\(\int f^{a+\frac {b}{x^3}} x \, dx\) [169]

Optimal result

Integrand size = 11, antiderivative size = 34 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{3} f^a x^2 \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2250} \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{3} x^2 f^a \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \]

[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 2250

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

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{3} f^a x^2 \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(28)=56\).

Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.09

method	result	size
meijerg	\(-\frac {f^{a} \left (-b \right )^{\frac {2}{3}} \ln \left (f \right )^{\frac {2}{3}} \left (\frac {\ln \left (f \right )^{\frac {1}{3}} b \pi \sqrt {3}}{x \left (-b \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {1}{3}}}-\frac {3 x^{2} {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{2 \left (-b \right )^{\frac {2}{3}} \ln \left (f \right )^{\frac {2}{3}}}-\frac {3 \ln \left (f \right )^{\frac {1}{3}} b \Gamma \left (\frac {1}{3}, -\frac {b \ln \left (f \right )}{x^{3}}\right )}{2 x \left (-b \right )^{\frac {2}{3}} \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {1}{3}}}\right )}{3}\)	\(105\)

[In]

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{2} \, f^{\frac {a x^{3} + b}{x^{3}}} x^{2} - \frac {1}{2} \, \left (-b \log \left (f\right )\right )^{\frac {2}{3}} f^{a} \Gamma \left (\frac {1}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right ) \]

[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)

Sympy [F]

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

[In]

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

[Out]

Integral(f**(a + b/x**3)*x, x)

Maxima [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{3} \, f^{a} x^{2} \left (-\frac {b \log \left (f\right )}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (-\frac {2}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right ) \]

[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)

Giac [F]

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

[In]

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

[Out]

integrate(f^(a + b/x^3)*x, x)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.06 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {f^a\,f^{\frac {b}{x^3}}\,x^2}{2}-\frac {f^a\,x^2\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \left (f\right )}{x^3}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{2/3}}{2}+\frac {\pi \,\sqrt {3}\,f^a\,x^2\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{2/3}}{3\,\Gamma \left (\frac {2}{3}\right )} \]

[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))