3.1524 \(\int \frac{1}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=170 \[ -\frac{6 b^2 e^2}{(a+b x) (b d-a e)^5}-\frac{10 b^2 e^3 \log (a+b x)}{(b d-a e)^6}+\frac{10 b^2 e^3 \log (d+e x)}{(b d-a e)^6}+\frac{3 b^2 e}{2 (a+b x)^2 (b d-a e)^4}-\frac{b^2}{3 (a+b x)^3 (b d-a e)^3}-\frac{4 b e^3}{(d+e x) (b d-a e)^5}-\frac{e^3}{2 (d+e x)^2 (b d-a e)^4} \]

[Out]

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

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Rubi [A]  time = 0.148281, antiderivative size = 170, 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} \[ -\frac{6 b^2 e^2}{(a+b x) (b d-a e)^5}-\frac{10 b^2 e^3 \log (a+b x)}{(b d-a e)^6}+\frac{10 b^2 e^3 \log (d+e x)}{(b d-a e)^6}+\frac{3 b^2 e}{2 (a+b x)^2 (b d-a e)^4}-\frac{b^2}{3 (a+b x)^3 (b d-a e)^3}-\frac{4 b e^3}{(d+e x) (b d-a e)^5}-\frac{e^3}{2 (d+e x)^2 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.214391, size = 154, normalized size = 0.91 \[ -\frac{\frac{36 b^2 e^2 (b d-a e)}{a+b x}-\frac{9 b^2 e (b d-a e)^2}{(a+b x)^2}+\frac{2 b^2 (b d-a e)^3}{(a+b x)^3}+60 b^2 e^3 \log (a+b x)+\frac{24 b e^3 (b d-a e)}{d+e x}+\frac{3 e^3 (b d-a e)^2}{(d+e x)^2}-60 b^2 e^3 \log (d+e x)}{6 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-((2*b^2*(b*d - a*e)^3)/(a + b*x)^3 - (9*b^2*e*(b*d - a*e)^2)/(a + b*x)^2 + (36*b^2*e^2*(b*d - a*e))/(a + b*x)
+ (3*e^3*(b*d - a*e)^2)/(d + e*x)^2 + (24*b*e^3*(b*d - a*e))/(d + e*x) + 60*b^2*e^3*Log[a + b*x] - 60*b^2*e^3
*Log[d + e*x])/(6*(b*d - a*e)^6)

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Maple [A]  time = 0.055, size = 165, normalized size = 1. \begin{align*} -{\frac{{e}^{3}}{2\, \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{b}^{2}{e}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{6}}}+4\,{\frac{{e}^{3}b}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}+{\frac{{b}^{2}}{3\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{3}}}-10\,{\frac{{b}^{2}{e}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{6}}}+6\,{\frac{{b}^{2}{e}^{2}}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }}+{\frac{3\,{b}^{2}e}{2\, \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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Maxima [B]  time = 1.36047, size = 1200, normalized size = 7.06 \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)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

-10*b^2*e^3*log(b*x + a)/(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) + 10*b^2*e^3*log(e*x + d)/(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) - 1/6*(60*b^4*e^4*x^4 + 2*b^4*d^4 - 13*a*b^3*d^3*
e + 47*a^2*b^2*d^2*e^2 + 27*a^3*b*d*e^3 - 3*a^4*e^4 + 30*(3*b^4*d*e^3 + 5*a*b^3*e^4)*x^3 + 10*(2*b^4*d^2*e^2 +
23*a*b^3*d*e^3 + 11*a^2*b^2*e^4)*x^2 - 5*(b^4*d^3*e - 11*a*b^3*d^2*e^2 - 35*a^2*b^2*d*e^3 - 3*a^3*b*e^4)*x)/(
a^3*b^5*d^7 - 5*a^4*b^4*d^6*e + 10*a^5*b^3*d^5*e^2 - 10*a^6*b^2*d^4*e^3 + 5*a^7*b*d^3*e^4 - a^8*d^2*e^5 + (b^8
*d^5*e^2 - 5*a*b^7*d^4*e^3 + 10*a^2*b^6*d^3*e^4 - 10*a^3*b^5*d^2*e^5 + 5*a^4*b^4*d*e^6 - a^5*b^3*e^7)*x^5 + (2
*b^8*d^6*e - 7*a*b^7*d^5*e^2 + 5*a^2*b^6*d^4*e^3 + 10*a^3*b^5*d^3*e^4 - 20*a^4*b^4*d^2*e^5 + 13*a^5*b^3*d*e^6
- 3*a^6*b^2*e^7)*x^4 + (b^8*d^7 + a*b^7*d^6*e - 17*a^2*b^6*d^5*e^2 + 35*a^3*b^5*d^4*e^3 - 25*a^4*b^4*d^3*e^4 -
a^5*b^3*d^2*e^5 + 9*a^6*b^2*d*e^6 - 3*a^7*b*e^7)*x^3 + (3*a*b^7*d^7 - 9*a^2*b^6*d^6*e + a^3*b^5*d^5*e^2 + 25*
a^4*b^4*d^4*e^3 - 35*a^5*b^3*d^3*e^4 + 17*a^6*b^2*d^2*e^5 - a^7*b*d*e^6 - a^8*e^7)*x^2 + (3*a^2*b^6*d^7 - 13*a
^3*b^5*d^6*e + 20*a^4*b^4*d^5*e^2 - 10*a^5*b^3*d^4*e^3 - 5*a^6*b^2*d^3*e^4 + 7*a^7*b*d^2*e^5 - 2*a^8*d*e^6)*x)

