3.1535 \(\int \frac{1}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^3} \, dx\)
Optimal. Leaf size=181 \[ -\frac{e^5}{(d+e x) (b d-a e)^6}-\frac{5 b e^4}{(a+b x) (b d-a e)^6}+\frac{2 b e^3}{(a+b x)^2 (b d-a e)^5}-\frac{b e^2}{(a+b x)^3 (b d-a e)^4}-\frac{6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac{6 b e^5 \log (d+e x)}{(b d-a e)^7}+\frac{b e}{2 (a+b x)^4 (b d-a e)^3}-\frac{b}{5 (a+b x)^5 (b d-a e)^2} \]
[Out]
-b/(5*(b*d - a*e)^2*(a + b*x)^5) + (b*e)/(2*(b*d - a*e)^3*(a + b*x)^4) - (b*e^2)/((b*d - a*e)^4*(a + b*x)^3) +
(2*b*e^3)/((b*d - a*e)^5*(a + b*x)^2) - (5*b*e^4)/((b*d - a*e)^6*(a + b*x)) - e^5/((b*d - a*e)^6*(d + e*x)) -
(6*b*e^5*Log[a + b*x])/(b*d - a*e)^7 + (6*b*e^5*Log[d + e*x])/(b*d - a*e)^7
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Rubi [A] time = 0.183939, antiderivative size = 181, 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, 44} \[ -\frac{e^5}{(d+e x) (b d-a e)^6}-\frac{5 b e^4}{(a+b x) (b d-a e)^6}+\frac{2 b e^3}{(a+b x)^2 (b d-a e)^5}-\frac{b e^2}{(a+b x)^3 (b d-a e)^4}-\frac{6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac{6 b e^5 \log (d+e x)}{(b d-a e)^7}+\frac{b e}{2 (a+b x)^4 (b d-a e)^3}-\frac{b}{5 (a+b x)^5 (b d-a e)^2} \]
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
-b/(5*(b*d - a*e)^2*(a + b*x)^5) + (b*e)/(2*(b*d - a*e)^3*(a + b*x)^4) - (b*e^2)/((b*d - a*e)^4*(a + b*x)^3) +
(2*b*e^3)/((b*d - a*e)^5*(a + b*x)^2) - (5*b*e^4)/((b*d - a*e)^6*(a + b*x)) - e^5/((b*d - a*e)^6*(d + e*x)) -
(6*b*e^5*Log[a + b*x])/(b*d - a*e)^7 + (6*b*e^5*Log[d + e*x])/(b*d - a*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 44
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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^6 (d+e x)^2} \, dx\\ &=\int \left (\frac{b^2}{(b d-a e)^2 (a+b x)^6}-\frac{2 b^2 e}{(b d-a e)^3 (a+b x)^5}+\frac{3 b^2 e^2}{(b d-a e)^4 (a+b x)^4}-\frac{4 b^2 e^3}{(b d-a e)^5 (a+b x)^3}+\frac{5 b^2 e^4}{(b d-a e)^6 (a+b x)^2}-\frac{6 b^2 e^5}{(b d-a e)^7 (a+b x)}+\frac{e^6}{(b d-a e)^6 (d+e x)^2}+\frac{6 b e^6}{(b d-a e)^7 (d+e x)}\right ) \, dx\\ &=-\frac{b}{5 (b d-a e)^2 (a+b x)^5}+\frac{b e}{2 (b d-a e)^3 (a+b x)^4}-\frac{b e^2}{(b d-a e)^4 (a+b x)^3}+\frac{2 b e^3}{(b d-a e)^5 (a+b x)^2}-\frac{5 b e^4}{(b d-a e)^6 (a+b x)}-\frac{e^5}{(b d-a e)^6 (d+e x)}-\frac{6 b e^5 \log (a+b x)}{(b d-a e)^7}+\frac{6 b e^5 \log (d+e x)}{(b d-a e)^7}\\ \end{align*}
Mathematica [A] time = 0.132863, size = 167, normalized size = 0.92 \[ \frac{\frac{10 e^5 (a e-b d)}{d+e x}-\frac{50 b e^4 (b d-a e)}{a+b x}+\frac{20 b e^3 (b d-a e)^2}{(a+b x)^2}-\frac{10 b e^2 (b d-a e)^3}{(a+b x)^3}+\frac{5 b e (b d-a e)^4}{(a+b x)^4}-\frac{2 b (b d-a e)^5}{(a+b x)^5}-60 b e^5 \log (a+b x)+60 b e^5 \log (d+e x)}{10 (b d-a e)^7} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
