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

Optimal. Leaf size=32 \[ \frac{1}{3} x f^a \sqrt [3]{-\frac{b \log (f)}{x^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (f)}{x^3}\right ) \]

[Out]

(f^a*x*Gamma[-1/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(1/3))/3

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Rubi [A]  time = 0.0047191, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2208} \[ \frac{1}{3} x f^a \sqrt [3]{-\frac{b \log (f)}{x^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3),x]

[Out]

(f^a*x*Gamma[-1/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(1/3))/3

Rule 2208

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

Rubi steps

\begin{align*} \int f^{a+\frac{b}{x^3}} \, dx &=\frac{1}{3} f^a x \Gamma \left (-\frac{1}{3},-\frac{b \log (f)}{x^3}\right ) \sqrt [3]{-\frac{b \log (f)}{x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0029922, size = 32, normalized size = 1. \[ \frac{1}{3} x f^a \sqrt [3]{-\frac{b \log (f)}{x^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3),x]

[Out]

(f^a*x*Gamma[-1/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(1/3))/3

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Maple [B]  time = 0.029, size = 98, normalized size = 3.1 \begin{align*} -{\frac{{f}^{a}}{3}\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) } \left ( 3\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2/3}b\Gamma \left ( 2/3 \right ) }{{x}^{2}\sqrt [3]{-b}} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-2/3}}-3\,{\frac{x}{\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }}{{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}-3\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2/3}b}{{x}^{2}\sqrt [3]{-b}}\Gamma \left ( 2/3,-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-2/3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*f^a*(-b)^(1/3)*ln(f)^(1/3)*(3/x^2/(-b)^(1/3)*ln(f)^(2/3)*b*GAMMA(2/3)/(-b*ln(f)/x^3)^(2/3)-3*x/(-b)^(1/3)
/ln(f)^(1/3)*exp(b*ln(f)/x^3)-3/x^2/(-b)^(1/3)*ln(f)^(2/3)*b/(-b*ln(f)/x^3)^(2/3)*GAMMA(2/3,-b*ln(f)/x^3))

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Maxima [A]  time = 1.24753, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{3} \, f^{a} x \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{1}{3}} \Gamma \left (-\frac{1}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.8089, size = 100, normalized size = 3.12 \begin{align*} -\left (-b \log \left (f\right )\right )^{\frac{1}{3}} f^{a} \Gamma \left (\frac{2}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right ) + f^{\frac{a x^{3} + b}{x^{3}}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^3), x)