3.1953 \(\int \frac{a+b x}{(d+e x)^4 (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=222 \[ \frac{15 b^2 e^4}{(d+e x) (b d-a e)^7}+\frac{20 b^3 e^3}{(a+b x) (b d-a e)^7}-\frac{5 b^3 e^2}{(a+b x)^2 (b d-a e)^6}+\frac{35 b^3 e^4 \log (a+b x)}{(b d-a e)^8}-\frac{35 b^3 e^4 \log (d+e x)}{(b d-a e)^8}+\frac{4 b^3 e}{3 (a+b x)^3 (b d-a e)^5}-\frac{b^3}{4 (a+b x)^4 (b d-a e)^4}+\frac{5 b e^4}{2 (d+e x)^2 (b d-a e)^6}+\frac{e^4}{3 (d+e x)^3 (b d-a e)^5} \]

[Out]

-b^3/(4*(b*d - a*e)^4*(a + b*x)^4) + (4*b^3*e)/(3*(b*d - a*e)^5*(a + b*x)^3) - (5*b^3*e^2)/((b*d - a*e)^6*(a +
b*x)^2) + (20*b^3*e^3)/((b*d - a*e)^7*(a + b*x)) + e^4/(3*(b*d - a*e)^5*(d + e*x)^3) + (5*b*e^4)/(2*(b*d - a*
e)^6*(d + e*x)^2) + (15*b^2*e^4)/((b*d - a*e)^7*(d + e*x)) + (35*b^3*e^4*Log[a + b*x])/(b*d - a*e)^8 - (35*b^3
*e^4*Log[d + e*x])/(b*d - a*e)^8

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Rubi [A]  time = 0.261491, antiderivative size = 222, 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} \[ \frac{15 b^2 e^4}{(d+e x) (b d-a e)^7}+\frac{20 b^3 e^3}{(a+b x) (b d-a e)^7}-\frac{5 b^3 e^2}{(a+b x)^2 (b d-a e)^6}+\frac{35 b^3 e^4 \log (a+b x)}{(b d-a e)^8}-\frac{35 b^3 e^4 \log (d+e x)}{(b d-a e)^8}+\frac{4 b^3 e}{3 (a+b x)^3 (b d-a e)^5}-\frac{b^3}{4 (a+b x)^4 (b d-a e)^4}+\frac{5 b e^4}{2 (d+e x)^2 (b d-a e)^6}+\frac{e^4}{3 (d+e x)^3 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

-b^3/(4*(b*d - a*e)^4*(a + b*x)^4) + (4*b^3*e)/(3*(b*d - a*e)^5*(a + b*x)^3) - (5*b^3*e^2)/((b*d - a*e)^6*(a +
b*x)^2) + (20*b^3*e^3)/((b*d - a*e)^7*(a + b*x)) + e^4/(3*(b*d - a*e)^5*(d + e*x)^3) + (5*b*e^4)/(2*(b*d - a*
e)^6*(d + e*x)^2) + (15*b^2*e^4)/((b*d - a*e)^7*(d + e*x)) + (35*b^3*e^4*Log[a + b*x])/(b*d - a*e)^8 - (35*b^3
*e^4*Log[d + e*x])/(b*d - a*e)^8

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)^4 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^5 (d+e x)^4} \, dx\\ &=\int \left (\frac{b^4}{(b d-a e)^4 (a+b x)^5}-\frac{4 b^4 e}{(b d-a e)^5 (a+b x)^4}+\frac{10 b^4 e^2}{(b d-a e)^6 (a+b x)^3}-\frac{20 b^4 e^3}{(b d-a e)^7 (a+b x)^2}+\frac{35 b^4 e^4}{(b d-a e)^8 (a+b x)}-\frac{e^5}{(b d-a e)^5 (d+e x)^4}-\frac{5 b e^5}{(b d-a e)^6 (d+e x)^3}-\frac{15 b^2 