3.108 \(\int f^{a+b x^3} x^4 \, dx\)

Optimal. Leaf size=34 \[ -\frac {x^5 f^a \Gamma \left (\frac {5}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{5/3}} \]

[Out]

-1/3*f^a*x^5*GAMMA(5/3,-b*x^3*ln(f))/(-b*x^3*ln(f))^(5/3)

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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} \[ -\frac {x^5 f^a \text {Gamma}\left (\frac {5}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{5/3}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)*x^4,x]

[Out]

-(f^a*x^5*Gamma[5/3, -(b*x^3*Log[f])])/(3*(-(b*x^3*Log[f]))^(5/3))

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^4 \, dx &=-\frac {f^a x^5 \Gamma \left (\frac {5}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{5/3}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ -\frac {x^5 f^a \Gamma \left (\frac {5}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)*x^4,x]

[Out]

-1/3*(f^a*x^5*Gamma[5/3, -(b*x^3*Log[f])])/(-(b*x^3*Log[f]))^(5/3)

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fricas [A]  time = 0.43, size = 49, normalized size = 1.44 \[ \frac {3 \, b f^{b x^{3} + a} x^{2} \log \relax (f) - 2 \, \left (-b \log \relax (f)\right )^{\frac {1}{3}} f^{a} \Gamma \left (\frac {2}{3}, -b x^{3} \log \relax (f)\right )}{9 \, b^{2} \log \relax (f)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)*x^4,x, algorithm="fricas")

[Out]

1/9*(3*b*f^(b*x^3 + a)*x^2*log(f) - 2*(-b*log(f))^(1/3)*f^a*gamma(2/3, -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 \[ \int f^{b x^{3} + a} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)*x^4,x, algorithm="giac")

[Out]

integrate(f^(b*x^3 + a)*x^4, x)

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maple [B]  time = 0.04, size = 106, normalized size = 3.12 \[ \frac {\left (\frac {2 \left (-b \right )^{\frac {5}{3}} x^{2} \Gamma \left (\frac {2}{3}, -b \,x^{3} \ln \relax (f )\right ) \ln \relax (f )^{\frac {2}{3}}}{3 \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {2}{3}} b}+\frac {\left (-b \right )^{\frac {5}{3}} x^{2} {\mathrm e}^{b \,x^{3} \ln \relax (f )} \ln \relax (f )^{\frac {2}{3}}}{b}-\frac {2 \left (-b \right )^{\frac {5}{3}} \Gamma \left (\frac {2}{3}\right ) x^{2} \ln \relax (f )^{\frac {2}{3}}}{3 \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {2}{3}} b}\right ) f^{a}}{3 \left (-b \right )^{\frac {5}{3}} \ln \relax (f )^{\frac {5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^3+a)*x^4,x)

[Out]

1/3*f^a/(-b)^(5/3)/ln(f)^(5/3)*(-2/3*x^2*(-b)^(5/3)*ln(f)^(2/3)/b*GAMMA(2/3)/(-b*x^3*ln(f))^(2/3)+x^2*(-b)^(5/
3)*ln(f)^(2/3)/b*exp(b*x^3*ln(f))+2/3*x^2*(-b)^(5/3)*ln(f)^(2/3)/b/(-b*x^3*ln(f))^(2/3)*GAMMA(2/3,-b*x^3*ln(f)
))

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maxima [A]  time = 1.27, size = 28, normalized size = 0.82 \[ -\frac {f^{a} x^{5} \Gamma \left (\frac {5}{3}, -b x^{3} \log \relax (f)\right )}{3 \, \left (-b x^{3} \log \relax (f)\right )^{\frac {5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^3+a)*x^4,x, algorithm="maxima")

[Out]

-1/3*f^a*x^5*gamma(5/3, -b*x^3*log(f))/(-b*x^3*log(f))^(5/3)

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mupad [B]  time = 3.56, size = 71, normalized size = 2.09 \[ \frac {2\,f^a\,x^5\,\Gamma \left (\frac {2}{3}\right )}{9\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{5/3}}-\frac {2\,f^a\,x^5\,\Gamma \left (\frac {2}{3},-b\,x^3\,\ln \relax (f)\right )}{9\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{5/3}}+\frac {f^a\,f^{b\,x^3}\,x^2}{3\,b\,\ln \relax (f)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^3)*x^4,x)

[Out]

(2*f^a*x^5*gamma(2/3))/(9*(-b*x^3*log(f))^(5/3)) - (2*f^a*x^5*igamma(2/3, -b*x^3*log(f)))/(9*(-b*x^3*log(f))^(
5/3)) + (f^a*f^(b*x^3)*x^2)/(3*b*log(f))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + b x^{3}} x^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**3+a)*x**4,x)

[Out]

Integral(f**(a + b*x**3)*x**4, x)

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