((-2*b*(b*d - a*e)^5)/(a + b*x)^5 + (5*b*e*(b*d - a*e)^4)/(a + b*x)^4 - (10*b*e^2*(b*d - a*e)^3)/(a + b*x)^3 +
(20*b*e^3*(b*d - a*e)^2)/(a + b*x)^2 - (50*b*e^4*(b*d - a*e))/(a + b*x) + (10*e^5*(-(b*d) + a*e))/(d + e*x) -
60*b*e^5*Log[a + b*x] + 60*b*e^5*Log[d + e*x])/(10*(b*d - a*e)^7)
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Maple [A] time = 0.059, size = 178, normalized size = 1. \begin{align*} -{\frac{{e}^{5}}{ \left ( ae-bd \right ) ^{6} \left ( ex+d \right ) }}-6\,{\frac{{e}^{5}b\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{7}}}-{\frac{b}{5\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{5}}}+6\,{\frac{{e}^{5}b\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{7}}}-5\,{\frac{b{e}^{4}}{ \left ( ae-bd \right ) ^{6} \left ( bx+a \right ) }}-2\,{\frac{b{e}^{3}}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) ^{2}}}-{\frac{b{e}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{3}}}-{\frac{be}{2\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
-e^5/(a*e-b*d)^6/(e*x+d)-6*e^5/(a*e-b*d)^7*b*ln(e*x+d)-1/5*b/(a*e-b*d)^2/(b*x+a)^5+6*e^5/(a*e-b*d)^7*b*ln(b*x+
a)-5*b/(a*e-b*d)^6*e^4/(b*x+a)-2*b/(a*e-b*d)^5*e^3/(b*x+a)^2-b/(a*e-b*d)^4*e^2/(b*x+a)^3-1/2*b/(a*e-b*d)^3*e/(
b*x+a)^4
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Maxima [B] time = 1.39119, size = 1558, normalized size = 8.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
[Out]
-6*b*e^5*log(b*x + a)/(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4
- 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7) + 6*b*e^5*log(e*x + d)/(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d
^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7) - 1/10*(60*b^
5*e^5*x^5 + 2*b^5*d^5 - 13*a*b^4*d^4*e + 37*a^2*b^3*d^3*e^2 - 63*a^3*b^2*d^2*e^3 + 87*a^4*b*d*e^4 + 10*a^5*e^5
+ 30*(b^5*d*e^4 + 9*a*b^4*e^5)*x^4 - 10*(b^5*d^2*e^3 - 14*a*b^4*d*e^4 - 47*a^2*b^3*e^5)*x^3 + 5*(b^5*d^3*e^2
- 9*a*b^4*d^2*e^3 + 51*a^2*b^3*d*e^4 + 77*a^3*b^2*e^5)*x^2 - (3*b^5*d^4*e - 22*a*b^4*d^3*e^2 + 78*a^2*b^3*d^2*
e^3 - 222*a^3*b^2*d*e^4 - 137*a^4*b*e^5)*x)/(a^5*b^6*d^7 - 6*a^6*b^5*d^6*e + 15*a^7*b^4*d^5*e^2 - 20*a^8*b^3*d
^4*e^3 + 15*a^9*b^2*d^3*e^4 - 6*a^10*b*d^2*e^5 + a^11*d*e^6 + (b^11*d^6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*
e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4*b^7*d^2*e^5 - 6*a^5*b^6*d*e^6 + a^6*b^5*e^7)*x^6 + (b^11*d^7 - a*b^10*d^6*e
- 15*a^2*b^9*d^5*e^2 + 55*a^3*b^8*d^4*e^3 - 85*a^4*b^7*d^3*e^4 + 69*a^5*b^6*d^2*e^5 - 29*a^6*b^5*d*e^6 + 5*a^7