e^5}{(b d-a e)^7 (d+e x)^2}-\frac{35 b^3 e^5}{(b d-a e)^8 (d+e x)}\right ) \, dx\\ &=-\frac{b^3}{4 (b d-a e)^4 (a+b x)^4}+\frac{4 b^3 e}{3 (b d-a e)^5 (a+b x)^3}-\frac{5 b^3 e^2}{(b d-a e)^6 (a+b x)^2}+\frac{20 b^3 e^3}{(b d-a e)^7 (a+b x)}+\frac{e^4}{3 (b d-a e)^5 (d+e x)^3}+\frac{5 b e^4}{2 (b d-a e)^6 (d+e x)^2}+\frac{15 b^2 e^4}{(b d-a e)^7 (d+e x)}+\frac{35 b^3 e^4 \log (a+b x)}{(b d-a e)^8}-\frac{35 b^3 e^4 \log (d+e x)}{(b d-a e)^8}\\ \end{align*}

Mathematica [A]  time = 0.139319, size = 204, normalized size = 0.92 \[ \frac{\frac{180 b^2 e^4 (b d-a e)}{d+e x}+\frac{240 b^3 e^3 (b d-a e)}{a+b x}-\frac{60 b^3 e^2 (b d-a e)^2}{(a+b x)^2}+\frac{16 b^3 e (b d-a e)^3}{(a+b x)^3}-\frac{3 b^3 (b d-a e)^4}{(a+b x)^4}+420 b^3 e^4 \log (a+b x)+\frac{30 b e^4 (b d-a e)^2}{(d+e x)^2}+\frac{4 e^4 (b d-a e)^3}{(d+e x)^3}-420 b^3 e^4 \log (d+e x)}{12 (b d-a e)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((-3*b^3*(b*d - a*e)^4)/(a + b*x)^4 + (16*b^3*e*(b*d - a*e)^3)/(a + b*x)^3 - (60*b^3*e^2*(b*d - a*e)^2)/(a + b
*x)^2 + (240*b^3*e^3*(b*d - a*e))/(a + b*x) + (4*e^4*(b*d - a*e)^3)/(d + e*x)^3 + (30*b*e^4*(b*d - a*e)^2)/(d
+ e*x)^2 + (180*b^2*e^4*(b*d - a*e))/(d + e*x) + 420*b^3*e^4*Log[a + b*x] - 420*b^3*e^4*Log[d + e*x])/(12*(b*d
- a*e)^8)

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Maple [A]  time = 0.017, size = 215, normalized size = 1. \begin{align*} -{\frac{{e}^{4}}{3\, \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) ^{3}}}-35\,{\frac{{e}^{4}{b}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{8}}}-15\,{\frac{{e}^{4}{b}^{2}}{ \left ( ae-bd \right ) ^{7} \left ( ex+d \right ) }}+{\frac{5\,{e}^{4}b}{2\, \left ( ae-bd \right ) ^{6} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{4}}}+35\,{\frac{{e}^{4}{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{8}}}-20\,{\frac{{b}^{3}{e}^{3}}{ \left ( ae-bd \right ) ^{7} \left ( bx+a \right ) }}-5\,{\frac{{b}^{3}{e}^{2}}{ \left ( ae-bd \right ) ^{6} \left ( bx+a \right ) ^{2}}}-{\frac{4\,{b}^{3}e}{3\, \left ( ae-bd \right ) ^{5} \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)^4/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-1/3*e^4/(a*e-b*d)^5/(e*x+d)^3-35*e^4/(a*e-b*d)^8*b^3*ln(e*x+d)-15*e^4/(a*e-b*d)^7*b^2/(e*x+d)+5/2*e^4/(a*e-b*
