3.13.89 \(\int \frac {1}{x^6 (2 b+b x^5)} \, dx\) [1289]

Optimal. Leaf size=33 \[ -\frac {1}{10 b x^5}-\frac {\log (x)}{4 b}+\frac {\log \left (2+x^5\right )}{20 b} \]

[Out]

-1/10/b/x^5-1/4*ln(x)/b+1/20*ln(x^5+2)/b

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 46} \begin {gather*} -\frac {1}{10 b x^5}+\frac {\log \left (x^5+2\right )}{20 b}-\frac {\log (x)}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(2*b + b*x^5)),x]

[Out]

-1/10*1/(b*x^5) - Log[x]/(4*b) + Log[2 + x^5]/(20*b)

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 272

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]]

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (2 b+b x^5\right )} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{x^2 (2 b+b x)} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {Subst}\left (\int \left (\frac {1}{2 b x^2}-\frac {1}{4 b x}+\frac {1}{4 b (2+x)}\right ) \, dx,x,x^5\right )\\ &=-\frac {1}{10 b x^5}-\frac {\log (x)}{4 b}+\frac {\log \left (2+x^5\right )}{20 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 33, normalized size = 1.00 \begin {gather*} -\frac {1}{10 b x^5}-\frac {\log (x)}{4 b}+\frac {\log \left (2+x^5\right )}{20 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(2*b + b*x^5)),x]

[Out]

-1/10*1/(b*x^5) - Log[x]/(4*b) + Log[2 + x^5]/(20*b)

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Maple [A]
time = 0.17, size = 23, normalized size = 0.70

method	result	size
default	\(\frac {-\frac {1}{10 x^{5}}-\frac {\ln \left (x \right )}{4}+\frac {\ln \left (x^{5}+2\right )}{20}}{b}\)	\(23\)
meijerg	\(\frac {\ln \left (1+\frac {x^{5}}{2}\right )-5 \ln \left (x \right )+\ln \left (2\right )-\frac {2}{x^{5}}}{20 b}\)	\(26\)
norman	\(-\frac {1}{10 b \,x^{5}}-\frac {\ln \left (x \right )}{4 b}+\frac {\ln \left (x^{5}+2\right )}{20 b}\)	\(28\)
risch	\(-\frac {1}{10 b \,x^{5}}-\frac {\ln \left (x \right )}{4 b}+\frac {\ln \left (-x^{5}-2\right )}{20 b}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^5+2*b),x,method=_RETURNVERBOSE)

[Out]

1/b*(-1/10/x^5-1/4*ln(x)+1/20*ln(x^5+2))

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Maxima [A]
time = 0.29, size = 29, normalized size = 0.88 \begin {gather*} \frac {\log \left (x^{5} + 2\right )}{20 \, b} - \frac {\log \left (x^{5}\right )}{20 \, b} - \frac {1}{10 \, b x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+2*b),x, algorithm="maxima")

[Out]

1/20*log(x^5 + 2)/b - 1/20*log(x^5)/b - 1/10/(b*x^5)

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Fricas [A]
time = 0.38, size = 27, normalized size = 0.82 \begin {gather*} \frac {x^{5} \log \left (x^{5} + 2\right ) - 5 \, x^{5} \log \left (x\right ) - 2}{20 \, b x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+2*b),x, algorithm="fricas")

[Out]

1/20*(x^5*log(x^5 + 2) - 5*x^5*log(x) - 2)/(b*x^5)

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Sympy [A]
time = 0.18, size = 24, normalized size = 0.73 \begin {gather*} - \frac {\log {\left (x \right )}}{4 b} + \frac {\log {\left (x^{5} + 2 \right )}}{20 b} - \frac {1}{10 b x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**5+2*b),x)

[Out]

-log(x)/(4*b) + log(x**5 + 2)/(20*b) - 1/(10*b*x**5)

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Giac [A]
time = 1.89, size = 34, normalized size = 1.03 \begin {gather*} \frac {\log \left ({\left | x^{5} + 2 \right |}\right )}{20 \, b} - \frac {\log \left ({\left | x \right |}\right )}{4 \, b} + \frac {x^{5} - 2}{20 \, b x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^5+2*b),x, algorithm="giac")

[Out]

1/20*log(abs(x^5 + 2))/b - 1/4*log(abs(x))/b + 1/20*(x^5 - 2)/(b*x^5)

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Mupad [B]
time = 1.07, size = 27, normalized size = 0.82 \begin {gather*} \frac {\ln \left (x^5+2\right )}{20\,b}-\frac {\ln \left (x\right )}{4\,b}-\frac {1}{10\,b\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(2*b + b*x^5)),x)

[Out]

log(x^5 + 2)/(20*b) - log(x)/(4*b) - 1/(10*b*x^5)

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