*b^4*e^7)*x^5 + 5*(a*b^10*d^7 - 4*a^2*b^9*d^6*e + 3*a^3*b^8*d^5*e^2 + 10*a^4*b^7*d^4*e^3 - 25*a^5*b^6*d^3*e^4
+ 24*a^6*b^5*d^2*e^5 - 11*a^7*b^4*d*e^6 + 2*a^8*b^3*e^7)*x^4 + 10*(a^2*b^9*d^7 - 5*a^3*b^8*d^6*e + 9*a^4*b^7*d
^5*e^2 - 5*a^5*b^6*d^4*e^3 - 5*a^6*b^5*d^3*e^4 + 9*a^7*b^4*d^2*e^5 - 5*a^8*b^3*d*e^6 + a^9*b^2*e^7)*x^3 + 5*(2
*a^3*b^8*d^7 - 11*a^4*b^7*d^6*e + 24*a^5*b^6*d^5*e^2 - 25*a^6*b^5*d^4*e^3 + 10*a^7*b^4*d^3*e^4 + 3*a^8*b^3*d^2
*e^5 - 4*a^9*b^2*d*e^6 + a^10*b*e^7)*x^2 + (5*a^4*b^7*d^7 - 29*a^5*b^6*d^6*e + 69*a^6*b^5*d^5*e^2 - 85*a^7*b^4
*d^4*e^3 + 55*a^8*b^3*d^3*e^4 - 15*a^9*b^2*d^2*e^5 - a^10*b*d*e^6 + a^11*e^7)*x)
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Fricas [B] time = 1.88807, size = 2889, normalized size = 15.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
[Out]
-1/10*(2*b^6*d^6 - 15*a*b^5*d^5*e + 50*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 150*a^4*b^2*d^2*e^4 - 77*a^5*b*
d*e^5 - 10*a^6*e^6 + 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 30*(b^6*d^2*e^4 + 8*a*b^5*d*e^5 - 9*a^2*b^4*e^6)*x^4 - 1
0*(b^6*d^3*e^3 - 15*a*b^5*d^2*e^4 - 33*a^2*b^4*d*e^5 + 47*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 10*a*b^5*d^3*e^3
+ 60*a^2*b^4*d^2*e^4 + 26*a^3*b^3*d*e^5 - 77*a^4*b^2*e^6)*x^2 - (3*b^6*d^5*e - 25*a*b^5*d^4*e^2 + 100*a^2*b^4
*d^3*e^3 - 300*a^3*b^3*d^2*e^4 + 85*a^4*b^2*d*e^5 + 137*a^5*b*e^6)*x + 60*(b^6*e^6*x^6 + a^5*b*d*e^5 + (b^6*d*
e^5 + 5*a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 + 2*a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 5*(2*a^3
*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + (5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*log(b*x + a) - 60*(b^6*e^6*x^6 + a^5*b*d*e^5
+ (b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 + 2*a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 +
5*(2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + (5*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*log(e*x + d))/(a^5*b^7*d^8 - 7*a^6*b
^6*d^7*e + 21*a^7*b^5*d^6*e^2 - 35*a^8*b^4*d^5*e^3 + 35*a^9*b^3*d^4*e^4 - 21*a^10*b^2*d^3*e^5 + 7*a^11*b*d^2*e
^6 - a^12*d*e^7 + (b^12*d^7*e - 7*a*b^11*d^6*e^2 + 21*a^2*b^10*d^5*e^3 - 35*a^3*b^9*d^4*e^4 + 35*a^4*b^8*d^3*e
^5 - 21*a^5*b^7*d^2*e^6 + 7*a^6*b^6*d*e^7 - a^7*b^5*e^8)*x^6 + (b^12*d^8 - 2*a*b^11*d^7*e - 14*a^2*b^10*d^6*e^
2 + 70*a^3*b^9*d^5*e^3 - 140*a^4*b^8*d^4*e^4 + 154*a^5*b^7*d^3*e^5 - 98*a^6*b^6*d^2*e^6 + 34*a^7*b^5*d*e^7 - 5
*a^8*b^4*e^8)*x^5 + 5*(a*b^11*d^8 - 5*a^2*b^10*d^7*e + 7*a^3*b^9*d^6*e^2 + 7*a^4*b^8*d^5*e^3 - 35*a^5*b^7*d^4*
e^4 + 49*a^6*b^6*d^3*e^5 - 35*a^7*b^5*d^2*e^6 + 13*a^8*b^4*d*e^7 - 2*a^9*b^3*e^8)*x^4 + 10*(a^2*b^10*d^8 - 6*a
^3*b^9*d^7*e + 14*a^4*b^8*d^6*e^2 - 14*a^5*b^7*d^5*e^3 + 14*a^7*b^5*d^3*e^5 - 14*a^8*b^4*d^2*e^6 + 6*a^9*b^3*d