d)^6*b/(e*x+d)^2-1/4*b^3/(a*e-b*d)^4/(b*x+a)^4+35*e^4/(a*e-b*d)^8*b^3*ln(b*x+a)-20*b^3/(a*e-b*d)^7*e^3/(b*x+a)
-5*b^3/(a*e-b*d)^6*e^2/(b*x+a)^2-4/3*b^3/(a*e-b*d)^5*e/(b*x+a)^3

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Maxima [B]  time = 1.66983, size = 2142, normalized size = 9.65 \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)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

35*b^3*e^4*log(b*x + a)/(b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^
4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8) - 35*b^3*e^4*log(e*x + d)/(b^8*d^8 - 8*
a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d
^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8) + 1/12*(420*b^6*e^6*x^6 - 3*b^6*d^6 + 25*a*b^5*d^5*e - 101*a^2*b^4*d^4*e^2 +
319*a^3*b^3*d^3*e^3 + 214*a^4*b^2*d^2*e^4 - 38*a^5*b*d*e^5 + 4*a^6*e^6 + 210*(5*b^6*d*e^5 + 7*a*b^5*e^6)*x^5
+ 70*(11*b^6*d^2*e^4 + 53*a*b^5*d*e^5 + 26*a^2*b^4*e^6)*x^4 + 35*(3*b^6*d^3*e^3 + 79*a*b^5*d^2*e^4 + 133*a^2*b
^4*d*e^5 + 25*a^3*b^3*e^6)*x^3 - 21*(b^6*d^4*e^2 - 19*a*b^5*d^3*e^3 - 169*a^2*b^4*d^2*e^4 - 109*a^3*b^3*d*e^5
- 4*a^4*b^2*e^6)*x^2 + 7*(b^6*d^5*e - 11*a*b^5*d^4*e^2 + 79*a^2*b^4*d^3*e^3 + 259*a^3*b^3*d^2*e^4 + 34*a^4*b^2
*d*e^5 - 2*a^5*b*e^6)*x)/(a^4*b^7*d^10 - 7*a^5*b^6*d^9*e + 21*a^6*b^5*d^8*e^2 - 35*a^7*b^4*d^7*e^3 + 35*a^8*b^
3*d^6*e^4 - 21*a^9*b^2*d^5*e^5 + 7*a^10*b*d^4*e^6 - a^11*d^3*e^7 + (b^11*d^7*e^3 - 7*a*b^10*d^6*e^4 + 21*a^2*b
^9*d^5*e^5 - 35*a^3*b^8*d^4*e^6 + 35*a^4*b^7*d^3*e^7 - 21*a^5*b^6*d^2*e^8 + 7*a^6*b^5*d*e^9 - a^7*b^4*e^10)*x^
7 + (3*b^11*d^8*e^2 - 17*a*b^10*d^7*e^3 + 35*a^2*b^9*d^6*e^4 - 21*a^3*b^8*d^5*e^5 - 35*a^4*b^7*d^4*e^6 + 77*a^
5*b^6*d^3*e^7 - 63*a^6*b^5*d^2*e^8 + 25*a^7*b^4*d*e^9 - 4*a^8*b^3*e^10)*x^6 + 3*(b^11*d^9*e - 3*a*b^10*d^8*e^2
- 5*a^2*b^9*d^7*e^3 + 35*a^3*b^8*d^6*e^4 - 63*a^4*b^7*d^5*e^5 + 49*a^5*b^6*d^4*e^6 - 7*a^6*b^5*d^3*e^7 - 15*a
^7*b^4*d^2*e^8 + 10*a^8*b^3*d*e^9 - 2*a^9*b^2*e^10)*x^5 + (b^11*d^10 + 5*a*b^10*d^9*e - 45*a^2*b^9*d^8*e^2 + 9
5*a^3*b^8*d^7*e^3 - 35*a^4*b^7*d^6*e^4 - 147*a^5*b^6*d^5*e^5 + 245*a^6*b^5*d^4*e^6 - 155*a^7*b^4*d^3*e^7 + 30*
