3.1491 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^2} \, dx\)

Optimal. Leaf size=156 \[ -\frac{3 b^5 (d+e x)^4 (b d-a e)}{2 e^7}+\frac{5 b^4 (d+e x)^3 (b d-a e)^2}{e^7}-\frac{10 b^3 (d+e x)^2 (b d-a e)^3}{e^7}+\frac{15 b^2 x (b d-a e)^4}{e^6}-\frac{(b d-a e)^6}{e^7 (d+e x)}-\frac{6 b (b d-a e)^5 \log (d+e x)}{e^7}+\frac{b^6 (d+e x)^5}{5 e^7} \]

[Out]

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

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Rubi [A]  time = 0.212552, 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} \[ -\frac{3 b^5 (d+e x)^4 (b d-a e)}{2 e^7}+\frac{5 b^4 (d+e x)^3 (b d-a e)^2}{e^7}-\frac{10 b^3 (d+e x)^2 (b d-a e)^3}{e^7}+\frac{15 b^2 x (b d-a e)^4}{e^6}-\frac{(b d-a e)^6}{e^7 (d+e x)}-\frac{6 b (b d-a e)^5 \log (d+e x)}{e^7}+\frac{b^6 (d+e x)^5}{5 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.103336, size = 302, normalized size = 1.94 \[ \frac{50 a^2 b^4 e^2 \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+100 a^3 b^3 e^3 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+150 a^4 b^2 e^4 \left (-d^2+d e x+e^2 x^2\right )+60 a^5 b d e^5-10 a^6 e^6+5 a b^5 e \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )-60 b (d+e x) (b d-a e)^5 \log (d+e x)+b^6 \left (30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4+50 d^5 e x-10 d^6-3 d e^5 x^5+2 e^6 x^6\right )}{10 e^7 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a^5*b*d*e^5 - 10*a^6*e^6 + 150*a^4*b^2*e^4*(-d^2 + d*e*x + e^2*x^2) + 100*a^3*b^3*e^3*(2*d^3 - 4*d^2*e*x -
3*d*e^2*x^2 + e^3*x^3) + 50*a^2*b^4*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b^
5*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + b^6*(-10*d^6 + 50*d^5*
e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) - 60*b*(b*d - a*e)^5*(d + e*x
)*Log[d + e*x])/(10*e^7*(d + e*x))

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Maple [B]  time = 0.049, size = 440, normalized size = 2.8 \begin{align*} -4\,{\frac{{b}^{5}{x}^{3}ad}{{e}^{3}}}+20\,{\frac{{a}^{3}{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-15\,{\frac{{a}^{2}{b}^{4}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}-15\,{\frac{{b}^{4}{x}^{2}{a}^{2}d}{{e}^{3}}}+6\,{\frac{a{b}^{5}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}-30\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{4}d}{{e}^{3}}}+60\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{3}{d}^{2}}{{e}^{4}}}-60\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{2}{d}^{3}}{{e}^{5}}}+30\,{\frac{{b}^{5}\ln \left ( ex+d \right ) a{d}^{4}}{{e}^{6}}}+6\,{\frac{d{a}^{5}b}{{e}^{2} \left ( ex+d \right ) }}-15\,{\frac{{d}^{2}{a}^{4}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{{b}^{6}{x}^{5}}{5\,{e}^{2}}}-{\frac{{a}^{6}}{e \left ( ex+d \right ) }}+9\,{\frac{{b}^{5}{x}^{2}a{d}^{2}}{{e}^{4}}}-40\,{\frac{{a}^{3}{b}^{3}dx}{{e}^{3}}}+45\,{\frac{{a}^{2}{b}^{4}{d}^{2}x}{{e}^{4}}}-24\,{\frac{a{b}^{5}{d}^{3}x}{{e}^{5}}}+5\,{\frac{{b}^{6}{d}^{4}x}{{e}^{6}}}+{\frac{3\,{b}^{5}{x}^{4}a}{2\,{e}^{2}}}-{\frac{{b}^{6}{x}^{4}d}{2\,{e}^{3}}}+5\,{\frac{{b}^{4}{x}^{3}{a}^{2}}{{e}^{2}}}+{\frac{{b}^{6}{x}^{3}{d}^{2}}{{e}^{4}}}+10\,{\frac{{x}^{2}{a}^{3}{b}^{3}}{{e}^{2}}}-2\,{\frac{{b}^{6}{x}^{2}{d}^{3}}{{e}^{5}}}+15\,{\frac{{a}^{4}{b}^{2}x}{{e}^{2}}}+6\,{\frac{b\ln \left ( ex+d \right ){a}^{5}}{{e}^{2}}}-6\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{5}}{{e}^{7}}}-{\frac{{d}^{6}{b}^{6}}{{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)^2,x)

[Out]

