\(\int f^{a+\frac {b}{x^3}} x^3 \, dx\) [168]

Optimal result

Integrand size = 13, antiderivative size = 34 \[ \int f^{a+\frac {b}{x^3}} x^3 \, dx=\frac {1}{3} f^a x^4 \Gamma \left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2250} \[ \int f^{a+\frac {b}{x^3}} x^3 \, dx=\frac {1}{3} x^4 f^a \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \]

[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 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^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] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^3}} x^3 \, dx=\frac {1}{3} f^a x^4 \Gamma \left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs. \(2(28)=56\).

Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.38

method	result	size
meijerg	\(\frac {f^{a} b \ln \left (f \right )^{\frac {4}{3}} \left (-b \right )^{\frac {1}{3}} \left (\frac {9 \ln \left (f \right )^{\frac {2}{3}} b^{2} \Gamma \left (\frac {2}{3}\right )}{4 x^{2} \left (-b \right )^{\frac {4}{3}} \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {2}{3}}}-\frac {3 x^{4} \left (\frac {3 b \ln \left (f \right )}{x^{3}}+1\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{4 \left (-b \right )^{\frac {4}{3}} \ln \left (f \right )^{\frac {4}{3}}}-\frac {9 \ln \left (f \right )^{\frac {2}{3}} b^{2} \Gamma \left (\frac {2}{3}, -\frac {b \ln \left (f \right )}{x^{3}}\right )}{4 x^{2} \left (-b \right )^{\frac {4}{3}} \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {2}{3}}}\right )}{3}\)	\(115\)

[In]

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int f^{a+\frac {b}{x^3}} x^3 \, dx=-\frac {3}{4} \, \left (-b \log \left (f\right )\right )^{\frac {1}{3}} b f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right ) + \frac {1}{4} \, {\left (x^{4} + 3 \, b x \log \left (f\right )\right )} f^{\frac {a x^{3} + b}{x^{3}}} \]

[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)

Sympy [F]

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

[In]

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

[Out]

Integral(f**(a + b/x**3)*x**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^3}} x^3 \, dx=\frac {1}{3} \, f^{a} x^{4} \left (-\frac {b \log \left (f\right )}{x^{3}}\right )^{\frac {4}{3}} \Gamma \left (-\frac {4}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right ) \]

[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)

Giac [F]

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

[In]

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

[Out]

integrate(f^(a + b/x^3)*x^3, x)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35 \[ \int f^{a+\frac {b}{x^3}} x^3 \, dx=\frac {f^a\,f^{\frac {b}{x^3}}\,x^4}{4}-\frac {3\,f^a\,x^4\,\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{4/3}}{4}+\frac {3\,f^a\,x^4\,\Gamma \left (\frac {2}{3},-\frac {b\,\ln \left (f\right )}{x^3}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{4/3}}{4}+\frac {3\,b\,f^a\,f^{\frac {b}{x^3}}\,x\,\ln \left (f\right )}{4} \]

[In]

int(f^(a + b/x^3)*x^3,x)

[Out]

(f^a*f^(b/x^3)*x^4)/4 - (3*f^a*x^4*gamma(2/3)*(-(b*log(f))/x^3)^(4/3))/4 + (3*f^a*x^4*igamma(2/3, -(b*log(f))/
x^3)*(-(b*log(f))/x^3)^(4/3))/4 + (3*b*f^a*f^(b/x^3)*x*log(f))/4