3.1952 \(\int \frac{a+b x}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=192 \[ \frac{10 b^2 e^3}{(a+b x) (b d-a e)^6}-\frac{3 b^2 e^2}{(a+b x)^2 (b d-a e)^5}+\frac{15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac{15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}+\frac{b^2 e}{(a+b x)^3 (b d-a e)^4}-\frac{b^2}{4 (a+b x)^4 (b d-a e)^3}+\frac{5 b e^4}{(d+e x) (b d-a e)^6}+\frac{e^4}{2 (d+e x)^2 (b d-a e)^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.191708, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 44} \[ \frac{10 b^2 e^3}{(a+b x) (b d-a e)^6}-\frac{3 b^2 e^2}{(a+b x)^2 (b d-a e)^5}+\frac{15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac{15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}+\frac{b^2 e}{(a+b x)^3 (b d-a e)^4}-\frac{b^2}{4 (a+b x)^4 (b d-a e)^3}+\frac{5 b e^4}{(d+e x) (b d-a e)^6}+\frac{e^4}{2 (d+e x)^2 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^2/(4*(b*d - a*e)^3*(a + b*x)^4) + (b^2*e)/((b*d - a*e)^4*(a + b*x)^3) - (3*b^2*e^2)/((b*d - a*e)^5*(a + b*x
)^2) + (10*b^2*e^3)/((b*d - a*e)^6*(a + b*x)) + e^4/(2*(b*d - a*e)^5*(d + e*x)^2) + (5*b*e^4)/((b*d - a*e)^6*(
d + e*x)) + (15*b^2*e^4*Log[a + b*x])/(b*d - a*e)^7 - (15*b^2*e^4*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{a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^5 (d+e x)^3} \, dx\\ &=\int \left (\frac{b^3}{(b d-a e)^3 (a+b x)^5}-\frac{3 b^3 e}{(b d-a e)^4 (a+b x)^4}+\frac{6 b^3 e^2}{(b d-a e)^5 (a+b x)^3}-\frac{10 b^3 e^3}{(b d-a e)^6 (a+b x)^2}+\frac{15 b^3 e^4}{(b d-a e)^7 (a+b x)}-\frac{e^5}{(b d-a e)^5 (d+e x)^3}-\frac{5 b e^5}{(b d-a e)^6 (d+e x)^2}-\frac{15 b^2 e^5}{(b d-a e)^7 (d+e x)}\right ) \, dx\\ &=-\frac{b^2}{4 (b d-a e)^3 (a+b x)^4}+\frac{b^2 e}{(b d-a e)^4 (a+b x)^3}-\frac{3 b^2 e^2}{(b d-a e)^5 (a+b x)^2}+\frac{10 b^2 e^3}{(b d-a e)^6 (a+b x)}+\frac{e^4}{2 (b d-a e)^5 (d+e x)^2}+\frac{5 b e^4}{(b d-a e)^6 (d+e x)}+\frac{15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac{15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}\\ \end{align*}

Mathematica [A]  time = 0.107841, size = 179, normalized size = 0.93 \[ \frac{\frac{40 b^2 e^3 (b d-a e)}{a+b x}-\frac{12 b^2 e^2 (b d-a e)^2}{(a+b x)^2}+\frac{4 b^2 e (b d-a e)^3}{(a+b x)^3}-\frac{b^2 (b d-a e)^4}{(a+b x)^4}+60 b^2 e^4 \log (a+b x)+\frac{20 b e^4 (b d-a e)}{d+e x}+\frac{2 e^4 (b d-a e)^2}{(d+e x)^2}-60 b^2 e^4 \log (d+e x)}{4 (b d-a e)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b^2*(b*d - a*e)^4)/(a + b*x)^4) + (4*b^2*e*(b*d - a*e)^3)/(a + b*x)^3 - (12*b^2*e^2*(b*d - a*e)^2)/(a + b*
x)^2 + (40*b^2*e^3*(b*d - a*e))/(a + b*x) + (2*e^4*(b*d - a*e)^2)/(d + e*x)^2 + (20*b*e^4*(b*d - a*e))/(d + e*
x) + 60*b^2*e^4*Log[a + b*x] - 60*b^2*e^4*Log[d + e*x])/(4*(b*d - a*e)^7)

