\(\int f^{a+\frac {b}{x^3}} \, dx\) [170]

Optimal result

Integrand size = 9, antiderivative size = 32 \[ \int f^{a+\frac {b}{x^3}} \, dx=\frac {1}{3} f^a x \Gamma \left (-\frac {1}{3},-\frac {b \log (f)}{x^3}\right ) \sqrt [3]{-\frac {b \log (f)}{x^3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2239} \[ \int f^{a+\frac {b}{x^3}} \, dx=\frac {1}{3} x f^a \sqrt [3]{-\frac {b \log (f)}{x^3}} \Gamma \left (-\frac {1}{3},-\frac {b \log (f)}{x^3}\right ) \]

[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 2239

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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^3}} \, dx=\frac {1}{3} f^a x \Gamma \left (-\frac {1}{3},-\frac {b \log (f)}{x^3}\right ) \sqrt [3]{-\frac {b \log (f)}{x^3}} \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(26)=52\).

Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.06

method	result	size
meijerg	\(-\frac {f^{a} \left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}} \left (\frac {3 \ln \left (f \right )^{\frac {2}{3}} b \Gamma \left (\frac {2}{3}\right )}{x^{2} \left (-b \right )^{\frac {1}{3}} \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {2}{3}}}-\frac {3 x \,{\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{\left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}}}-\frac {3 \ln \left (f \right )^{\frac {2}{3}} b \Gamma \left (\frac {2}{3}, -\frac {b \ln \left (f \right )}{x^{3}}\right )}{x^{2} \left (-b \right )^{\frac {1}{3}} \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {2}{3}}}\right )}{3}\)	\(98\)

[In]

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int f^{a+\frac {b}{x^3}} \, dx=-\left (-b \log \left (f\right )\right )^{\frac {1}{3}} f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right ) + f^{\frac {a x^{3} + b}{x^{3}}} x \]

[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

Sympy [F]

\[ \int f^{a+\frac {b}{x^3}} \, dx=\int f^{a + \frac {b}{x^{3}}}\, dx \]

[In]

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

[Out]

Integral(f**(a + b/x**3), x)

Maxima [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int f^{a+\frac {b}{x^3}} \, dx=\frac {1}{3} \, f^{a} x \left (-\frac {b \log \left (f\right )}{x^{3}}\right )^{\frac {1}{3}} \Gamma \left (-\frac {1}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right ) \]

[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)

Giac [F]

\[ \int f^{a+\frac {b}{x^3}} \, dx=\int { f^{a + \frac {b}{x^{3}}} \,d x } \]

[In]

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

[Out]

integrate(f^(a + b/x^3), x)

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int f^{a+\frac {b}{x^3}} \, dx=f^a\,x\,\left (f^{\frac {b}{x^3}}+\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{1/3}-\Gamma \left (\frac {2}{3},-\frac {b\,\ln \left (f\right )}{x^3}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{1/3}\right ) \]

[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))