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Fricas [B]  time = 2.03205, size = 2313, normalized size = 13.61 \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)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

-1/6*(2*b^5*d^5 - 15*a*b^4*d^4*e + 60*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 30*a^4*b*d*e^4 + 3*a^5*e^5 + 60*(
b^5*d*e^4 - a*b^4*e^5)*x^4 + 30*(3*b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 5*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 + 21*a
*b^4*d^2*e^3 - 12*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 12*a*b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3 +
32*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x + 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + (b^
5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 + (3*a^
2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*log(b*x + a) - 60*(b^5*e^5*x^5 + a^3*b^2*d^2*e^3 + (2*b^5*d*e^4 + 3*a*b^4*
e^5)*x^4 + (b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + (3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^
5)*x^2 + (3*a^2*b^3*d^2*e^3 + 2*a^3*b^2*d*e^4)*x)*log(e*x + d))/(a^3*b^6*d^8 - 6*a^4*b^5*d^7*e + 15*a^5*b^4*d^
6*e^2 - 20*a^6*b^3*d^5*e^3 + 15*a^7*b^2*d^4*e^4 - 6*a^8*b*d^3*e^5 + a^9*d^2*e^6 + (b^9*d^6*e^2 - 6*a*b^8*d^5*e
^3 + 15*a^2*b^7*d^4*e^4 - 20*a^3*b^6*d^3*e^5 + 15*a^4*b^5*d^2*e^6 - 6*a^5*b^4*d*e^7 + a^6*b^3*e^8)*x^5 + (2*b^
9*d^7*e - 9*a*b^8*d^6*e^2 + 12*a^2*b^7*d^5*e^3 + 5*a^3*b^6*d^4*e^4 - 30*a^4*b^5*d^3*e^5 + 33*a^5*b^4*d^2*e^6 -
16*a^6*b^3*d*e^7 + 3*a^7*b^2*e^8)*x^4 + (b^9*d^8 - 18*a^2*b^7*d^6*e^2 + 52*a^3*b^6*d^5*e^3 - 60*a^4*b^5*d^4*e
^4 + 24*a^5*b^4*d^3*e^5 + 10*a^6*b^3*d^2*e^6 - 12*a^7*b^2*d*e^7 + 3*a^8*b*e^8)*x^3 + (3*a*b^8*d^8 - 12*a^2*b^7
*d^7*e + 10*a^3*b^6*d^6*e^2 + 24*a^4*b^5*d^5*e^3 - 60*a^5*b^4*d^4*e^4 + 52*a^6*b^3*d^3*e^5 - 18*a^7*b^2*d^2*e^
6 + a^9*e^8)*x^2 + (3*a^2*b^7*d^8 - 16*a^3*b^6*d^7*e + 33*a^4*b^5*d^6*e^2 - 30*a^5*b^4*d^5*e^3 + 5*a^6*b^3*d^4
*e^4 + 12*a^7*b^2*d^3*e^5 - 9*a^8*b*d^2*e^6 + 2*a^9*d*e^7)*x)

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Sympy [B]  time = 5.09614, size = 1217, normalized size = 7.16 \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)**3/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