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 189, normalized size = 1. \begin{align*} -{\frac{{e}^{4}}{2\, \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) ^{2}}}+15\,{\frac{{e}^{4}{b}^{2}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{7}}}+5\,{\frac{{e}^{4}b}{ \left ( ae-bd \right ) ^{6} \left ( ex+d \right ) }}+{\frac{{b}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{4}}}-15\,{\frac{{e}^{4}{b}^{2}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{7}}}+10\,{\frac{{b}^{2}{e}^{3}}{ \left ( ae-bd \right ) ^{6} \left ( bx+a \right ) }}+3\,{\frac{{b}^{2}{e}^{2}}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) ^{2}}}+{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.3732, size = 1620, normalized size = 8.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

15*b^2*e^4*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) - 15*b^2*e^4*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) + 1/4*(6
0*b^5*e^5*x^5 - b^5*d^5 + 7*a*b^4*d^4*e - 23*a^2*b^3*d^3*e^2 + 57*a^3*b^2*d^2*e^3 + 22*a^4*b*d*e^4 - 2*a^5*e^5
+ 30*(3*b^5*d*e^4 + 7*a*b^4*e^5)*x^4 + 20*(b^5*d^2*e^3 + 16*a*b^4*d*e^4 + 13*a^2*b^3*e^5)*x^3 - 5*(b^5*d^3*e^
2 - 15*a*b^4*d^2*e^3 - 81*a^2*b^3*d*e^4 - 25*a^3*b^2*e^5)*x^2 + 2*(b^5*d^4*e - 9*a*b^4*d^3*e^2 + 51*a^2*b^3*d^
2*e^3 + 101*a^3*b^2*d*e^4 + 6*a^4*b*e^5)*x)/(a^4*b^6*d^8 - 6*a^5*b^5*d^7*e + 15*a^6*b^4*d^6*e^2 - 20*a^7*b^3*d
^5*e^3 + 15*a^8*b^2*d^4*e^4 - 6*a^9*b*d^3*e^5 + a^10*d^2*e^6 + (b^10*d^6*e^2 - 6*a*b^9*d^5*e^3 + 15*a^2*b^8*d^
4*e^4 - 20*a^3*b^7*d^3*e^5 + 15*a^4*b^6*d^2*e^6 - 6*a^5*b^5*d*e^7 + a^6*b^4*e^8)*x^6 + 2*(b^10*d^7*e - 4*a*b^9
*d^6*e^2 + 3*a^2*b^8*d^5*e^3 + 10*a^3*b^7*d^4*e^4 - 25*a^4*b^6*d^3*e^5 + 24*a^5*b^5*d^2*e^6 - 11*a^6*b^4*d*e^7
+ 2*a^7*b^3*e^8)*x^5 + (b^10*d^8 + 2*a*b^9*d^7*e - 27*a^2*b^8*d^6*e^2 + 64*a^3*b^7*d^5*e^3 - 55*a^4*b^6*d^4*e
^4 - 6*a^5*b^5*d^3*e^5 + 43*a^6*b^4*d^2*e^6 - 28*a^7*b^3*d*e^7 + 6*a^8*b^2*e^8)*x^4 + 4*(a*b^9*d^8 - 3*a^2*b^8
*d^7*e - 2*a^3*b^7*d^6*e^2 + 19*a^4*b^6*d^5*e^3 - 30*a^5*b^5*d^4*e^4 + 19*a^6*b^4*d^3*e^5 - 2*a^7*b^3*d^2*e^6
- 3*a^8*b^2*d*e^7 + a^9*b*e^8)*x^3 + (6*a^2*b^8*d^8 - 28*a^3*b^7*d^7*e + 43*a^4*b^6*d^6*e^2 - 6*a^5*b^5*d^5*e^
3 - 55*a^6*b^4*d^4*e^4 + 64*a^7*b^3*d^3*e^5 - 27*a^8*b^2*d^2*e^6 + 2*a^9*b*d*e^7 + a^10*e^8)*x^2 + 2*(2*a^3*b^
7*d^8 - 11*a^4*b^6*d^7*e + 24*a^5*b^5*d^6*e^2 - 25*a^6*b^4*d^5*e^3 + 10*a^7*b^3*d^4*e^4 + 3*a^8*b^2*d^3*e^5 -
4*a^9*b*d^2*e^6 + a^10*d*e^7)*x)

