### 3.1513 $$\int \frac{1}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)} \, dx$$

Optimal. Leaf size=110 $-\frac{b^2}{(a+b x) (b d-a e)^3}-\frac{3 b^2 e \log (a+b x)}{(b d-a e)^4}+\frac{3 b^2 e \log (d+e x)}{(b d-a e)^4}-\frac{2 b e}{(d+e x) (b d-a e)^3}-\frac{e}{2 (d+e x)^2 (b d-a e)^2}$

[Out]

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

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Rubi [A]  time = 0.0786082, antiderivative size = 110, 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} $-\frac{b^2}{(a+b x) (b d-a e)^3}-\frac{3 b^2 e \log (a+b x)}{(b d-a e)^4}+\frac{3 b^2 e \log (d+e x)}{(b d-a e)^4}-\frac{2 b e}{(d+e x) (b d-a e)^3}-\frac{e}{2 (d+e x)^2 (b d-a e)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0995215, size = 97, normalized size = 0.88 $-\frac{\frac{2 b^2 (b d-a e)}{a+b x}+6 b^2 e \log (a+b x)+\frac{4 b e (b d-a e)}{d+e x}+\frac{e (b d-a e)^2}{(d+e x)^2}-6 b^2 e \log (d+e x)}{2 (b d-a e)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.055, size = 108, normalized size = 1. \begin{align*} -{\frac{e}{2\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{2}e\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{4}}}+2\,{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) }}+{\frac{{b}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-3\,{\frac{{b}^{2}e\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)^3/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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Maxima [B]  time = 1.16565, size = 521, normalized size = 4.74 \begin{align*} -\frac{3 \, b^{2} e \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{3 \, b^{2} e \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} + 5 \, a b d e - a^{2} e^{2} + 3 \,{\left (3 \, b^{2} d e + a b e^{2}\right )} x}{2 \,{\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} +{\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} +{\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} +{\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \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),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.78047, size = 991, normalized size = 9.01 \begin{align*} -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e +{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} +{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e +{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} +{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a b^{4} d^{6} - 4 \, a^{2} b^{3} d^{5} e + 6 \, a^{3} b^{2} d^{4} e^{2} - 4 \, a^{4} b d^{3} e^{3} + a^{5} d^{2} e^{4} +{\left (b^{5} d^{4} e^{2} - 4 \, a b^{4} d^{3} e^{3} + 6 \, a^{2} b^{3} d^{2} e^{4} - 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{3} +{\left (2 \, b^{5} d^{5} e - 7 \, a b^{4} d^{4} e^{2} + 8 \, a^{2} b^{3} d^{3} e^{3} - 2 \, a^{3} b^{2} d^{2} e^{4} - 2 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{2} +{\left (b^{5} d^{6} - 2 \, a b^{4} d^{5} e - 2 \, a^{2} b^{3} d^{4} e^{2} + 8 \, a^{3} b^{2} d^{3} e^{3} - 7 \, a^{4} b d^{2} e^{4} + 2 \, a^{5} d e^{5}\right )} x\right )}} \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),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.1331, size = 632, normalized size = 5.75 \begin{align*} \frac{3 b^{2} e \log{\left (x + \frac{- \frac{3 a^{5} b^{2} e^{6}}{\left (a e - b d\right )^{4}} + \frac{15 a^{4} b^{3} d e^{5}}{\left (a e - b d\right )^{4}} - \frac{30 a^{3} b^{4} d^{2} e^{4}}{\left (a e - b d\right )^{4}} + \frac{30 a^{2} b^{5} d^{3} e^{3}}{\left (a e - b d\right )^{4}} - \frac{15 a b^{6} d^{4} e^{2}}{\left (a e - b d\right )^{4}} + 3 a b^{2} e^{2} + \frac{3 b^{7} d^{5} e}{\left (a e - b d\right )^{4}} + 3 b^{3} d e}{6 b^{3} e^{2}} \right )}}{\left (a e - b d\right )^{4}} - \frac{3 b^{2} e \log{\left (x + \frac{\frac{3 a^{5} b^{2} e^{6}}{\left (a e - b d\right )^{4}} - \frac{15 a^{4} b^{3} d e^{5}}{\left (a e - b d\right )^{4}} + \frac{30 a^{3} b^{4} d^{2} e^{4}}{\left (a e - b d\right )^{4}} - \frac{30 a^{2} b^{5} d^{3} e^{3}}{\left (a e - b d\right )^{4}} + \frac{15 a b^{6} d^{4} e^{2}}{\left (a e - b d\right )^{4}} + 3 a b^{2} e^{2} - \frac{3 b^{7} d^{5} e}{\left (a e - b d\right )^{4}} + 3 b^{3} d e}{6 b^{3} e^{2}} \right )}}{\left (a e - b d\right )^{4}} + \frac{- a^{2} e^{2} + 5 a b d e + 2 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} + 9 b^{2} d e\right )}{2 a^{4} d^{2} e^{3} - 6 a^{3} b d^{3} e^{2} + 6 a^{2} b^{2} d^{4} e - 2 a b^{3} d^{5} + x^{3} \left (2 a^{3} b e^{5} - 6 a^{2} b^{2} d e^{4} + 6 a b^{3} d^{2} e^{3} - 2 b^{4} d^{3} e^{2}\right ) + x^{2} \left (2 a^{4} e^{5} - 2 a^{3} b d e^{4} - 6 a^{2} b^{2} d^{2} e^{3} + 10 a b^{3} d^{3} e^{2} - 4 b^{4} d^{4} e\right ) + x \left (4 a^{4} d e^{4} - 10 a^{3} b d^{2} e^{3} + 6 a^{2} b^{2} d^{3} e^{2} + 2 a b^{3} d^{4} e - 2 b^{4} d^{5}\right )} \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),x)

