### 3.1522 $$\int \frac{1}{(d+e x) (a^2+2 a b x+b^2 x^2)^2} \, dx$$

Optimal. Leaf size=107 $-\frac{e^2}{(a+b x) (b d-a e)^3}-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (b d-a e)}$

[Out]

-1/(3*(b*d - a*e)*(a + b*x)^3) + e/(2*(b*d - a*e)^2*(a + b*x)^2) - e^2/((b*d - a*e)^3*(a + b*x)) - (e^3*Log[a
+ b*x])/(b*d - a*e)^4 + (e^3*Log[d + e*x])/(b*d - a*e)^4

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Rubi [A]  time = 0.0680284, antiderivative size = 107, 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} $-\frac{e^2}{(a+b x) (b d-a e)^3}-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(3*(b*d - a*e)*(a + b*x)^3) + e/(2*(b*d - a*e)^2*(a + b*x)^2) - e^2/((b*d - a*e)^3*(a + b*x)) - (e^3*Log[a
+ b*x])/(b*d - a*e)^4 + (e^3*Log[d + e*x])/(b*d - a*e)^4

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

Mathematica [A]  time = 0.0448062, size = 107, normalized size = 1. $-\frac{e^2}{(a+b x) (b d-a e)^3}-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}+\frac{1}{3 (a+b x)^3 (a e-b d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1/(3*(-(b*d) + a*e)*(a + b*x)^3) + e/(2*(b*d - a*e)^2*(a + b*x)^2) - e^2/((b*d - a*e)^3*(a + b*x)) - (e^3*Log[
a + b*x])/(b*d - a*e)^4 + (e^3*Log[d + e*x])/(b*d - a*e)^4

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Maple [A]  time = 0.05, size = 103, normalized size = 1. \begin{align*}{\frac{{e}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{4}}}+{\frac{1}{ \left ( 3\,ae-3\,bd \right ) \left ( bx+a \right ) ^{3}}}+{\frac{e}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{e}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{{e}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.21259, size = 487, normalized size = 4.55 \begin{align*} -\frac{e^{3} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac{e^{3} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac{6 \, b^{2} e^{2} x^{2} + 2 \, b^{2} d^{2} - 7 \, a b d e + 11 \, a^{2} e^{2} - 3 \,{\left (b^{2} d e - 5 \, a b e^{2}\right )} x}{6 \,{\left (a^{3} b^{3} d^{3} - 3 \, a^{4} b^{2} d^{2} e + 3 \, a^{5} b d e^{2} - a^{6} e^{3} +{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{3} + 3 \,{\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^3*log(b*x + a)/(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4) + e^3*log(e*x + d)/(
b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4) - 1/6*(6*b^2*e^2*x^2 + 2*b^2*d^2 - 7*a*
b*d*e + 11*a^2*e^2 - 3*(b^2*d*e - 5*a*b*e^2)*x)/(a^3*b^3*d^3 - 3*a^4*b^2*d^2*e + 3*a^5*b*d*e^2 - a^6*e^3 + (b^
6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*x^3 + 3*(a*b^5*d^3 - 3*a^2*b^4*d^2*e + 3*a^3*b^3*d*e^2
- a^4*b^2*e^3)*x^2 + 3*(a^2*b^4*d^3 - 3*a^3*b^3*d^2*e + 3*a^4*b^2*d*e^2 - a^5*b*e^3)*x)

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Fricas [B]  time = 1.76155, size = 856, normalized size = 8. \begin{align*} -\frac{2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (a^{3} b^{4} d^{4} - 4 \, a^{4} b^{3} d^{3} e + 6 \, a^{5} b^{2} d^{2} e^{2} - 4 \, a^{6} b d e^{3} + a^{7} e^{4} +{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{6} d^{4} - 4 \, a^{2} b^{5} d^{3} e + 6 \, a^{3} b^{4} d^{2} e^{2} - 4 \, a^{4} b^{3} d e^{3} + a^{5} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} d^{4} - 4 \, a^{3} b^{4} d^{3} e + 6 \, a^{4} b^{3} d^{2} e^{2} - 4 \, a^{5} b^{2} d e^{3} + a^{6} b e^{4}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^3*d^3 - 9*a*b^2*d^2*e + 18*a^2*b*d*e^2 - 11*a^3*e^3 + 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 - 3*(b^3*d^2*e -
6*a*b^2*d*e^2 + 5*a^2*b*e^3)*x + 6*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*log(b*x + a) - 6
*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*log(e*x + d))/(a^3*b^4*d^4 - 4*a^4*b^3*d^3*e + 6*a^
5*b^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*e^4 + (b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4
*b^3*e^4)*x^3 + 3*(a*b^6*d^4 - 4*a^2*b^5*d^3*e + 6*a^3*b^4*d^2*e^2 - 4*a^4*b^3*d*e^3 + a^5*b^2*e^4)*x^2 + 3*(a
^2*b^5*d^4 - 4*a^3*b^4*d^3*e + 6*a^4*b^3*d^2*e^2 - 4*a^5*b^2*d*e^3 + a^6*b*e^4)*x)

