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

Optimal. Leaf size=34 \[ -\frac {f^a \left (-b x^3 \log (f)\right )^{2/3} \Gamma \left (-\frac {2}{3},-b x^3 \log (f)\right )}{3 x^2} \]

[Out]

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

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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} \[ -\frac {f^a \left (-b x^3 \log (f)\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},-b x^3 \log (f)\right )}{3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)/x^3,x]

[Out]

-(f^a*Gamma[-2/3, -(b*x^3*Log[f])]*(-(b*x^3*Log[f]))^(2/3))/(3*x^2)

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 \frac {f^{a+b x^3}}{x^3} \, dx &=-\frac {f^a \Gamma \left (-\frac {2}{3},-b x^3 \log (f)\right ) \left (-b x^3 \log (f)\right )^{2/3}}{3 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 34, normalized size = 1.00 \[ -\frac {f^a \left (-b x^3 \log (f)\right )^{2/3} \Gamma \left (-\frac {2}{3},-b x^3 \log (f)\right )}{3 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)/x^3,x]

[Out]

-1/3*(f^a*Gamma[-2/3, -(b*x^3*Log[f])]*(-(b*x^3*Log[f]))^(2/3))/x^2

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fricas [A]  time = 0.45, size = 41, normalized size = 1.21 \[ \frac {\left (-b \log \relax (f)\right )^{\frac {2}{3}} f^{a} x^{2} \Gamma \left (\frac {1}{3}, -b x^{3} \log \relax (f)\right ) - f^{b x^{3} + a}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^3 + a)/x^3, x)

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maple [B]  time = 0.04, size = 102, normalized size = 3.00 \[ -\frac {\left (-\frac {3 b x \Gamma \left (\frac {1}{3}, -b \,x^{3} \ln \relax (f )\right ) \ln \relax (f )^{\frac {1}{3}}}{2 \left (-b \right )^{\frac {2}{3}} \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {1}{3}}}+\frac {\pi \sqrt {3}\, b x \ln \relax (f )^{\frac {1}{3}}}{\left (-b \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {1}{3}}}-\frac {3 \,{\mathrm e}^{b \,x^{3} \ln \relax (f )}}{2 \left (-b \right )^{\frac {2}{3}} x^{2} \ln \relax (f )^{\frac {2}{3}}}\right ) b \,f^{a} \ln \relax (f )^{\frac {2}{3}}}{3 \left (-b \right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.65, size = 28, normalized size = 0.82 \[ -\frac {\left (-b x^{3} \log \relax (f)\right )^{\frac {2}{3}} f^{a} \Gamma \left (-\frac {2}{3}, -b x^{3} \log \relax (f)\right )}{3 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.58, size = 70, normalized size = 2.06 \[ \frac {f^a\,\Gamma \left (\frac {1}{3},-b\,x^3\,\ln \relax (f)\right )\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{2/3}}{2\,x^2}-\frac {f^a\,f^{b\,x^3}}{2\,x^2}-\frac {\pi \,\sqrt {3}\,f^a\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{2/3}}{3\,x^2\,\Gamma \left (\frac {2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^3)/x^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**3)/x**3, x)

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