-4*b^5/e^3*x^3*a*d+20/e^4/(e*x+d)*a^3*b^3*d^3-15/e^5/(e*x+d)*a^2*b^4*d^4-15*b^4/e^3*x^2*a^2*d+6/e^6/(e*x+d)*a*
b^5*d^5-30*b^2/e^3*ln(e*x+d)*a^4*d+60*b^3/e^4*ln(e*x+d)*a^3*d^2-60*b^4/e^5*ln(e*x+d)*a^2*d^3+30*b^5/e^6*ln(e*x
+d)*a*d^4+6/e^2/(e*x+d)*d*a^5*b-15/e^3/(e*x+d)*d^2*a^4*b^2+1/5*b^6/e^2*x^5-1/e/(e*x+d)*a^6+9*b^5/e^4*x^2*a*d^2
-40*b^3/e^3*a^3*d*x+45*b^4/e^4*d^2*a^2*x-24*b^5/e^5*a*d^3*x+5*b^6/e^6*d^4*x+3/2*b^5/e^2*x^4*a-1/2*b^6/e^3*x^4*
d+5*b^4/e^2*x^3*a^2+b^6/e^4*x^3*d^2+10*b^3/e^2*x^2*a^3-2*b^6/e^5*x^2*d^3+15*b^2/e^2*a^4*x+6*b/e^2*ln(e*x+d)*a^
5-6*b^6/e^7*ln(e*x+d)*d^5-1/e^7/(e*x+d)*d^6*b^6

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Maxima [B]  time = 1.20369, size = 482, normalized size = 3.09 \begin{align*} -\frac{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, 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} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}}{e^{8} x + d e^{7}} + \frac{2 \, b^{6} e^{4} x^{5} - 5 \,{\left (b^{6} d e^{3} - 3 \, a b^{5} e^{4}\right )} x^{4} + 10 \,{\left (b^{6} d^{2} e^{2} - 4 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{3} - 10 \,{\left (2 \, b^{6} d^{3} e - 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} - 10 \, a^{3} b^{3} e^{4}\right )} x^{2} + 10 \,{\left (5 \, b^{6} d^{4} - 24 \, a b^{5} d^{3} e + 45 \, a^{2} b^{4} d^{2} e^{2} - 40 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x}{10 \, e^{6}} - \frac{6 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\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)^2,x, algorithm="maxima")

[Out]

-(b^6*d^6 - 6*a*b^5*d^5*e + 15*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 - 6*a^5*b*d*e^5 + a^6
*e^6)/(e^8*x + d*e^7) + 1/10*(2*b^6*e^4*x^5 - 5*(b^6*d*e^3 - 3*a*b^5*e^4)*x^4 + 10*(b^6*d^2*e^2 - 4*a*b^5*d*e^
3 + 5*a^2*b^4*e^4)*x^3 - 10*(2*b^6*d^3*e - 9*a*b^5*d^2*e^2 + 15*a^2*b^4*d*e^3 - 10*a^3*b^3*e^4)*x^2 + 10*(5*b^
6*d^4 - 24*a*b^5*d^3*e + 45*a^2*b^4*d^2*e^2 - 40*a^3*b^3*d*e^3 + 15*a^4*b^2*e^4)*x)/e^6 - 6*(b^6*d^5 - 5*a*b^5
*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*log(e*x + d)/e^7

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Fricas [B]  time = 1.81108, size = 1018, normalized size = 6.53 \begin{align*} \frac{2 \, b^{6} e^{6} x^{6} - 10 \, b^{6} d^{6} + 60 \, a b^{5} d^{5} e - 150 \, a^{2} b^{4} d^{4} e^{2} + 200 \, a^{3} b^{3} d^{3} e^{3} - 150 \, a^{4} b^{2} d^{2} e^{4} + 60 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - 5 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} - 10 \,{\left (b^{6} d^{3} e^{3} - 5 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} - 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (b^{6} d^{4} e^{2} - 5 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} - 10 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (5 \, b^{6} d^{5} e - 24 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} - 40 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5}\right )} x - 60 \,{\left (b^{6} d^{6} - 5 \, a b^{5} d^{5} e + 10 \, 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} - a^{5} b d e^{5} +{\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 )} \log \left (e x + d\right )}{10 \,{\left (e^{8} x + d 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)^2,x, algorithm="fricas")

[Out]