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Sympy [B]  time = 2.20426, size = 570, normalized size = 5.33 \begin{align*} \frac{e^{3} \log{\left (x + \frac{- \frac{a^{5} e^{8}}{\left (a e - b d\right )^{4}} + \frac{5 a^{4} b d e^{7}}{\left (a e - b d\right )^{4}} - \frac{10 a^{3} b^{2} d^{2} e^{6}}{\left (a e - b d\right )^{4}} + \frac{10 a^{2} b^{3} d^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac{5 a b^{4} d^{4} e^{4}}{\left (a e - b d\right )^{4}} + a e^{4} + \frac{b^{5} d^{5} e^{3}}{\left (a e - b d\right )^{4}} + b d e^{3}}{2 b e^{4}} \right )}}{\left (a e - b d\right )^{4}} - \frac{e^{3} \log{\left (x + \frac{\frac{a^{5} e^{8}}{\left (a e - b d\right )^{4}} - \frac{5 a^{4} b d e^{7}}{\left (a e - b d\right )^{4}} + \frac{10 a^{3} b^{2} d^{2} e^{6}}{\left (a e - b d\right )^{4}} - \frac{10 a^{2} b^{3} d^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac{5 a b^{4} d^{4} e^{4}}{\left (a e - b d\right )^{4}} + a e^{4} - \frac{b^{5} d^{5} e^{3}}{\left (a e - b d\right )^{4}} + b d e^{3}}{2 b e^{4}} \right )}}{\left (a e - b d\right )^{4}} + \frac{11 a^{2} e^{2} - 7 a b d e + 2 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (15 a b e^{2} - 3 b^{2} d e\right )}{6 a^{6} e^{3} - 18 a^{5} b d e^{2} + 18 a^{4} b^{2} d^{2} e - 6 a^{3} b^{3} d^{3} + x^{3} \left (6 a^{3} b^{3} e^{3} - 18 a^{2} b^{4} d e^{2} + 18 a b^{5} d^{2} e - 6 b^{6} d^{3}\right ) + x^{2} \left (18 a^{4} b^{2} e^{3} - 54 a^{3} b^{3} d e^{2} + 54 a^{2} b^{4} d^{2} e - 18 a b^{5} d^{3}\right ) + x \left (18 a^{5} b e^{3} - 54 a^{4} b^{2} d e^{2} + 54 a^{3} b^{3} d^{2} e - 18 a^{2} b^{4} d^{3}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

e**3*log(x + (-a**5*e**8/(a*e - b*d)**4 + 5*a**4*b*d*e**7/(a*e - b*d)**4 - 10*a**3*b**2*d**2*e**6/(a*e - b*d)*
*4 + 10*a**2*b**3*d**3*e**5/(a*e - b*d)**4 - 5*a*b**4*d**4*e**4/(a*e - b*d)**4 + a*e**4 + b**5*d**5*e**3/(a*e
- b*d)**4 + b*d*e**3)/(2*b*e**4))/(a*e - b*d)**4 - e**3*log(x + (a**5*e**8/(a*e - b*d)**4 - 5*a**4*b*d*e**7/(a
*e - b*d)**4 + 10*a**3*b**2*d**2*e**6/(a*e - b*d)**4 - 10*a**2*b**3*d**3*e**5/(a*e - b*d)**4 + 5*a*b**4*d**4*e
**4/(a*e - b*d)**4 + a*e**4 - b**5*d**5*e**3/(a*e - b*d)**4 + b*d*e**3)/(2*b*e**4))/(a*e - b*d)**4 + (11*a**2*
e**2 - 7*a*b*d*e + 2*b**2*d**2 + 6*b**2*e**2*x**2 + x*(15*a*b*e**2 - 3*b**2*d*e))/(6*a**6*e**3 - 18*a**5*b*d*e
**2 + 18*a**4*b**2*d**2*e - 6*a**3*b**3*d**3 + x**3*(6*a**3*b**3*e**3 - 18*a**2*b**4*d*e**2 + 18*a*b**5*d**2*e
- 6*b**6*d**3) + x**2*(18*a**4*b**2*e**3 - 54*a**3*b**3*d*e**2 + 54*a**2*b**4*d**2*e - 18*a*b**5*d**3) + x*(1
8*a**5*b*e**3 - 54*a**4*b**2*d*e**2 + 54*a**3*b**3*d**2*e - 18*a**2*b**4*d**3))

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Giac [B]  time = 1.17333, size = 316, normalized size = 2.95 \begin{align*} -\frac{b e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac{e^{4} \log \left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac{2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x}{6 \,{\left (b d - a e\right )}^{4}{\left (b x + a\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*e^3*log(abs(b*x + a))/(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4) + e^4*log
(abs(x*e + d))/(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5) - 1/6*(2*b^3*d^3 -
9*a*b^2*d^2*e + 18*a^2*b*d*e^2 - 11*a^3*e^3 + 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 - 3*(b^3*d^2*e - 6*a*b^2*d*e^2 + 5
*a^2*b*e^3)*x)/((b*d - a*e)^4*(b*x + a)^3)