\frac{{b}^{6}{x}^{2}{d}^{2}}{{e}^{5}}}+20\,{\frac{x{a}^{3}{b}^{3}}{{e}^{3}}}-10\,{\frac{{b}^{6}{d}^{3}x}{{e}^{6}}}-{\frac{{d}^{6}{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+15\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{4}}{{e}^{3}}}+15\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{4}}{{e}^{7}}}-6\,{\frac{{a}^{5}b}{{e}^{2} \left ( ex+d \right ) }}+6\,{\frac{{b}^{6}{d}^{5}}{{e}^{7} \left ( ex+d \right ) }} \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)^3,x)

[Out]

1/4*b^6/e^3*x^4-1/2/e/(e*x+d)^2*a^6-9*b^5/e^4*x^2*a*d-45*b^4/e^4*a^2*d*x-15/2/e^3/(e*x+d)^2*d^2*a^4*b^2+10/e^4
/(e*x+d)^2*a^3*b^3*d^3-15/2/e^5/(e*x+d)^2*a^2*b^4*d^4+3/e^6/(e*x+d)^2*a*b^5*d^5-60*b^3/e^4*ln(e*x+d)*a^3*d+90*
b^4/e^5*ln(e*x+d)*d^2*a^2-60*b^5/e^6*ln(e*x+d)*a*d^3+30*b^2/e^3/(e*x+d)*a^4*d-60*b^3/e^4/(e*x+d)*a^3*d^2+60*b^
4/e^5/(e*x+d)*a^2*d^3-30*b^5/e^6/(e*x+d)*a*d^4+36*b^5/e^5*a*d^2*x+3/e^2/(e*x+d)^2*d*a^5*b+2*b^5/e^3*x^3*a-b^6/
e^4*x^3*d+15/2*b^4/e^3*x^2*a^2+3*b^6/e^5*x^2*d^2+20*b^3/e^3*a^3*x-10*b^6/e^6*d^3*x-1/2/e^7/(e*x+d)^2*d^6*b^6+1
5*b^2/e^3*ln(e*x+d)*a^4+15*b^6/e^7*ln(e*x+d)*d^4-6*b/e^2/(e*x+d)*a^5+6*b^6/e^7/(e*x+d)*d^5

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Maxima [B]  time = 1.16169, size = 491, normalized size = 3.11 \begin{align*} \frac{11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \,{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, 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} - a^{5} b e^{6}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{b^{6} e^{3} x^{4} - 4 \,{\left (b^{6} d e^{2} - 2 \, a b^{5} e^{3}\right )} x^{3} + 6 \,{\left (2 \, b^{6} d^{2} e - 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{2} - 4 \,{\left (10 \, b^{6} d^{3} - 36 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} - 20 \, a^{3} b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac{15 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\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)^3,x, algorithm="maxima")

[Out]

1/2*(11*b^6*d^6 - 54*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 45*a^4*b^2*d^2*e^4 - 6*a^5*b*d*
e^5 - a^6*e^6 + 12*(b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*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 -
a^5*b*e^6)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(b^6*e^3*x^4 - 4*(b^6*d*e^2 - 2*a*b^5*e^3)*x^3 + 6*(2*b^6*
d^2*e - 6*a*b^5*d*e^2 + 5*a^2*b^4*e^3)*x^2 - 4*(10*b^6*d^3 - 36*a*b^5*d^2*e + 45*a^2*b^4*d*e^2 - 20*a^3*b^3*e^
3)*x)/e^6 + 15*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*log(e*x + d)/e^7

