3.1990 \(\int \frac{1}{(a+\frac{b}{x^3})^2 x^7} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\log \left (a x^3+b\right )}{3 b^2}+\frac{1}{3 b \left (a x^3+b\right )}+\frac{\log (x)}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0252161, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 44} \[ -\frac{\log \left (a x^3+b\right )}{3 b^2}+\frac{1}{3 b \left (a x^3+b\right )}+\frac{\log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)^2*x^7),x]

[Out]

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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}{\left (a+\frac{b}{x^3}\right )^2 x^7} \, dx &=\int \frac{1}{x \left (b+a x^3\right )^2} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x (b+a x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{b^2 x}-\frac{a}{b (b+a x)^2}-\frac{a}{b^2 (b+a x)}\right ) \, dx,x,x^3\right )\\ &=\frac{1}{3 b \left (b+a x^3\right )}+\frac{\log (x)}{b^2}-\frac{\log \left (b+a x^3\right )}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0129779, size = 33, normalized size = 0.87 \[ \frac{\frac{b}{a x^3+b}-\log \left (a x^3+b\right )+3 \log (x)}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)^2*x^7),x]

[Out]

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

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Maple [A]  time = 0.006, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{3\,b \left ( a{x}^{3}+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\ln \left ( a{x}^{3}+b \right ) }{3\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^3)^2/x^7,x)

[Out]

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

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Maxima [A]  time = 0.976622, size = 50, normalized size = 1.32 \begin{align*} \frac{1}{3 \,{\left (a b x^{3} + b^{2}\right )}} - \frac{\log \left (a x^{3} + b\right )}{3 \, b^{2}} + \frac{\log \left (x^{3}\right )}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^2/x^7,x, algorithm="maxima")

[Out]

1/3/(a*b*x^3 + b^2) - 1/3*log(a*x^3 + b)/b^2 + 1/3*log(x^3)/b^2

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Fricas [A]  time = 1.45785, size = 108, normalized size = 2.84 \begin{align*} -\frac{{\left (a x^{3} + b\right )} \log \left (a x^{3} + b\right ) - 3 \,{\left (a x^{3} + b\right )} \log \left (x\right ) - b}{3 \,{\left (a b^{2} x^{3} + b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^2/x^7,x, algorithm="fricas")

[Out]

-1/3*((a*x^3 + b)*log(a*x^3 + b) - 3*(a*x^3 + b)*log(x) - b)/(a*b^2*x^3 + b^3)

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Sympy [A]  time = 0.729122, size = 34, normalized size = 0.89 \begin{align*} \frac{1}{3 a b x^{3} + 3 b^{2}} + \frac{\log{\left (x \right )}}{b^{2}} - \frac{\log{\left (x^{3} + \frac{b}{a} \right )}}{3 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**3)**2/x**7,x)

[Out]

1/(3*a*b*x**3 + 3*b**2) + log(x)/b**2 - log(x**3 + b/a)/(3*b**2)

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Giac [A]  time = 1.22704, size = 61, normalized size = 1.61 \begin{align*} -\frac{\log \left ({\left | a x^{3} + b \right |}\right )}{3 \, b^{2}} + \frac{\log \left ({\left | x \right |}\right )}{b^{2}} + \frac{a x^{3} + 2 \, b}{3 \,{\left (a x^{3} + b\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^2/x^7,x, algorithm="giac")

[Out]

-1/3*log(abs(a*x^3 + b))/b^2 + log(abs(x))/b^2 + 1/3*(a*x^3 + 2*b)/((a*x^3 + b)*b^2)