[Out]

3*b**2*e*log(x + (-3*a**5*b**2*e**6/(a*e - b*d)**4 + 15*a**4*b**3*d*e**5/(a*e - b*d)**4 - 30*a**3*b**4*d**2*e*
*4/(a*e - b*d)**4 + 30*a**2*b**5*d**3*e**3/(a*e - b*d)**4 - 15*a*b**6*d**4*e**2/(a*e - b*d)**4 + 3*a*b**2*e**2
+ 3*b**7*d**5*e/(a*e - b*d)**4 + 3*b**3*d*e)/(6*b**3*e**2))/(a*e - b*d)**4 - 3*b**2*e*log(x + (3*a**5*b**2*e*
*6/(a*e - b*d)**4 - 15*a**4*b**3*d*e**5/(a*e - b*d)**4 + 30*a**3*b**4*d**2*e**4/(a*e - b*d)**4 - 30*a**2*b**5*
d**3*e**3/(a*e - b*d)**4 + 15*a*b**6*d**4*e**2/(a*e - b*d)**4 + 3*a*b**2*e**2 - 3*b**7*d**5*e/(a*e - b*d)**4 +
3*b**3*d*e)/(6*b**3*e**2))/(a*e - b*d)**4 + (-a**2*e**2 + 5*a*b*d*e + 2*b**2*d**2 + 6*b**2*e**2*x**2 + x*(3*a
*b*e**2 + 9*b**2*d*e))/(2*a**4*d**2*e**3 - 6*a**3*b*d**3*e**2 + 6*a**2*b**2*d**4*e - 2*a*b**3*d**5 + x**3*(2*a
**3*b*e**5 - 6*a**2*b**2*d*e**4 + 6*a*b**3*d**2*e**3 - 2*b**4*d**3*e**2) + x**2*(2*a**4*e**5 - 2*a**3*b*d*e**4
- 6*a**2*b**2*d**2*e**3 + 10*a*b**3*d**3*e**2 - 4*b**4*d**4*e) + x*(4*a**4*d*e**4 - 10*a**3*b*d**2*e**3 + 6*a
**2*b**2*d**3*e**2 + 2*a*b**3*d**4*e - 2*b**4*d**5))

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Giac [B]  time = 1.16789, size = 335, normalized size = 3.05 \begin{align*} -\frac{3 \, b^{3} e \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{3 \, b^{2} e^{2} \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} + 3 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{2 \,{\left (b d - a e\right )}^{4}{\left (b x + a\right )}{\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),x, algorithm="giac")

[Out]

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