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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 34, normalized size = 1.00 \[ \frac {1}{3} x^2 f^a \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \]

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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fricas [A]  time = 0.43, size = 41, normalized size = 1.21 \[ \frac {1}{2} \, f^{\frac {a x^{3} + b}{x^{3}}} x^{2} - \frac {1}{2} \, \left (-b \log \relax (f)\right )^{\frac {2}{3}} f^{a} \Gamma \left (\frac {1}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{3}}} x\,{d x} \]

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)

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maple [B]  time = 0.06, size = 105, normalized size = 3.09 \[ -\frac {\left (-b \right )^{\frac {2}{3}} \left (-\frac {3 x^{2} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{2 \left (-b \right )^{\frac {2}{3}} \ln \relax (f )^{\frac {2}{3}}}-\frac {3 b \Gamma \left (\frac {1}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {1}{3}}}{2 \left (-b \right )^{\frac {2}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} x}+\frac {\pi \sqrt {3}\, b \ln \relax (f )^{\frac {1}{3}}}{\left (-b \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} x}\right ) f^{a} \ln \relax (f )^{\frac {2}{3}}}{3} \]

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

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maxima [A]  time = 1.30, size = 28, normalized size = 0.82 \[ \frac {1}{3} \, f^{a} x^{2} \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (-\frac {2}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]

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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mupad [B]  time = 3.57, size = 70, normalized size = 2.06 \[ \frac {f^a\,f^{\frac {b}{x^3}}\,x^2}{2}-\frac {f^a\,x^2\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{2/3}}{2}+\frac {\pi \,\sqrt {3}\,f^a\,x^2\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{2/3}}{3\,\Gamma \left (\frac {2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^3)*x,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{3}}} x\, dx \]

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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