________________________________________________________________________________________

Fricas [B]  time = 1.89758, size = 3136, normalized size = 16.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 24*a^5*b*d*e^5
- 2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x^4 - 20*(b^
6*d^3*e^3 + 15*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^3*e^3 - 66*
a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*e - 10*a*b^5*d^4*e^2 + 60*a^2*b^4*d^3*e^
3 + 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x - 60*(b^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5
+ 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 +
a^3*b^3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^2*e^4 + a^4*b^2*d*
e^5)*x)*log(b*x + a) + 60*(b^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*
a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4
+ 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(e*x + d))/(a^4*b^7*d^9 - 7
*a^5*b^6*d^8*e + 21*a^6*b^5*d^7*e^2 - 35*a^7*b^4*d^6*e^3 + 35*a^8*b^3*d^5*e^4 - 21*a^9*b^2*d^4*e^5 + 7*a^10*b*
d^3*e^6 - a^11*d^2*e^7 + (b^11*d^7*e^2 - 7*a*b^10*d^6*e^3 + 21*a^2*b^9*d^5*e^4 - 35*a^3*b^8*d^4*e^5 + 35*a^4*b
^7*d^3*e^6 - 21*a^5*b^6*d^2*e^7 + 7*a^6*b^5*d*e^8 - a^7*b^4*e^9)*x^6 + 2*(b^11*d^8*e - 5*a*b^10*d^7*e^2 + 7*a^
2*b^9*d^6*e^3 + 7*a^3*b^8*d^5*e^4 - 35*a^4*b^7*d^4*e^5 + 49*a^5*b^6*d^3*e^6 - 35*a^6*b^5*d^2*e^7 + 13*a^7*b^4*
d*e^8 - 2*a^8*b^3*e^9)*x^5 + (b^11*d^9 + a*b^10*d^8*e - 29*a^2*b^9*d^7*e^2 + 91*a^3*b^8*d^6*e^3 - 119*a^4*b^7*
d^5*e^4 + 49*a^5*b^6*d^4*e^5 + 49*a^6*b^5*d^3*e^6 - 71*a^7*b^4*d^2*e^7 + 34*a^8*b^3*d*e^8 - 6*a^9*b^2*e^9)*x^4
+ 4*(a*b^10*d^9 - 4*a^2*b^9*d^8*e + a^3*b^8*d^7*e^2 + 21*a^4*b^7*d^6*e^3 - 49*a^5*b^6*d^5*e^4 + 49*a^6*b^5*d^
4*e^5 - 21*a^7*b^4*d^3*e^6 - a^8*b^3*d^2*e^7 + 4*a^9*b^2*d*e^8 - a^10*b*e^9)*x^3 + (6*a^2*b^9*d^9 - 34*a^3*b^8
*d^8*e + 71*a^4*b^7*d^7*e^2 - 49*a^5*b^6*d^6*e^3 - 49*a^6*b^5*d^5*e^4 + 119*a^7*b^4*d^4*e^5 - 91*a^8*b^3*d^3*e
^6 + 29*a^9*b^2*d^2*e^7 - a^10*b*d*e^8 - a^11*e^9)*x^2 + 2*(2*a^3*b^8*d^9 - 13*a^4*b^7*d^8*e + 35*a^5*b^6*d^7*
e^2 - 49*a^6*b^5*d^6*e^3 + 35*a^7*b^4*d^5*e^4 - 7*a^8*b^3*d^4*e^5 - 7*a^9*b^2*d^3*e^6 + 5*a^10*b*d^2*e^7 - a^1
1*d*e^8)*x)