*e^7 - a^10*b^2*e^8)*x^3 + 5*(2*a^3*b^9*d^8 - 13*a^4*b^8*d^7*e + 35*a^5*b^7*d^6*e^2 - 49*a^6*b^6*d^5*e^3 + 35*
a^7*b^5*d^4*e^4 - 7*a^8*b^4*d^3*e^5 - 7*a^9*b^3*d^2*e^6 + 5*a^10*b^2*d*e^7 - a^11*b*e^8)*x^2 + (5*a^4*b^8*d^8
- 34*a^5*b^7*d^7*e + 98*a^6*b^6*d^6*e^2 - 154*a^7*b^5*d^5*e^3 + 140*a^8*b^4*d^4*e^4 - 70*a^9*b^3*d^3*e^5 + 14*
a^10*b^2*d^2*e^6 + 2*a^11*b*d*e^7 - a^12*e^8)*x)
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Sympy [B] time = 9.33575, size = 1516, normalized size = 8.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
-6*b*e**5*log(x + (-6*a**8*b*e**13/(a*e - b*d)**7 + 48*a**7*b**2*d*e**12/(a*e - b*d)**7 - 168*a**6*b**3*d**2*e
**11/(a*e - b*d)**7 + 336*a**5*b**4*d**3*e**10/(a*e - b*d)**7 - 420*a**4*b**5*d**4*e**9/(a*e - b*d)**7 + 336*a
**3*b**6*d**5*e**8/(a*e - b*d)**7 - 168*a**2*b**7*d**6*e**7/(a*e - b*d)**7 + 48*a*b**8*d**7*e**6/(a*e - b*d)**
7 + 6*a*b*e**6 - 6*b**9*d**8*e**5/(a*e - b*d)**7 + 6*b**2*d*e**5)/(12*b**2*e**6))/(a*e - b*d)**7 + 6*b*e**5*lo
g(x + (6*a**8*b*e**13/(a*e - b*d)**7 - 48*a**7*b**2*d*e**12/(a*e - b*d)**7 + 168*a**6*b**3*d**2*e**11/(a*e - b
*d)**7 - 336*a**5*b**4*d**3*e**10/(a*e - b*d)**7 + 420*a**4*b**5*d**4*e**9/(a*e - b*d)**7 - 336*a**3*b**6*d**5
*e**8/(a*e - b*d)**7 + 168*a**2*b**7*d**6*e**7/(a*e - b*d)**7 - 48*a*b**8*d**7*e**6/(a*e - b*d)**7 + 6*a*b*e**
6 + 6*b**9*d**8*e**5/(a*e - b*d)**7 + 6*b**2*d*e**5)/(12*b**2*e**6))/(a*e - b*d)**7 - (10*a**5*e**5 + 87*a**4*
b*d*e**4 - 63*a**3*b**2*d**2*e**3 + 37*a**2*b**3*d**3*e**2 - 13*a*b**4*d**4*e + 2*b**5*d**5 + 60*b**5*e**5*x**
5 + x**4*(270*a*b**4*e**5 + 30*b**5*d*e**4) + x**3*(470*a**2*b**3*e**5 + 140*a*b**4*d*e**4 - 10*b**5*d**2*e**3
) + x**2*(385*a**3*b**2*e**5 + 255*a**2*b**3*d*e**4 - 45*a*b**4*d**2*e**3 + 5*b**5*d**3*e**2) + x*(137*a**4*b*
e**5 + 222*a**3*b**2*d*e**4 - 78*a**2*b**3*d**2*e**3 + 22*a*b**4*d**3*e**2 - 3*b**5*d**4*e))/(10*a**11*d*e**6
- 60*a**10*b*d**2*e**5 + 150*a**9*b**2*d**3*e**4 - 200*a**8*b**3*d**4*e**3 + 150*a**7*b**4*d**5*e**2 - 60*a**6
*b**5*d**6*e + 10*a**5*b**6*d**7 + x**6*(10*a**6*b**5*e**7 - 60*a**5*b**6*d*e**6 + 150*a**4*b**7*d**2*e**5 - 2
00*a**3*b**8*d**3*e**4 + 150*a**2*b**9*d**4*e**3 - 60*a*b**10*d**5*e**2 + 10*b**11*d**6*e) + x**5*(50*a**7*b**
4*e**7 - 290*a**6*b**5*d*e**6 + 690*a**5*b**6*d**2*e**5 - 850*a**4*b**7*d**3*e**4 + 550*a**3*b**8*d**4*e**3 -
150*a**2*b**9*d**5*e**2 - 10*a*b**10*d**6*e + 10*b**11*d**7) + x**4*(100*a**8*b**3*e**7 - 550*a**7*b**4*d*e**6
+ 1200*a**6*b**5*d**2*e**5 - 1250*a**5*b**6*d**3*e**4 + 500*a**4*b**7*d**4*e**3 + 150*a**3*b**8*d**5*e**2 - 2
00*a**2*b**9*d**6*e + 50*a*b**10*d**7) + x**3*(100*a**9*b**2*e**7 - 500*a**8*b**3*d*e**6 + 900*a**7*b**4*d**2*