a^8*b^3*d^2*e^8 + 10*a^9*b^2*d*e^9 - 4*a^10*b*e^10)*x^4 + (4*a*b^10*d^10 - 10*a^2*b^9*d^9*e - 30*a^3*b^8*d^8*e
^2 + 155*a^4*b^7*d^7*e^3 - 245*a^5*b^6*d^6*e^4 + 147*a^6*b^5*d^5*e^5 + 35*a^7*b^4*d^4*e^6 - 95*a^8*b^3*d^3*e^7
+ 45*a^9*b^2*d^2*e^8 - 5*a^10*b*d*e^9 - a^11*e^10)*x^3 + 3*(2*a^2*b^9*d^10 - 10*a^3*b^8*d^9*e + 15*a^4*b^7*d^
8*e^2 + 7*a^5*b^6*d^7*e^3 - 49*a^6*b^5*d^6*e^4 + 63*a^7*b^4*d^5*e^5 - 35*a^8*b^3*d^4*e^6 + 5*a^9*b^2*d^3*e^7 +
3*a^10*b*d^2*e^8 - a^11*d*e^9)*x^2 + (4*a^3*b^8*d^10 - 25*a^4*b^7*d^9*e + 63*a^5*b^6*d^8*e^2 - 77*a^6*b^5*d^7
*e^3 + 35*a^7*b^4*d^6*e^4 + 21*a^8*b^3*d^5*e^5 - 35*a^9*b^2*d^4*e^6 + 17*a^10*b*d^3*e^7 - 3*a^11*d^2*e^8)*x)

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Fricas [B]  time = 2.10146, size = 4320, normalized size = 19.46 \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)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/12*(3*b^7*d^7 - 28*a*b^6*d^6*e + 126*a^2*b^5*d^5*e^2 - 420*a^3*b^4*d^4*e^3 + 105*a^4*b^3*d^3*e^4 + 252*a^5*
b^2*d^2*e^5 - 42*a^6*b*d*e^6 + 4*a^7*e^7 - 420*(b^7*d*e^6 - a*b^6*e^7)*x^6 - 210*(5*b^7*d^2*e^5 + 2*a*b^6*d*e^
6 - 7*a^2*b^5*e^7)*x^5 - 70*(11*b^7*d^3*e^4 + 42*a*b^6*d^2*e^5 - 27*a^2*b^5*d*e^6 - 26*a^3*b^4*e^7)*x^4 - 35*(
3*b^7*d^4*e^3 + 76*a*b^6*d^3*e^4 + 54*a^2*b^5*d^2*e^5 - 108*a^3*b^4*d*e^6 - 25*a^4*b^3*e^7)*x^3 + 21*(b^7*d^5*
e^2 - 20*a*b^6*d^4*e^3 - 150*a^2*b^5*d^3*e^4 + 60*a^3*b^4*d^2*e^5 + 105*a^4*b^3*d*e^6 + 4*a^5*b^2*e^7)*x^2 - 7
*(b^7*d^6*e - 12*a*b^6*d^5*e^2 + 90*a^2*b^5*d^4*e^3 + 180*a^3*b^4*d^3*e^4 - 225*a^4*b^3*d^2*e^5 - 36*a^5*b^2*d
*e^6 + 2*a^6*b*e^7)*x - 420*(b^7*e^7*x^7 + a^4*b^3*d^3*e^4 + (3*b^7*d*e^6 + 4*a*b^6*e^7)*x^6 + 3*(b^7*d^2*e^5
+ 4*a*b^6*d*e^6 + 2*a^2*b^5*e^7)*x^5 + (b^7*d^3*e^4 + 12*a*b^6*d^2*e^5 + 18*a^2*b^5*d*e^6 + 4*a^3*b^4*e^7)*x^4
+ (4*a*b^6*d^3*e^4 + 18*a^2*b^5*d^2*e^5 + 12*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + 3*(2*a^2*b^5*d^3*e^4 + 4*a^3*
b^4*d^2*e^5 + a^4*b^3*d*e^6)*x^2 + (4*a^3*b^4*d^3*e^4 + 3*a^4*b^3*d^2*e^5)*x)*log(b*x + a) + 420*(b^7*e^7*x^7
+ a^4*b^3*d^3*e^4 + (3*b^7*d*e^6 + 4*a*b^6*e^7)*x^6 + 3*(b^7*d^2*e^5 + 4*a*b^6*d*e^6 + 2*a^2*b^5*e^7)*x^5 + (b
^7*d^3*e^4 + 12*a*b^6*d^2*e^5 + 18*a^2*b^5*d*e^6 + 4*a^3*b^4*e^7)*x^4 + (4*a*b^6*d^3*e^4 + 18*a^2*b^5*d^2*e^5