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Fricas [B]  time = 1.75028, size = 1110, normalized size = 7.03 \begin{align*} \frac{b^{6} e^{6} x^{6} + 22 \, b^{6} d^{6} - 108 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} d^{4} e^{2} - 200 \, a^{3} b^{3} d^{3} e^{3} + 90 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 2 \,{\left (b^{6} d e^{5} - 4 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{3} - 4 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} - 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 2 \,{\left (34 \, b^{6} d^{4} e^{2} - 126 \, a b^{5} d^{3} e^{3} + 165 \, a^{2} b^{4} d^{2} e^{4} - 80 \, a^{3} b^{3} d e^{5}\right )} x^{2} - 4 \,{\left (4 \, b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} - 15 \, a^{2} b^{4} d^{3} e^{3} + 40 \, a^{3} b^{3} d^{2} e^{4} - 30 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 4 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} - 4 \, a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} +{\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} + 2 \,{\left (b^{6} d^{5} e - 4 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} - 4 \, a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*e^6*x^6 + 22*b^6*d^6 - 108*a*b^5*d^5*e + 210*a^2*b^4*d^4*e^2 - 200*a^3*b^3*d^3*e^3 + 90*a^4*b^2*d^2*e
^4 - 12*a^5*b*d*e^5 - 2*a^6*e^6 - 2*(b^6*d*e^5 - 4*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 4*a*b^5*d*e^5 + 6*a^2*b^4
*e^6)*x^4 - 20*(b^6*d^3*e^3 - 4*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 - 4*a^3*b^3*e^6)*x^3 - 2*(34*b^6*d^4*e^2 - 126
*a*b^5*d^3*e^3 + 165*a^2*b^4*d^2*e^4 - 80*a^3*b^3*d*e^5)*x^2 - 4*(4*b^6*d^5*e - 6*a*b^5*d^4*e^2 - 15*a^2*b^4*d
^3*e^3 + 40*a^3*b^3*d^2*e^4 - 30*a^4*b^2*d*e^5 + 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 4*a*b^5*d^5*e + 6*a^2*b^4*d^4*
e^2 - 4*a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + (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 + 2*(b^6*d^5*e - 4*a*b^5*d^4*e^2 + 6*a^2*b^4*d^3*e^3 - 4*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5
)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [B]  time = 5.16562, size = 335, normalized size = 2.12 \begin{align*} \frac{b^{6} x^{4}}{4 e^{3}} + \frac{15 b^{2} \left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} + 6 a^{5} b d e^{5} - 45 a^{4} b^{2} d^{2} e^{4} + 100 a^{3} b^{3} d^{3} e^{3} - 105 a^{2} b^{4} d^{4} e^{2} + 54 a b^{5} d^{5} e - 11 b^{6} d^{6} + x \left (12 a^{5} b e^{6} - 60 a^{4} b^{2} d e^{5} + 120 a^{3} b^{3} d^{2} e^{4} - 120 a^{2} b^{4} d^{3} e^{3} + 60 a b^{5} d^{4} e^{2} - 12 b^{6} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{3} \left (2 a b^{5} e - b^{6} d\right )}{e^{4}} + \frac{x^{2} \left (15 a^{2} b^{4} e^{2} - 18 a b^{5} d e + 6 b^{6} d^{2}\right )}{2 e^{5}} + \frac{x \left (20 a^{3} b^{3} e^{3} - 45 a^{2} b^{4} d e^{2} + 36 a b^{5} d^{2} e - 10 b^{6} d^{3}\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)**3,x)

[Out]

b**6*x**4/(4*e**3) + 15*b**2*(a*e - b*d)**4*log(d + e*x)/e**7 - (a**6*e**6 + 6*a**5*b*d*e**5 - 45*a**4*b**2*d*
*2*e**4 + 100*a**3*b**3*d**3*e**3 - 105*a**2*b**4*d**4*e**2 + 54*a*b**5*d**5*e - 11*b**6*d**6 + x*(12*a**5*b*e
**6 - 60*a**4*b**2*d*e**5 + 120*a**3*b**3*d**2*e**4 - 120*a**2*b**4*d**3*e**3 + 60*a*b**5*d**4*e**2 - 12*b**6*
d**5*e))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2) + x**3*(2*a*b**5*e - b**6*d)/e**4 + x**2*(15*a**2*b**4*e**2
- 18*a*b**5*d*e + 6*b**6*d**2)/(2*e**5) + x*(20*a**3*b**3*e**3 - 45*a**2*b**4*d*e**2 + 36*a*b**5*d**2*e - 10*b
**6*d**3)/e**6

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Giac [B]  time = 1.14652, size = 460, normalized size = 2.91 \begin{align*} 15 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (b^{6} x^{4} e^{9} - 4 \, b^{6} d x^{3} e^{8} + 12 \, b^{6} d^{2} x^{2} e^{7} - 40 \, b^{6} d^{3} x e^{6} + 8 \, a b^{5} x^{3} e^{9} - 36 \, a b^{5} d x^{2} e^{8} + 144 \, a b^{5} d^{2} x e^{7} + 30 \, a^{2} b^{4} x^{2} e^{9} - 180 \, a^{2} b^{4} d x e^{8} + 80 \, a^{3} b^{3} x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \,{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, 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} - a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

15*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*e^(-7)*log(abs(x*e + d)) + 1/
4*(b^6*x^4*e^9 - 4*b^6*d*x^3*e^8 + 12*b^6*d^2*x^2*e^7 - 40*b^6*d^3*x*e^6 + 8*a*b^5*x^3*e^9 - 36*a*b^5*d*x^2*e^
8 + 144*a*b^5*d^2*x*e^7 + 30*a^2*b^4*x^2*e^9 - 180*a^2*b^4*d*x*e^8 + 80*a^3*b^3*x*e^9)*e^(-12) + 1/2*(11*b^6*d
^6 - 54*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 45*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 - a^6*e^6
+ 12*(b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*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 - a^5*b*e^6)*x)
*e^(-7)/(x*e + d)^2