3.164 \(\int \frac{f^{a+\frac{b}{x^3}}}{x^{13}} \, dx\)

Optimal. Leaf size=83 \[ \frac{f^{a+\frac{b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac{2 f^{a+\frac{b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac{2 f^{a+\frac{b}{x^3}}}{b^4 \log ^4(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^9 \log (f)} \]

[Out]

(2*f^(a + b/x^3))/(b^4*Log[f]^4) - (2*f^(a + b/x^3))/(b^3*x^3*Log[f]^3) + f^(a + b/x^3)/(b^2*x^6*Log[f]^2) - f
^(a + b/x^3)/(3*b*x^9*Log[f])

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Rubi [A]  time = 0.0942408, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac{f^{a+\frac{b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac{2 f^{a+\frac{b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac{2 f^{a+\frac{b}{x^3}}}{b^4 \log ^4(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^9 \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^13,x]

[Out]

(2*f^(a + b/x^3))/(b^4*Log[f]^4) - (2*f^(a + b/x^3))/(b^3*x^3*Log[f]^3) + f^(a + b/x^3)/(b^2*x^6*Log[f]^2) - f
^(a + b/x^3)/(3*b*x^9*Log[f])

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{f^{a+\frac{b}{x^3}}}{x^{13}} \, dx &=-\frac{f^{a+\frac{b}{x^3}}}{3 b x^9 \log (f)}-\frac{3 \int \frac{f^{a+\frac{b}{x^3}}}{x^{10}} \, dx}{b \log (f)}\\ &=\frac{f^{a+\frac{b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^9 \log (f)}+\frac{6 \int \frac{f^{a+\frac{b}{x^3}}}{x^7} \, dx}{b^2 \log ^2(f)}\\ &=-\frac{2 f^{a+\frac{b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac{f^{a+\frac{b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^9 \log (f)}-\frac{6 \int \frac{f^{a+\frac{b}{x^3}}}{x^4} \, dx}{b^3 \log ^3(f)}\\ &=\frac{2 f^{a+\frac{b}{x^3}}}{b^4 \log ^4(f)}-\frac{2 f^{a+\frac{b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac{f^{a+\frac{b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^9 \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0105382, size = 58, normalized size = 0.7 \[ \frac{f^{a+\frac{b}{x^3}} \left (3 b^2 x^3 \log ^2(f)-b^3 \log ^3(f)-6 b x^6 \log (f)+6 x^9\right )}{3 b^4 x^9 \log ^4(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^13,x]

[Out]

(f^(a + b/x^3)*(6*x^9 - 6*b*x^6*Log[f] + 3*b^2*x^3*Log[f]^2 - b^3*Log[f]^3))/(3*b^4*x^9*Log[f]^4)

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Maple [A]  time = 0.022, size = 97, normalized size = 1.2 \begin{align*}{\frac{1}{{x}^{12}} \left ({\frac{{x}^{6}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}+2\,{\frac{{x}^{12}}{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}-2\,{\frac{{x}^{9}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}-{\frac{{x}^{3}}{3\,b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x^3)/x^13,x)

[Out]

(1/b^2/ln(f)^2*x^6*exp((a+b/x^3)*ln(f))+2/b^4/ln(f)^4*x^12*exp((a+b/x^3)*ln(f))-2/b^3/ln(f)^3*x^9*exp((a+b/x^3
)*ln(f))-1/3/b/ln(f)*x^3*exp((a+b/x^3)*ln(f)))/x^12

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Maxima [C]  time = 1.2797, size = 30, normalized size = 0.36 \begin{align*} \frac{f^{a} \Gamma \left (4, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{4} \log \left (f\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^13,x, algorithm="maxima")

[Out]

1/3*f^a*gamma(4, -b*log(f)/x^3)/(b^4*log(f)^4)

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Fricas [A]  time = 1.73014, size = 142, normalized size = 1.71 \begin{align*} \frac{{\left (6 \, x^{9} - 6 \, b x^{6} \log \left (f\right ) + 3 \, b^{2} x^{3} \log \left (f\right )^{2} - b^{3} \log \left (f\right )^{3}\right )} f^{\frac{a x^{3} + b}{x^{3}}}}{3 \, b^{4} x^{9} \log \left (f\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^13,x, algorithm="fricas")

[Out]

1/3*(6*x^9 - 6*b*x^6*log(f) + 3*b^2*x^3*log(f)^2 - b^3*log(f)^3)*f^((a*x^3 + b)/x^3)/(b^4*x^9*log(f)^4)

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Sympy [A]  time = 0.14431, size = 58, normalized size = 0.7 \begin{align*} \frac{f^{a + \frac{b}{x^{3}}} \left (- b^{3} \log{\left (f \right )}^{3} + 3 b^{2} x^{3} \log{\left (f \right )}^{2} - 6 b x^{6} \log{\left (f \right )} + 6 x^{9}\right )}{3 b^{4} x^{9} \log{\left (f \right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)/x**13,x)

[Out]

f**(a + b/x**3)*(-b**3*log(f)**3 + 3*b**2*x**3*log(f)**2 - 6*b*x**6*log(f) + 6*x**9)/(3*b**4*x**9*log(f)**4)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{13}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^13,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^13, x)