3.163 \(\int \frac{f^{a+\frac{b}{x^3}}}{x^{10}} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 f^{a+\frac{b}{x^3}}}{3 b^2 x^3 \log ^2(f)}-\frac{2 f^{a+\frac{b}{x^3}}}{3 b^3 \log ^3(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^6 \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.0703232, antiderivative size = 67, 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 = {2212, 2209} \[ \frac{2 f^{a+\frac{b}{x^3}}}{3 b^2 x^3 \log ^2(f)}-\frac{2 f^{a+\frac{b}{x^3}}}{3 b^3 \log ^3(f)}-\frac{f^{a+\frac{b}{x^3}}}{3 b x^6 \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)/x^10,x]

[Out]

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

Mathematica [A]  time = 0.0087973, size = 45, normalized size = 0.67 \[ -\frac{f^{a+\frac{b}{x^3}} \left (b^2 \log ^2(f)-2 b x^3 \log (f)+2 x^6\right )}{3 b^3 x^6 \log ^3(f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)/x^10,x]

[Out]

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

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Maple [A]  time = 0.018, size = 75, normalized size = 1.1 \begin{align*}{\frac{1}{{x}^{9}} \left ( -{\frac{2\,{x}^{9}}{3\, \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{{x}^{3}}} \right ) \ln \left ( f \right ) }}}+{\frac{2\,{x}^{6}}{3\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\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^10,x)

[Out]

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

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Maxima [C]  time = 1.12247, size = 30, normalized size = 0.45 \begin{align*} -\frac{f^{a} \Gamma \left (3, -\frac{b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{3} \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^10,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76628, size = 115, normalized size = 1.72 \begin{align*} -\frac{{\left (2 \, x^{6} - 2 \, b x^{3} \log \left (f\right ) + b^{2} \log \left (f\right )^{2}\right )} f^{\frac{a x^{3} + b}{x^{3}}}}{3 \, b^{3} x^{6} \log \left (f\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.129388, size = 44, normalized size = 0.66 \begin{align*} \frac{f^{a + \frac{b}{x^{3}}} \left (- b^{2} \log{\left (f \right )}^{2} + 2 b x^{3} \log{\left (f \right )} - 2 x^{6}\right )}{3 b^{3} x^{6} \log{\left (f \right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x**3)/x**10,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x^{3}}}}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(a+b/x^3)/x^10,x, algorithm="giac")

[Out]

integrate(f^(a + b/x^3)/x^10, x)