+ 12*a^3*b^4*d*e^6 + a^4*b^3*e^7)*x^3 + 3*(2*a^2*b^5*d^3*e^4 + 4*a^3*b^4*d^2*e^5 + a^4*b^3*d*e^6)*x^2 + (4*a^3
*b^4*d^3*e^4 + 3*a^4*b^3*d^2*e^5)*x)*log(e*x + d))/(a^4*b^8*d^11 - 8*a^5*b^7*d^10*e + 28*a^6*b^6*d^9*e^2 - 56*
a^7*b^5*d^8*e^3 + 70*a^8*b^4*d^7*e^4 - 56*a^9*b^3*d^6*e^5 + 28*a^10*b^2*d^5*e^6 - 8*a^11*b*d^4*e^7 + a^12*d^3*
e^8 + (b^12*d^8*e^3 - 8*a*b^11*d^7*e^4 + 28*a^2*b^10*d^6*e^5 - 56*a^3*b^9*d^5*e^6 + 70*a^4*b^8*d^4*e^7 - 56*a^
5*b^7*d^3*e^8 + 28*a^6*b^6*d^2*e^9 - 8*a^7*b^5*d*e^10 + a^8*b^4*e^11)*x^7 + (3*b^12*d^9*e^2 - 20*a*b^11*d^8*e^
3 + 52*a^2*b^10*d^7*e^4 - 56*a^3*b^9*d^6*e^5 - 14*a^4*b^8*d^5*e^6 + 112*a^5*b^7*d^4*e^7 - 140*a^6*b^6*d^3*e^8
+ 88*a^7*b^5*d^2*e^9 - 29*a^8*b^4*d*e^10 + 4*a^9*b^3*e^11)*x^6 + 3*(b^12*d^10*e - 4*a*b^11*d^9*e^2 - 2*a^2*b^1
0*d^8*e^3 + 40*a^3*b^9*d^7*e^4 - 98*a^4*b^8*d^6*e^5 + 112*a^5*b^7*d^5*e^6 - 56*a^6*b^6*d^4*e^7 - 8*a^7*b^5*d^3
*e^8 + 25*a^8*b^4*d^2*e^9 - 12*a^9*b^3*d*e^10 + 2*a^10*b^2*e^11)*x^5 + (b^12*d^11 + 4*a*b^11*d^10*e - 50*a^2*b
^10*d^9*e^2 + 140*a^3*b^9*d^8*e^3 - 130*a^4*b^8*d^7*e^4 - 112*a^5*b^7*d^6*e^5 + 392*a^6*b^6*d^5*e^6 - 400*a^7*
b^5*d^4*e^7 + 185*a^8*b^4*d^3*e^8 - 20*a^9*b^3*d^2*e^9 - 14*a^10*b^2*d*e^10 + 4*a^11*b*e^11)*x^4 + (4*a*b^11*d
^11 - 14*a^2*b^10*d^10*e - 20*a^3*b^9*d^9*e^2 + 185*a^4*b^8*d^8*e^3 - 400*a^5*b^7*d^7*e^4 + 392*a^6*b^6*d^6*e^
5 - 112*a^7*b^5*d^5*e^6 - 130*a^8*b^4*d^4*e^7 + 140*a^9*b^3*d^3*e^8 - 50*a^10*b^2*d^2*e^9 + 4*a^11*b*d*e^10 +
a^12*e^11)*x^3 + 3*(2*a^2*b^10*d^11 - 12*a^3*b^9*d^10*e + 25*a^4*b^8*d^9*e^2 - 8*a^5*b^7*d^8*e^3 - 56*a^6*b^6*
d^7*e^4 + 112*a^7*b^5*d^6*e^5 - 98*a^8*b^4*d^5*e^6 + 40*a^9*b^3*d^4*e^7 - 2*a^10*b^2*d^3*e^8 - 4*a^11*b*d^2*e^
9 + a^12*d*e^10)*x^2 + (4*a^3*b^9*d^11 - 29*a^4*b^8*d^10*e + 88*a^5*b^7*d^9*e^2 - 140*a^6*b^6*d^8*e^3 + 112*a^
7*b^5*d^7*e^4 - 14*a^8*b^4*d^6*e^5 - 56*a^9*b^3*d^5*e^6 + 52*a^10*b^2*d^4*e^7 - 20*a^11*b*d^3*e^8 + 3*a^12*d^2
*e^9)*x)

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Sympy [B]  time = 12.316, size = 2004, normalized size = 9.03 \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)**4/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-35*b**3*e**4*log(x + (-35*a**9*b**3*e**13/(a*e - b*d)**8 + 315*a**8*b**4*d*e**12/(a*e - b*d)**8 - 1260*a**7*b
