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

Optimal. Leaf size=34 \[ \frac{1}{3} x^2 f^a \left (-\frac{b \log (f)}{x^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{b \log (f)}{x^3}\right ) \]

[Out]

(f^a*x^2*Gamma[-2/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(2/3))/3

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Rubi [A]  time = 0.0142921, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2218} \[ \frac{1}{3} x^2 f^a \left (-\frac{b \log (f)}{x^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)*x,x]

[Out]

(f^a*x^2*Gamma[-2/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(2/3))/3

Rule 2218

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

Rubi steps

\begin{align*} \int f^{a+\frac{b}{x^3}} x \, dx &=\frac{1}{3} f^a x^2 \Gamma \left (-\frac{2}{3},-\frac{b \log (f)}{x^3}\right ) \left (-\frac{b \log (f)}{x^3}\right )^{2/3}\\ \end{align*}

Mathematica [A]  time = 0.0035972, size = 34, normalized size = 1. \[ \frac{1}{3} x^2 f^a \left (-\frac{b \log (f)}{x^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)*x,x]

[Out]

(f^a*x^2*Gamma[-2/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(2/3))/3

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Maple [B]  time = 0.029, size = 105, normalized size = 3.1 \begin{align*} -{\frac{{f}^{a}}{3} \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}} \left ({\frac{b\pi \,\sqrt{3}}{x\Gamma \left ({\frac{2}{3}} \right ) }\sqrt [3]{\ln \left ( f \right ) } \left ( -b \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{-{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}}-{\frac{3\,{x}^{2}}{2}{{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}} \left ( -b \right ) ^{-{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}}-{\frac{3\,b}{2\,x}\sqrt [3]{\ln \left ( f \right ) }\Gamma \left ({\frac{1}{3}},-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) \left ( -b \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{-{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1695, size = 38, normalized size = 1.12 \begin{align*} \frac{1}{3} \, f^{a} x^{2} \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{2}{3}} \Gamma \left (-\frac{2}{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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.13962, size = 112, normalized size = 3.29 \begin{align*} \frac{1}{2} \, f^{\frac{a x^{3} + b}{x^{3}}} x^{2} - \frac{1}{2} \, \left (-b \log \left (f\right )\right )^{\frac{2}{3}} f^{a} \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,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**3)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^3)*x, x)