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

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

[Out]

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

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Rubi [A]  time = 0.013284, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2218} \[ -\frac{x^2 f^a \text{Gamma}\left (\frac{2}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)*x,x]

[Out]

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

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

Mathematica [A]  time = 0.0049437, size = 34, normalized size = 1. \[ -\frac{x^2 f^a \text{Gamma}\left (\frac{2}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3)*x,x]

[Out]

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

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Maple [B]  time = 0.016, size = 75, normalized size = 2.2 \begin{align*}{\frac{{f}^{a}}{3} \left ({{x}^{2}\Gamma \left ({\frac{2}{3}} \right ) \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}} \left ( -b{x}^{3}\ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}}-{{x}^{2} \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}}\Gamma \left ({\frac{2}{3}},-b{x}^{3}\ln \left ( f \right ) \right ) \left ( -b{x}^{3}\ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}} \right ) \left ( -b \right ) ^{-{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.29837, size = 38, normalized size = 1.12 \begin{align*} -\frac{f^{a} x^{2} \Gamma \left (\frac{2}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.70051, size = 86, normalized size = 2.53 \begin{align*} \frac{\left (-b \log \left (f\right )\right )^{\frac{1}{3}} f^{a} \Gamma \left (\frac{2}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, b \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x^{3}} x\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**3)*x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{b x^{3} + a} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^3 + a)*x, x)