**5*d**2*e**11/(a*e - b*d)**8 + 2940*a**6*b**6*d**3*e**10/(a*e - b*d)**8 - 4410*a**5*b**7*d**4*e**9/(a*e - b*d
)**8 + 4410*a**4*b**8*d**5*e**8/(a*e - b*d)**8 - 2940*a**3*b**9*d**6*e**7/(a*e - b*d)**8 + 1260*a**2*b**10*d**
7*e**6/(a*e - b*d)**8 - 315*a*b**11*d**8*e**5/(a*e - b*d)**8 + 35*a*b**3*e**5 + 35*b**12*d**9*e**4/(a*e - b*d)
**8 + 35*b**4*d*e**4)/(70*b**4*e**5))/(a*e - b*d)**8 + 35*b**3*e**4*log(x + (35*a**9*b**3*e**13/(a*e - b*d)**8
- 315*a**8*b**4*d*e**12/(a*e - b*d)**8 + 1260*a**7*b**5*d**2*e**11/(a*e - b*d)**8 - 2940*a**6*b**6*d**3*e**10
/(a*e - b*d)**8 + 4410*a**5*b**7*d**4*e**9/(a*e - b*d)**8 - 4410*a**4*b**8*d**5*e**8/(a*e - b*d)**8 + 2940*a**
3*b**9*d**6*e**7/(a*e - b*d)**8 - 1260*a**2*b**10*d**7*e**6/(a*e - b*d)**8 + 315*a*b**11*d**8*e**5/(a*e - b*d)
**8 + 35*a*b**3*e**5 - 35*b**12*d**9*e**4/(a*e - b*d)**8 + 35*b**4*d*e**4)/(70*b**4*e**5))/(a*e - b*d)**8 - (4
*a**6*e**6 - 38*a**5*b*d*e**5 + 214*a**4*b**2*d**2*e**4 + 319*a**3*b**3*d**3*e**3 - 101*a**2*b**4*d**4*e**2 +
25*a*b**5*d**5*e - 3*b**6*d**6 + 420*b**6*e**6*x**6 + x**5*(1470*a*b**5*e**6 + 1050*b**6*d*e**5) + x**4*(1820*
a**2*b**4*e**6 + 3710*a*b**5*d*e**5 + 770*b**6*d**2*e**4) + x**3*(875*a**3*b**3*e**6 + 4655*a**2*b**4*d*e**5 +
2765*a*b**5*d**2*e**4 + 105*b**6*d**3*e**3) + x**2*(84*a**4*b**2*e**6 + 2289*a**3*b**3*d*e**5 + 3549*a**2*b**
4*d**2*e**4 + 399*a*b**5*d**3*e**3 - 21*b**6*d**4*e**2) + x*(-14*a**5*b*e**6 + 238*a**4*b**2*d*e**5 + 1813*a**
3*b**3*d**2*e**4 + 553*a**2*b**4*d**3*e**3 - 77*a*b**5*d**4*e**2 + 7*b**6*d**5*e))/(12*a**11*d**3*e**7 - 84*a*
*10*b*d**4*e**6 + 252*a**9*b**2*d**5*e**5 - 420*a**8*b**3*d**6*e**4 + 420*a**7*b**4*d**7*e**3 - 252*a**6*b**5*
d**8*e**2 + 84*a**5*b**6*d**9*e - 12*a**4*b**7*d**10 + x**7*(12*a**7*b**4*e**10 - 84*a**6*b**5*d*e**9 + 252*a*
*5*b**6*d**2*e**8 - 420*a**4*b**7*d**3*e**7 + 420*a**3*b**8*d**4*e**6 - 252*a**2*b**9*d**5*e**5 + 84*a*b**10*d
**6*e**4 - 12*b**11*d**7*e**3) + x**6*(48*a**8*b**3*e**10 - 300*a**7*b**4*d*e**9 + 756*a**6*b**5*d**2*e**8 - 9
24*a**5*b**6*d**3*e**7 + 420*a**4*b**7*d**4*e**6 + 252*a**3*b**8*d**5*e**5 - 420*a**2*b**9*d**6*e**4 + 204*a*b