1/10*(2*b^6*e^6*x^6 - 10*b^6*d^6 + 60*a*b^5*d^5*e - 150*a^2*b^4*d^4*e^2 + 200*a^3*b^3*d^3*e^3 - 150*a^4*b^2*d^
2*e^4 + 60*a^5*b*d*e^5 - 10*a^6*e^6 - 3*(b^6*d*e^5 - 5*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 5*a*b^5*d*e^5 + 10*a^
2*b^4*e^6)*x^4 - 10*(b^6*d^3*e^3 - 5*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 - 10*a^3*b^3*e^6)*x^3 + 30*(b^6*d^4*e^2
- 5*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 - 10*a^3*b^3*d*e^5 + 5*a^4*b^2*e^6)*x^2 + 10*(5*b^6*d^5*e - 24*a*b^5*d^
4*e^2 + 45*a^2*b^4*d^3*e^3 - 40*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5)*x - 60*(b^6*d^6 - 5*a*b^5*d^5*e + 10*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 - a^5*b*d*e^5 + (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)*log(e*x + d))/(e^8*x + d*e^7)

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Sympy [B]  time = 1.83105, size = 303, normalized size = 1.94 \begin{align*} \frac{b^{6} x^{5}}{5 e^{2}} + \frac{6 b \left (a e - b d\right )^{5} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}}{d e^{7} + e^{8} x} + \frac{x^{4} \left (3 a b^{5} e - b^{6} d\right )}{2 e^{3}} + \frac{x^{3} \left (5 a^{2} b^{4} e^{2} - 4 a b^{5} d e + b^{6} d^{2}\right )}{e^{4}} + \frac{x^{2} \left (10 a^{3} b^{3} e^{3} - 15 a^{2} b^{4} d e^{2} + 9 a b^{5} d^{2} e - 2 b^{6} d^{3}\right )}{e^{5}} + \frac{x \left (15 a^{4} b^{2} e^{4} - 40 a^{3} b^{3} d e^{3} + 45 a^{2} b^{4} d^{2} e^{2} - 24 a b^{5} d^{3} e + 5 b^{6} d^{4}\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)**2,x)

[Out]

b**6*x**5/(5*e**2) + 6*b*(a*e - b*d)**5*log(d + e*x)/e**7 - (a**6*e**6 - 6*a**5*b*d*e**5 + 15*a**4*b**2*d**2*e
**4 - 20*a**3*b**3*d**3*e**3 + 15*a**2*b**4*d**4*e**2 - 6*a*b**5*d**5*e + b**6*d**6)/(d*e**7 + e**8*x) + x**4*
(3*a*b**5*e - b**6*d)/(2*e**3) + x**3*(5*a**2*b**4*e**2 - 4*a*b**5*d*e + b**6*d**2)/e**4 + x**2*(10*a**3*b**3*
e**3 - 15*a**2*b**4*d*e**2 + 9*a*b**5*d**2*e - 2*b**6*d**3)/e**5 + x*(15*a**4*b**2*e**4 - 40*a**3*b**3*d*e**3
+ 45*a**2*b**4*d**2*e**2 - 24*a*b**5*d**3*e + 5*b**6*d**4)/e**6

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Giac [B]  time = 1.18799, size = 579, normalized size = 3.71 \begin{align*} \frac{1}{10} \,{\left (2 \, b^{6} - \frac{15 \,{\left (b^{6} d e - a b^{5} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{50 \,{\left (b^{6} d^{2} e^{2} - 2 \, a b^{5} d e^{3} + a^{2} b^{4} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{100 \,{\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 )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{150 \,{\left (b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )}{\left (x e + d\right )}^{5} e^{\left (-7\right )} + 6 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{b^{6} d^{6} e^{5}}{x e + d} - \frac{6 \, a b^{5} d^{5} e^{6}}{x e + d} + \frac{15 \, a^{2} b^{4} d^{4} e^{7}}{x e + d} - \frac{20 \, a^{3} b^{3} d^{3} e^{8}}{x e + d} + \frac{15 \, a^{4} b^{2} d^{2} e^{9}}{x e + d} - \frac{6 \, a^{5} b d e^{10}}{x e + d} + \frac{a^{6} e^{11}}{x e + d}\right )} e^{\left (-12\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)^2,x, algorithm="giac")

[Out]

1/10*(2*b^6 - 15*(b^6*d*e - a*b^5*e^2)*e^(-1)/(x*e + d) + 50*(b^6*d^2*e^2 - 2*a*b^5*d*e^3 + a^2*b^4*e^4)*e^(-2
)/(x*e + d)^2 - 100*(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)*e^(-3)/(x*e + d)^3 + 150*(
b^6*d^4*e^4 - 4*a*b^5*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 4*a^3*b^3*d*e^7 + a^4*b^2*e^8)*e^(-4)/(x*e + d)^4)*(x*e +
d)^5*e^(-7) + 6*(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e
^5)*e^(-7)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (b^6*d^6*e^5/(x*e + d) - 6*a*b^5*d^5*e^6/(x*e + d) + 15*a^2*
b^4*d^4*e^7/(x*e + d) - 20*a^3*b^3*d^3*e^8/(x*e + d) + 15*a^4*b^2*d^2*e^9/(x*e + d) - 6*a^5*b*d*e^10/(x*e + d)
+ a^6*e^11/(x*e + d))*e^(-12)