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

Optimal. Leaf size=34 \[ \frac{1}{3} x^4 f^a \left (-\frac{b \log (f)}{x^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{b \log (f)}{x^3}\right ) \]

[Out]

(f^a*x^4*Gamma[-4/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(4/3))/3

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Rubi [A]  time = 0.0247654, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ \frac{1}{3} x^4 f^a \left (-\frac{b \log (f)}{x^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x^3)*x^3,x]

[Out]

(f^a*x^4*Gamma[-4/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(4/3))/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^3 \, dx &=\frac{1}{3} f^a x^4 \Gamma \left (-\frac{4}{3},-\frac{b \log (f)}{x^3}\right ) \left (-\frac{b \log (f)}{x^3}\right )^{4/3}\\ \end{align*}

Mathematica [A]  time = 0.0035358, size = 34, normalized size = 1. \[ \frac{1}{3} x^4 f^a \left (-\frac{b \log (f)}{x^3}\right )^{4/3} \text{Gamma}\left (-\frac{4}{3},-\frac{b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x^3)*x^3,x]

[Out]

(f^a*x^4*Gamma[-4/3, -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(4/3))/3

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Maple [B]  time = 0.034, size = 115, normalized size = 3.4 \begin{align*}{\frac{{f}^{a}b}{3} \left ( \ln \left ( f \right ) \right ) ^{{\frac{4}{3}}}\sqrt [3]{-b} \left ({\frac{9\,{b}^{2}\Gamma \left ( 2/3 \right ) }{4\,{x}^{2}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}} \left ( -b \right ) ^{-{\frac{4}{3}}} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-{\frac{2}{3}}}}-{\frac{3\,{x}^{4}}{4} \left ( 3\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}}+1 \right ){{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}} \left ( -b \right ) ^{-{\frac{4}{3}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{4}{3}}}}-{\frac{9\,{b}^{2}}{4\,{x}^{2}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}}\Gamma \left ({\frac{2}{3}},-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) \left ( -b \right ) ^{-{\frac{4}{3}}} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-{\frac{2}{3}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.18673, size = 38, normalized size = 1.12 \begin{align*} \frac{1}{3} \, f^{a} x^{4} \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{4}{3}} \Gamma \left (-\frac{4}{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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.04674, size = 149, normalized size = 4.38 \begin{align*} -\frac{3}{4} \, \left (-b \log \left (f\right )\right )^{\frac{1}{3}} b f^{a} \Gamma \left (\frac{2}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right ) + \frac{1}{4} \,{\left (x^{4} + 3 \, b x \log \left (f\right )\right )} f^{\frac{a x^{3} + b}{x^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(-b*log(f))^(1/3)*b*f^a*gamma(2/3, -b*log(f)/x^3)*log(f) + 1/4*(x^4 + 3*b*x*log(f))*f^((a*x^3 + b)/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^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**3)*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}}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^3)*x^3, x)