**10*d**7*e**3 - 36*b**11*d**8*e**2) + x**5*(72*a**9*b**2*e**10 - 360*a**8*b**3*d*e**9 + 540*a**7*b**4*d**2*e*
*8 + 252*a**6*b**5*d**3*e**7 - 1764*a**5*b**6*d**4*e**6 + 2268*a**4*b**7*d**5*e**5 - 1260*a**3*b**8*d**6*e**4
+ 180*a**2*b**9*d**7*e**3 + 108*a*b**10*d**8*e**2 - 36*b**11*d**9*e) + x**4*(48*a**10*b*e**10 - 120*a**9*b**2*
d*e**9 - 360*a**8*b**3*d**2*e**8 + 1860*a**7*b**4*d**3*e**7 - 2940*a**6*b**5*d**4*e**6 + 1764*a**5*b**6*d**5*e
**5 + 420*a**4*b**7*d**6*e**4 - 1140*a**3*b**8*d**7*e**3 + 540*a**2*b**9*d**8*e**2 - 60*a*b**10*d**9*e - 12*b*
*11*d**10) + x**3*(12*a**11*e**10 + 60*a**10*b*d*e**9 - 540*a**9*b**2*d**2*e**8 + 1140*a**8*b**3*d**3*e**7 - 4
20*a**7*b**4*d**4*e**6 - 1764*a**6*b**5*d**5*e**5 + 2940*a**5*b**6*d**6*e**4 - 1860*a**4*b**7*d**7*e**3 + 360*
a**3*b**8*d**8*e**2 + 120*a**2*b**9*d**9*e - 48*a*b**10*d**10) + x**2*(36*a**11*d*e**9 - 108*a**10*b*d**2*e**8
- 180*a**9*b**2*d**3*e**7 + 1260*a**8*b**3*d**4*e**6 - 2268*a**7*b**4*d**5*e**5 + 1764*a**6*b**5*d**6*e**4 -
252*a**5*b**6*d**7*e**3 - 540*a**4*b**7*d**8*e**2 + 360*a**3*b**8*d**9*e - 72*a**2*b**9*d**10) + x*(36*a**11*d
**2*e**8 - 204*a**10*b*d**3*e**7 + 420*a**9*b**2*d**4*e**6 - 252*a**8*b**3*d**5*e**5 - 420*a**7*b**4*d**6*e**4
+ 924*a**6*b**5*d**7*e**3 - 756*a**5*b**6*d**8*e**2 + 300*a**4*b**7*d**9*e - 48*a**3*b**8*d**10))

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Giac [B]  time = 1.19614, size = 909, normalized size = 4.09 \begin{align*} \frac{35 \, b^{4} e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{9} d^{8} - 8 \, a b^{8} d^{7} e + 28 \, a^{2} b^{7} d^{6} e^{2} - 56 \, a^{3} b^{6} d^{5} e^{3} + 70 \, a^{4} b^{5} d^{4} e^{4} - 56 \, a^{5} b^{4} d^{3} e^{5} + 28 \, a^{6} b^{3} d^{2} e^{6} - 8 \, a^{7} b^{2} d e^{7} + a^{8} b e^{8}} - \frac{35 \, b^{3} e^{5} \log \left ({\left | x e + d \right |}\right )}{b^{8} d^{8} e - 8 \, a b^{7} d^{7} e^{2} + 28 \, a^{2} b^{6} d^{6} e^{3} - 56 \, a^{3} b^{5} d^{5} e^{4} + 70 \, a^{4} b^{4} d^{4} e^{5} - 56 \, a^{5} b^{3} d^{3} e^{6} + 28 \, a^{6} b^{2} d^{2} e^{7} - 8 \, a^{7} b d e^{8} + a^{8} e^{9}} - \frac{3 \, b^{7} d^{7} - 28 \, a b^{6} d^{6} e + 126 \, a^{2} b^{5} d^{5} e^{2} - 420 \, a^{3} b^{4} d^{4} e^{3} + 105 \, a^{4} b^{3} d^{3} e^{4} + 