________________________________________________________________________________________

Sympy [B]  time = 8.30439, size = 1571, normalized size = 8.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

15*b**2*e**4*log(x + (-15*a**8*b**2*e**12/(a*e - b*d)**7 + 120*a**7*b**3*d*e**11/(a*e - b*d)**7 - 420*a**6*b**
4*d**2*e**10/(a*e - b*d)**7 + 840*a**5*b**5*d**3*e**9/(a*e - b*d)**7 - 1050*a**4*b**6*d**4*e**8/(a*e - b*d)**7
+ 840*a**3*b**7*d**5*e**7/(a*e - b*d)**7 - 420*a**2*b**8*d**6*e**6/(a*e - b*d)**7 + 120*a*b**9*d**7*e**5/(a*e
- b*d)**7 + 15*a*b**2*e**5 - 15*b**10*d**8*e**4/(a*e - b*d)**7 + 15*b**3*d*e**4)/(30*b**3*e**5))/(a*e - b*d)*
*7 - 15*b**2*e**4*log(x + (15*a**8*b**2*e**12/(a*e - b*d)**7 - 120*a**7*b**3*d*e**11/(a*e - b*d)**7 + 420*a**6
*b**4*d**2*e**10/(a*e - b*d)**7 - 840*a**5*b**5*d**3*e**9/(a*e - b*d)**7 + 1050*a**4*b**6*d**4*e**8/(a*e - b*d
)**7 - 840*a**3*b**7*d**5*e**7/(a*e - b*d)**7 + 420*a**2*b**8*d**6*e**6/(a*e - b*d)**7 - 120*a*b**9*d**7*e**5/
(a*e - b*d)**7 + 15*a*b**2*e**5 + 15*b**10*d**8*e**4/(a*e - b*d)**7 + 15*b**3*d*e**4)/(30*b**3*e**5))/(a*e - b
*d)**7 + (-2*a**5*e**5 + 22*a**4*b*d*e**4 + 57*a**3*b**2*d**2*e**3 - 23*a**2*b**3*d**3*e**2 + 7*a*b**4*d**4*e
- b**5*d**5 + 60*b**5*e**5*x**5 + x**4*(210*a*b**4*e**5 + 90*b**5*d*e**4) + x**3*(260*a**2*b**3*e**5 + 320*a*b
**4*d*e**4 + 20*b**5*d**2*e**3) + x**2*(125*a**3*b**2*e**5 + 405*a**2*b**3*d*e**4 + 75*a*b**4*d**2*e**3 - 5*b*
*5*d**3*e**2) + x*(12*a**4*b*e**5 + 202*a**3*b**2*d*e**4 + 102*a**2*b**3*d**2*e**3 - 18*a*b**4*d**3*e**2 + 2*b
**5*d**4*e))/(4*a**10*d**2*e**6 - 24*a**9*b*d**3*e**5 + 60*a**8*b**2*d**4*e**4 - 80*a**7*b**3*d**5*e**3 + 60*a
**6*b**4*d**6*e**2 - 24*a**5*b**5*d**7*e + 4*a**4*b**6*d**8 + x**6*(4*a**6*b**4*e**8 - 24*a**5*b**5*d*e**7 + 6
0*a**4*b**6*d**2*e**6 - 80*a**3*b**7*d**3*e**5 + 60*a**2*b**8*d**4*e**4 - 24*a*b**9*d**5*e**3 + 4*b**10*d**6*e
**2) + x**5*(16*a**7*b**3*e**8 - 88*a**6*b**4*d*e**7 + 192*a**5*b**5*d**2*e**6 - 200*a**4*b**6*d**3*e**5 + 80*
a**3*b**7*d**4*e**4 + 24*a**2*b**8*d**5*e**3 - 32*a*b**9*d**6*e**2 + 8*b**10*d**7*e) + x**4*(24*a**8*b**2*e**8
- 112*a**7*b**3*d*e**7 + 172*a**6*b**4*d**2*e**6 - 24*a**5*b**5*d**3*e**5 - 220*a**4*b**6*d**4*e**4 + 256*a**
3*b**7*d**5*e**3 - 108*a**2*b**8*d**6*e**2 + 8*a*b**9*d**7*e + 4*b**10*d**8) + x**3*(16*a**9*b*e**8 - 48*a**8*
b**2*d*e**7 - 32*a**7*b**3*d**2*e**6 + 304*a**6*b**4*d**3*e**5 - 480*a**5*b**5*d**4*e**4 + 304*a**4*b**6*d**5*
e**3 - 32*a**3*b**7*d**6*e**2 - 48*a**2*b**8*d**7*e + 16*a*b**9*d**8) + x**2*(4*a**10*e**8 + 8*a**9*b*d*e**7 -
108*a**8*b**2*d**2*e**6 + 256*a**7*b**3*d**3*e**5 - 220*a**6*b**4*d**4*e**4 - 24*a**5*b**5*d**5*e**3 + 172*a*
*4*b**6*d**6*e**2 - 112*a**3*b**7*d**7*e + 24*a**2*b**8*d**8) + x*(8*a**10*d*e**7 - 32*a**9*b*d**2*e**6 + 24*a
**8*b**2*d**3*e**5 + 80*a**7*b**3*d**4*e**4 - 200*a**6*b**4*d**5*e**3 + 192*a**5*b**5*d**6*e**2 - 88*a**4*b**6
*d**7*e + 16*a**3*b**7*d**8))

