3.13.94 \(\int \frac {1}{x^6 (3+b x^5)} \, dx\) [1294]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {272, 46} \begin {gather*} \frac {1}{45} b \log \left (b x^5+3\right )-\frac {1}{9} b \log (x)-\frac {1}{15 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(3 + b*x^5)),x]

[Out]

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

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 (3+b x^5\right )} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{x^2 (3+b x)} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {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*}

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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {1}{15 x^5}-\frac {1}{9} b \log (x)+\frac {1}{45} b \log \left (3+b x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(3 + b*x^5)),x]

[Out]

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

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Maple [A]
time = 0.18, size = 23, normalized size = 0.82

method	result	size
default	\(-\frac {1}{15 x^{5}}-\frac {b \ln \left (x \right )}{9}+\frac {b \ln \left (b \,x^{5}+3\right )}{45}\)	\(23\)
norman	\(-\frac {1}{15 x^{5}}-\frac {b \ln \left (x \right )}{9}+\frac {b \ln \left (b \,x^{5}+3\right )}{45}\)	\(23\)
risch	\(-\frac {1}{15 x^{5}}-\frac {b \ln \left (x \right )}{9}+\frac {b \ln \left (-b \,x^{5}-3\right )}{45}\)	\(24\)
meijerg	\(\frac {b \left (\ln \left (1+\frac {b \,x^{5}}{3}\right )-5 \ln \left (x \right )+\ln \left (3\right )-\ln \left (b \right )-\frac {3}{b \,x^{5}}\right )}{45}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 24, normalized size = 0.86 \begin {gather*} \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 {gather*}

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 = 0.36, size = 28, normalized size = 1.00 \begin {gather*} \frac {b x^{5} \log \left (b x^{5} + 3\right ) - 5 \, b x^{5} \log \left (x\right ) - 3}{45 \, x^{5}} \end {gather*}

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 = 0.16, size = 24, normalized size = 0.86 \begin {gather*} - \frac {b \log {\left (x \right )}}{9} + \frac {b \log {\left (x^{5} + \frac {3}{b} \right )}}{45} - \frac {1}{15 x^{5}} \end {gather*}

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 = 2.26, size = 31, normalized size = 1.11 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.06, size = 22, normalized size = 0.79 \begin {gather*} \frac {b\,\ln \left (b\,x^5+3\right )}{45}-\frac {b\,\ln \left (x\right )}{9}-\frac {1}{15\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(b*x^5 + 3)),x)

[Out]

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

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