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

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

[Out]

-1/3*f^a*x^4*GAMMA(4/3,-b*x^3*ln(f))/(-b*x^3*ln(f))^(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} \[ -\frac {x^4 f^a \text {Gamma}\left (\frac {4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)*x^3,x]

[Out]

-(f^a*x^4*Gamma[4/3, -(b*x^3*Log[f])])/(3*(-(b*x^3*Log[f]))^(4/3))

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+b x^3} x^3 \, dx &=-\frac {f^a x^4 \Gamma \left (\frac {4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ -\frac {x^4 f^a \Gamma \left (\frac {4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)*x^3,x]

[Out]

-1/3*(f^a*x^4*Gamma[4/3, -(b*x^3*Log[f])])/(-(b*x^3*Log[f]))^(4/3)

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fricas [A]  time = 0.44, size = 47, normalized size = 1.38 \[ \frac {3 \, b f^{b x^{3} + a} x \log \relax (f) - \left (-b \log \relax (f)\right )^{\frac {2}{3}} f^{a} \Gamma \left (\frac {1}{3}, -b x^{3} \log \relax (f)\right )}{9 \, b^{2} \log \relax (f)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*b*f^(b*x^3 + a)*x*log(f) - (-b*log(f))^(2/3)*f^a*gamma(1/3, -b*x^3*log(f)))/(b^2*log(f)^2)

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

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 = 109, normalized size = 3.21 \[ -\frac {\left (\frac {\left (-b \right )^{\frac {4}{3}} x \Gamma \left (\frac {1}{3}, -b \,x^{3} \ln \relax (f )\right ) \ln \relax (f )^{\frac {1}{3}}}{3 \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {1}{3}} b}+\frac {\left (-b \right )^{\frac {4}{3}} x \,{\mathrm e}^{b \,x^{3} \ln \relax (f )} \ln \relax (f )^{\frac {1}{3}}}{b}-\frac {2 \left (-b \right )^{\frac {4}{3}} \pi \sqrt {3}\, x \ln \relax (f )^{\frac {1}{3}}}{9 \Gamma \left (\frac {2}{3}\right ) \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {1}{3}} b}\right ) f^{a}}{3 \left (-b \right )^{\frac {1}{3}} b \ln \relax (f )^{\frac {4}{3}}} \]

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)^(4/3)/(-b)^(1/3)*(-2/9*x*(-b)^(4/3)*ln(f)^(1/3)/b*Pi*3^(1/2)/GAMMA(2/3)/(-b*x^3*ln(f))^(1/3)+
x*(-b)^(4/3)*ln(f)^(1/3)/b*exp(b*x^3*ln(f))+1/3*x*(-b)^(4/3)*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 = 1.33, size = 28, normalized size = 0.82 \[ -\frac {f^{a} x^{4} \Gamma \left (\frac {4}{3}, -b x^{3} \log \relax (f)\right )}{3 \, \left (-b x^{3} \log \relax (f)\right )^{\frac {4}{3}}} \]

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

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mupad [B]  time = 3.18, size = 75, normalized size = 2.21 \[ \frac {f^a\,f^{b\,x^3}\,x}{3\,b\,\ln \relax (f)}-\frac {f^a\,x^4\,\Gamma \left (\frac {1}{3},-b\,x^3\,\ln \relax (f)\right )}{9\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{4/3}}+\frac {2\,\pi \,\sqrt {3}\,f^a\,x^4}{27\,\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{4/3}} \]

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)/(3*b*log(f)) - (f^a*x^4*igamma(1/3, -b*x^3*log(f)))/(9*(-b*x^3*log(f))^(4/3)) + (2*3^(1/2)*f
^a*x^4*pi)/(27*gamma(2/3)*(-b*x^3*log(f))^(4/3))

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

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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