3.1977 \(\int \frac{1}{(a+\frac{b}{x^3}) x^7} \, dx\)

Optimal. Leaf size=35 \[ \frac{a \log \left (a x^3+b\right )}{3 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{3 b x^3} \]

[Out]

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

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Rubi [A]  time = 0.0231146, antiderivative size = 35, 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} \[ \frac{a \log \left (a x^3+b\right )}{3 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)*x^7),x]

[Out]

-1/(3*b*x^3) - (a*Log[x])/b^2 + (a*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 ) x^7} \, dx &=\int \frac{1}{x^4 \left (b+a x^3\right )} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 (b+a x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{b x^2}-\frac{a}{b^2 x}+\frac{a^2}{b^2 (b+a x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{3 b x^3}-\frac{a \log (x)}{b^2}+\frac{a \log \left (b+a x^3\right )}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0069067, size = 35, normalized size = 1. \[ \frac{a \log \left (a x^3+b\right )}{3 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)*x^7),x]

[Out]

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

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Maple [A]  time = 0.005, size = 32, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,b{x}^{3}}}-{\frac{a\ln \left ( x \right ) }{{b}^{2}}}+{\frac{a\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)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.00295, size = 45, normalized size = 1.29 \begin{align*} \frac{a \log \left (a x^{3} + b\right )}{3 \, b^{2}} - \frac{a \log \left (x^{3}\right )}{3 \, b^{2}} - \frac{1}{3 \, b x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44066, size = 80, normalized size = 2.29 \begin{align*} \frac{a x^{3} \log \left (a x^{3} + b\right ) - 3 \, a x^{3} \log \left (x\right ) - b}{3 \, b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.655607, size = 31, normalized size = 0.89 \begin{align*} - \frac{a \log{\left (x \right )}}{b^{2}} + \frac{a \log{\left (x^{3} + \frac{b}{a} \right )}}{3 b^{2}} - \frac{1}{3 b x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19547, size = 57, normalized size = 1.63 \begin{align*} \frac{a \log \left ({\left | a x^{3} + b \right |}\right )}{3 \, b^{2}} - \frac{a \log \left ({\left | x \right |}\right )}{b^{2}} + \frac{a x^{3} - b}{3 \, b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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