e**5 - 500*a**6*b**5*d**3*e**4 - 500*a**5*b**6*d**4*e**3 + 900*a**4*b**7*d**5*e**2 - 500*a**3*b**8*d**6*e + 10
0*a**2*b**9*d**7) + x**2*(50*a**10*b*e**7 - 200*a**9*b**2*d*e**6 + 150*a**8*b**3*d**2*e**5 + 500*a**7*b**4*d**
3*e**4 - 1250*a**6*b**5*d**4*e**3 + 1200*a**5*b**6*d**5*e**2 - 550*a**4*b**7*d**6*e + 100*a**3*b**8*d**7) + x*
(10*a**11*e**7 - 10*a**10*b*d*e**6 - 150*a**9*b**2*d**2*e**5 + 550*a**8*b**3*d**3*e**4 - 850*a**7*b**4*d**4*e*
*3 + 690*a**6*b**5*d**5*e**2 - 290*a**5*b**6*d**6*e + 50*a**4*b**7*d**7))
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Giac [B] time = 1.15345, size = 609, normalized size = 3.36 \begin{align*} -\frac{6 \, b e^{6} \log \left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{x e + d} \right |}\right )}{b^{7} d^{7} e - 7 \, a b^{6} d^{6} e^{2} + 21 \, a^{2} b^{5} d^{5} e^{3} - 35 \, a^{3} b^{4} d^{4} e^{4} + 35 \, a^{4} b^{3} d^{3} e^{5} - 21 \, a^{5} b^{2} d^{2} e^{6} + 7 \, a^{6} b d e^{7} - a^{7} e^{8}} - \frac{e^{11}}{{\left (b^{6} d^{6} e^{6} - 6 \, a b^{5} d^{5} e^{7} + 15 \, a^{2} b^{4} d^{4} e^{8} - 20 \, a^{3} b^{3} d^{3} e^{9} + 15 \, a^{4} b^{2} d^{2} e^{10} - 6 \, a^{5} b d e^{11} + a^{6} e^{12}\right )}{\left (x e + d\right )}} - \frac{87 \, b^{6} e^{5} - \frac{385 \,{\left (b^{6} d e^{6} - a b^{5} e^{7}\right )} e^{\left (-1\right )}}{x e + d} + \frac{650 \,{\left (b^{6} d^{2} e^{7} - 2 \, a b^{5} d e^{8} + a^{2} b^{4} e^{9}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{500 \,{\left (b^{6} d^{3} e^{8} - 3 \, a b^{5} d^{2} e^{9} + 3 \, a^{2} b^{4} d e^{10} - a^{3} b^{3} e^{11}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{150 \,{\left (b^{6} d^{4} e^{9} - 4 \, a b^{5} d^{3} e^{10} + 6 \, a^{2} b^{4} d^{2} e^{11} - 4 \, a^{3} b^{3} d e^{12} + a^{4} b^{2} e^{13}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}}{10 \,{\left (b d - a e\right )}^{7}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
[Out]
-6*b*e^6*log(abs(b - b*d/(x*e + d) + a*e/(x*e + d)))/(b^7*d^7*e - 7*a*b^6*d^6*e^2 + 21*a^2*b^5*d^5*e^3 - 35*a^
3*b^4*d^4*e^4 + 35*a^4*b^3*d^3*e^5 - 21*a^5*b^2*d^2*e^6 + 7*a^6*b*d*e^7 - a^7*e^8) - e^11/((b^6*d^6*e^6 - 6*a*
b^5*d^5*e^7 + 15*a^2*b^4*d^4*e^8 - 20*a^3*b^3*d^3*e^9 + 15*a^4*b^2*d^2*e^10 - 6*a^5*b*d*e^11 + a^6*e^12)*(x*e
+ d)) - 1/10*(87*b^6*e^5 - 385*(b^6*d*e^6 - a*b^5*e^7)*e^(-1)/(x*e + d) + 650*(b^6*d^2*e^7 - 2*a*b^5*d*e^8 + a
^2*b^4*e^9)*e^(-2)/(x*e + d)^2 - 500*(b^6*d^3*e^8 - 3*a*b^5*d^2*e^9 + 3*a^2*b^4*d*e^10 - a^3*b^3*e^11)*e^(-3)/
(x*e + d)^3 + 150*(b^6*d^4*e^9 - 4*a*b^5*d^3*e^10 + 6*a^2*b^4*d^2*e^11 - 4*a^3*b^3*d*e^12 + a^4*b^2*e^13)*e^(-
4)/(x*e + d)^4)/((b*d - a*e)^7*(b - b*d/(x*e + d) + a*e/(x*e + d))^5)