________________________________________________________________________________________

Giac [B]  time = 1.27115, size = 738, normalized size = 3.84 \begin{align*} \frac{15 \, b^{3} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8} d^{7} - 7 \, a b^{7} d^{6} e + 21 \, a^{2} b^{6} d^{5} e^{2} - 35 \, a^{3} b^{5} d^{4} e^{3} + 35 \, a^{4} b^{4} d^{3} e^{4} - 21 \, a^{5} b^{3} d^{2} e^{5} + 7 \, a^{6} b^{2} d e^{6} - a^{7} b e^{7}} - \frac{15 \, b^{2} e^{5} \log \left ({\left | 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{b^{6} d^{6} - 8 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} - 80 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 24 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 60 \,{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} - 30 \,{\left (3 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} - 7 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{3} + 15 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 13 \, a^{3} b^{3} e^{6}\right )} x^{3} + 5 \,{\left (b^{6} d^{4} e^{2} - 16 \, a b^{5} d^{3} e^{3} - 66 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 25 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \,{\left (b^{6} d^{5} e - 10 \, a b^{5} d^{4} e^{2} + 60 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 95 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x}{4 \,{\left (b d - a e\right )}^{7}{\left (b x + a\right )}^{4}{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

15*b^3*e^4*log(abs(b*x + a))/(b^8*d^7 - 7*a*b^7*d^6*e + 21*a^2*b^6*d^5*e^2 - 35*a^3*b^5*d^4*e^3 + 35*a^4*b^4*d
^3*e^4 - 21*a^5*b^3*d^2*e^5 + 7*a^6*b^2*d*e^6 - a^7*b*e^7) - 15*b^2*e^5*log(abs(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) - 1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 24*a
^5*b*d*e^5 - 2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x
^4 - 20*(b^6*d^3*e^3 + 15*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^
3*e^3 - 66*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*e - 10*a*b^5*d^4*e^2 + 60*a^2
*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x)/((b*d - a*e)^7*(b*x + a)^4*(x*e + d)^2)