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

Optimal. Leaf size=34 \[ \frac {1}{3} x^4 f^a \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2218} \[ \frac {1}{3} x^4 f^a \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \text {Gamma}\left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[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 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^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}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 34, normalized size = 1.00 \[ \frac {1}{3} x^4 f^a \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \]

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.44, size = 51, normalized size = 1.50 \[ -\frac {3}{4} \, \left (-b \log \relax (f)\right )^{\frac {1}{3}} b f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f) + \frac {1}{4} \, {\left (x^{4} + 3 \, b x \log \relax (f)\right )} f^{\frac {a x^{3} + b}{x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x^3)*x^3, x)

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maple [B]  time = 0.06, size = 115, normalized size = 3.38 \[ \frac {\left (-b \right )^{\frac {1}{3}} \left (-\frac {3 \left (\frac {3 b \ln \relax (f )}{x^{3}}+1\right ) x^{4} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{4 \left (-b \right )^{\frac {4}{3}} \ln \relax (f )^{\frac {4}{3}}}-\frac {9 b^{2} \Gamma \left (\frac {2}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {2}{3}}}{4 \left (-b \right )^{\frac {4}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}+\frac {9 \Gamma \left (\frac {2}{3}\right ) b^{2} \ln \relax (f )^{\frac {2}{3}}}{4 \left (-b \right )^{\frac {4}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}\right ) b \,f^{a} \ln \relax (f )^{\frac {4}{3}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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/x^3*ln(f))^(2/3)-3/4*x^4/(
-b)^(4/3)/ln(f)^(4/3)*(3*b/x^3*ln(f)+1)*exp(b/x^3*ln(f))-9/4/x^2/(-b)^(4/3)*ln(f)^(2/3)*b^2/(-b/x^3*ln(f))^(2/
3)*GAMMA(2/3,-b/x^3*ln(f)))

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maxima [A]  time = 1.28, size = 28, normalized size = 0.82 \[ \frac {1}{3} \, f^{a} x^{4} \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {4}{3}} \Gamma \left (-\frac {4}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 3.59, size = 80, normalized size = 2.35 \[ \frac {f^a\,f^{\frac {b}{x^3}}\,x^4}{4}-\frac {3\,f^a\,x^4\,\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{4/3}}{4}+\frac {3\,f^a\,x^4\,\Gamma \left (\frac {2}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{4/3}}{4}+\frac {3\,b\,f^a\,f^{\frac {b}{x^3}}\,x\,\ln \relax (f)}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b/x**3)*x**3, x)

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