252 \, a^{5} b^{2} d^{2} e^{5} - 42 \, a^{6} b d e^{6} + 4 \, a^{7} e^{7} - 420 \,{\left (b^{7} d e^{6} - a b^{6} e^{7}\right )} x^{6} - 210 \,{\left (5 \, b^{7} d^{2} e^{5} + 2 \, a b^{6} d e^{6} - 7 \, a^{2} b^{5} e^{7}\right )} x^{5} - 70 \,{\left (11 \, b^{7} d^{3} e^{4} + 42 \, a b^{6} d^{2} e^{5} - 27 \, a^{2} b^{5} d e^{6} - 26 \, a^{3} b^{4} e^{7}\right )} x^{4} - 35 \,{\left (3 \, b^{7} d^{4} e^{3} + 76 \, a b^{6} d^{3} e^{4} + 54 \, a^{2} b^{5} d^{2} e^{5} - 108 \, a^{3} b^{4} d e^{6} - 25 \, a^{4} b^{3} e^{7}\right )} x^{3} + 21 \,{\left (b^{7} d^{5} e^{2} - 20 \, a b^{6} d^{4} e^{3} - 150 \, a^{2} b^{5} d^{3} e^{4} + 60 \, a^{3} b^{4} d^{2} e^{5} + 105 \, a^{4} b^{3} d e^{6} + 4 \, a^{5} b^{2} e^{7}\right )} x^{2} - 7 \,{\left (b^{7} d^{6} e - 12 \, a b^{6} d^{5} e^{2} + 90 \, a^{2} b^{5} d^{4} e^{3} + 180 \, a^{3} b^{4} d^{3} e^{4} - 225 \, a^{4} b^{3} d^{2} e^{5} - 36 \, a^{5} b^{2} d e^{6} + 2 \, a^{6} b e^{7}\right )} x}{12 \,{\left (b d - a e\right )}^{8}{\left (b x + a\right )}^{4}{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

35*b^4*e^4*log(abs(b*x + a))/(b^9*d^8 - 8*a*b^8*d^7*e + 28*a^2*b^7*d^6*e^2 - 56*a^3*b^6*d^5*e^3 + 70*a^4*b^5*d
^4*e^4 - 56*a^5*b^4*d^3*e^5 + 28*a^6*b^3*d^2*e^6 - 8*a^7*b^2*d*e^7 + a^8*b*e^8) - 35*b^3*e^5*log(abs(x*e + d))
/(b^8*d^8*e - 8*a*b^7*d^7*e^2 + 28*a^2*b^6*d^6*e^3 - 56*a^3*b^5*d^5*e^4 + 70*a^4*b^4*d^4*e^5 - 56*a^5*b^3*d^3*
e^6 + 28*a^6*b^2*d^2*e^7 - 8*a^7*b*d*e^8 + a^8*e^9) - 1/12*(3*b^7*d^7 - 28*a*b^6*d^6*e + 126*a^2*b^5*d^5*e^2 -
420*a^3*b^4*d^4*e^3 + 105*a^4*b^3*d^3*e^4 + 252*a^5*b^2*d^2*e^5 - 42*a^6*b*d*e^6 + 4*a^7*e^7 - 420*(b^7*d*e^6
- a*b^6*e^7)*x^6 - 210*(5*b^7*d^2*e^5 + 2*a*b^6*d*e^6 - 7*a^2*b^5*e^7)*x^5 - 70*(11*b^7*d^3*e^4 + 42*a*b^6*d^
2*e^5 - 27*a^2*b^5*d*e^6 - 26*a^3*b^4*e^7)*x^4 - 35*(3*b^7*d^4*e^3 + 76*a*b^6*d^3*e^4 + 54*a^2*b^5*d^2*e^5 - 1
08*a^3*b^4*d*e^6 - 25*a^4*b^3*e^7)*x^3 + 21*(b^7*d^5*e^2 - 20*a*b^6*d^4*e^3 - 150*a^2*b^5*d^3*e^4 + 60*a^3*b^4
*d^2*e^5 + 105*a^4*b^3*d*e^6 + 4*a^5*b^2*e^7)*x^2 - 7*(b^7*d^6*e - 12*a*b^6*d^5*e^2 + 90*a^2*b^5*d^4*e^3 + 180
*a^3*b^4*d^3*e^4 - 225*a^4*b^3*d^2*e^5 - 36*a^5*b^2*d*e^6 + 2*a^6*b*e^7)*x)/((b*d - a*e)^8*(b*x + a)^4*(x*e +
d)^3)