10*b**2*e**3*log(x + (-10*a**7*b**2*e**10/(a*e - b*d)**6 + 70*a**6*b**3*d*e**9/(a*e - b*d)**6 - 210*a**5*b**4*
d**2*e**8/(a*e - b*d)**6 + 350*a**4*b**5*d**3*e**7/(a*e - b*d)**6 - 350*a**3*b**6*d**4*e**6/(a*e - b*d)**6 + 2
10*a**2*b**7*d**5*e**5/(a*e - b*d)**6 - 70*a*b**8*d**6*e**4/(a*e - b*d)**6 + 10*a*b**2*e**4 + 10*b**9*d**7*e**
3/(a*e - b*d)**6 + 10*b**3*d*e**3)/(20*b**3*e**4))/(a*e - b*d)**6 - 10*b**2*e**3*log(x + (10*a**7*b**2*e**10/(
a*e - b*d)**6 - 70*a**6*b**3*d*e**9/(a*e - b*d)**6 + 210*a**5*b**4*d**2*e**8/(a*e - b*d)**6 - 350*a**4*b**5*d*
*3*e**7/(a*e - b*d)**6 + 350*a**3*b**6*d**4*e**6/(a*e - b*d)**6 - 210*a**2*b**7*d**5*e**5/(a*e - b*d)**6 + 70*
a*b**8*d**6*e**4/(a*e - b*d)**6 + 10*a*b**2*e**4 - 10*b**9*d**7*e**3/(a*e - b*d)**6 + 10*b**3*d*e**3)/(20*b**3
*e**4))/(a*e - b*d)**6 + (-3*a**4*e**4 + 27*a**3*b*d*e**3 + 47*a**2*b**2*d**2*e**2 - 13*a*b**3*d**3*e + 2*b**4
*d**4 + 60*b**4*e**4*x**4 + x**3*(150*a*b**3*e**4 + 90*b**4*d*e**3) + x**2*(110*a**2*b**2*e**4 + 230*a*b**3*d*
e**3 + 20*b**4*d**2*e**2) + x*(15*a**3*b*e**4 + 175*a**2*b**2*d*e**3 + 55*a*b**3*d**2*e**2 - 5*b**4*d**3*e))/(
6*a**8*d**2*e**5 - 30*a**7*b*d**3*e**4 + 60*a**6*b**2*d**4*e**3 - 60*a**5*b**3*d**5*e**2 + 30*a**4*b**4*d**6*e
- 6*a**3*b**5*d**7 + x**5*(6*a**5*b**3*e**7 - 30*a**4*b**4*d*e**6 + 60*a**3*b**5*d**2*e**5 - 60*a**2*b**6*d**
3*e**4 + 30*a*b**7*d**4*e**3 - 6*b**8*d**5*e**2) + x**4*(18*a**6*b**2*e**7 - 78*a**5*b**3*d*e**6 + 120*a**4*b*
*4*d**2*e**5 - 60*a**3*b**5*d**3*e**4 - 30*a**2*b**6*d**4*e**3 + 42*a*b**7*d**5*e**2 - 12*b**8*d**6*e) + x**3*
(18*a**7*b*e**7 - 54*a**6*b**2*d*e**6 + 6*a**5*b**3*d**2*e**5 + 150*a**4*b**4*d**3*e**4 - 210*a**3*b**5*d**4*e
**3 + 102*a**2*b**6*d**5*e**2 - 6*a*b**7*d**6*e - 6*b**8*d**7) + x**2*(6*a**8*e**7 + 6*a**7*b*d*e**6 - 102*a**
6*b**2*d**2*e**5 + 210*a**5*b**3*d**3*e**4 - 150*a**4*b**4*d**4*e**3 - 6*a**3*b**5*d**5*e**2 + 54*a**2*b**6*d*
*6*e - 18*a*b**7*d**7) + x*(12*a**8*d*e**6 - 42*a**7*b*d**2*e**5 + 30*a**6*b**2*d**3*e**4 + 60*a**5*b**3*d**4*
e**3 - 120*a**4*b**4*d**5*e**2 + 78*a**3*b**5*d**6*e - 18*a**2*b**6*d**7))

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Giac [B]  time = 1.12635, size = 587, normalized size = 3.45 \begin{align*} -\frac{10 \, b^{3} e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac{10 \, b^{2} e^{4} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac{2 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 20 \, a^{3} b^{2} d^{2} e^{3} - 30 \, a^{4} b d e^{4} + 3 \, a^{5} e^{5} + 60 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 30 \,{\left (3 \, b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 5 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \,{\left (2 \, b^{5} d^{3} e^{2} + 21 \, a b^{4} d^{2} e^{3} - 12 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \,{\left (b^{5} d^{4} e - 12 \, a b^{4} d^{3} e^{2} - 24 \, a^{2} b^{3} d^{2} e^{3} + 32 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x}{6 \,{\left (b d - a e\right )}^{6}{\left (b x + a\right )}^{3}{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

-10*b^3*e^3*log(abs(b*x + a))/(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*
d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6) + 10*b^2*e^4*log(abs(x*e + d))/(b^6*d^6*e - 6*a*b^5*d^5*e^2 + 15*a^2*b^
4*d^4*e^3 - 20*a^3*b^3*d^3*e^4 + 15*a^4*b^2*d^2*e^5 - 6*a^5*b*d*e^6 + a^6*e^7) - 1/6*(2*b^5*d^5 - 15*a*b^4*d^4
*e + 60*a^2*b^3*d^3*e^2 - 20*a^3*b^2*d^2*e^3 - 30*a^4*b*d*e^4 + 3*a^5*e^5 + 60*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 3
0*(3*b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 5*a^2*b^3*e^5)*x^3 + 10*(2*b^5*d^3*e^2 + 21*a*b^4*d^2*e^3 - 12*a^2*b^3*d*e^
4 - 11*a^3*b^2*e^5)*x^2 - 5*(b^5*d^4*e - 12*a*b^4*d^3*e^2 - 24*a^2*b^3*d^2*e^3 + 32*a^3*b^2*d*e^4 + 3*a^4*b*e^
5)*x)/((b*d - a*e)^6*(b*x + a)^3*(x*e + d)^2)