3.1294 \(\int \frac{1}{x^6 (3+b x^5)} \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{45} b \log \left (b x^5+3\right )-\frac{1}{9} b \log (x)-\frac{1}{15 x^5} \]

[Out]

-1/(15*x^5) - (b*Log[x])/9 + (b*Log[3 + b*x^5])/45

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Rubi [A]  time = 0.0185472, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac{1}{45} b \log \left (b x^5+3\right )-\frac{1}{9} b \log (x)-\frac{1}{15 x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(3 + b*x^5)),x]

[Out]

-1/(15*x^5) - (b*Log[x])/9 + (b*Log[3 + b*x^5])/45

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}{x^6 \left (3+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x^2 (3+b x)} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{1}{3 x^2}-\frac{b}{9 x}+\frac{b^2}{9 (3+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{1}{15 x^5}-\frac{1}{9} b \log (x)+\frac{1}{45} b \log \left (3+b x^5\right )\\ \end{align*}

Mathematica [A]  time = 0.0044286, size = 28, normalized size = 1. \[ \frac{1}{45} b \log \left (b x^5+3\right )-\frac{1}{9} b \log (x)-\frac{1}{15 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(3 + b*x^5)),x]

[Out]

-1/(15*x^5) - (b*Log[x])/9 + (b*Log[3 + b*x^5])/45

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Maple [A]  time = 0.005, size = 23, normalized size = 0.8 \begin{align*} -{\frac{1}{15\,{x}^{5}}}-{\frac{b\ln \left ( x \right ) }{9}}+{\frac{b\ln \left ( b{x}^{5}+3 \right ) }{45}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^5+3),x)

[Out]

-1/15/x^5-1/9*b*ln(x)+1/45*b*ln(b*x^5+3)

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Maxima [A]  time = 1.06359, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{45} \, b \log \left (b x^{5} + 3\right ) - \frac{1}{45} \, b \log \left (x^{5}\right ) - \frac{1}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+3),x, algorithm="maxima")

[Out]

1/45*b*log(b*x^5 + 3) - 1/45*b*log(x^5) - 1/15/x^5

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Fricas [A]  time = 1.71815, size = 73, normalized size = 2.61 \begin{align*} \frac{b x^{5} \log \left (b x^{5} + 3\right ) - 5 \, b x^{5} \log \left (x\right ) - 3}{45 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+3),x, algorithm="fricas")

[Out]

1/45*(b*x^5*log(b*x^5 + 3) - 5*b*x^5*log(x) - 3)/x^5

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Sympy [A]  time = 0.82268, size = 24, normalized size = 0.86 \begin{align*} - \frac{b \log{\left (x \right )}}{9} + \frac{b \log{\left (x^{5} + \frac{3}{b} \right )}}{45} - \frac{1}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**5+3),x)

[Out]

-b*log(x)/9 + b*log(x**5 + 3/b)/45 - 1/(15*x**5)

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Giac [A]  time = 1.38326, size = 42, normalized size = 1.5 \begin{align*} \frac{1}{45} \, b \log \left ({\left | b x^{5} + 3 \right |}\right ) - \frac{1}{9} \, b \log \left ({\left | x \right |}\right ) + \frac{b x^{5} - 3}{45 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+3),x, algorithm="giac")

[Out]

1/45*b*log(abs(b*x^5 + 3)) - 1/9*b*log(abs(x)) + 1/45*(b*x^5 - 3)/x^5