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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = 0.00, size = 32, normalized size = 1.00 \[ \frac {1}{3} x f^a \sqrt [3]{-\frac {b \log (f)}{x^3}} \Gamma \left (-\frac {1}{3},-\frac {b \log (f)}{x^3}\right ) \]

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

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 38, normalized size = 1.19 \[ -\left (-b \log \relax (f)\right )^{\frac {1}{3}} f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) + f^{\frac {a x^{3} + b}{x^{3}}} x \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 98, normalized size = 3.06 \[ -\frac {\left (-b \right )^{\frac {1}{3}} \left (-\frac {3 x \,{\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{\left (-b \right )^{\frac {1}{3}} \ln \relax (f )^{\frac {1}{3}}}-\frac {3 b \Gamma \left (\frac {2}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {2}{3}}}{\left (-b \right )^{\frac {1}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}+\frac {3 \Gamma \left (\frac {2}{3}\right ) b \ln \relax (f )^{\frac {2}{3}}}{\left (-b \right )^{\frac {1}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}\right ) f^{a} \ln \relax (f )^{\frac {1}{3}}}{3} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.24, size = 26, normalized size = 0.81 \[ \frac {1}{3} \, f^{a} x \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {1}{3}} \Gamma \left (-\frac {1}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.59, size = 48, normalized size = 1.50 \[ f^a\,x\,\left (f^{\frac {b}{x^3}}+\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{1/3}-\Gamma \left (\frac {2}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{1/3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x^3),x)

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________