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

Optimal. Leaf size=32 \[ -\frac{x f^a \text{Gamma}\left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.0038035, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2208} \[ -\frac{x f^a \text{Gamma}\left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3),x]

[Out]

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

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin{align*} \int f^{a+b x^3} \, dx &=-\frac{f^a x \Gamma \left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.0043517, size = 32, normalized size = 1. \[ -\frac{x f^a \text{Gamma}\left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^3),x]

[Out]

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

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Maple [B]  time = 0.015, size = 78, normalized size = 2.4 \begin{align*}{\frac{{f}^{a}}{3} \left ({\frac{2\,x\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) }\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}}-{x\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }\Gamma \left ({\frac{1}{3}},-b{x}^{3}\ln \left ( f \right ) \right ){\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt [3]{-b}}}{\frac{1}{\sqrt [3]{\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.28956, size = 35, normalized size = 1.09 \begin{align*} -\frac{f^{a} x \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78871, size = 86, normalized size = 2.69 \begin{align*} \frac{\left (-b \log \left (f\right )\right )^{\frac{2}{3}} f^{a} \Gamma \left (\frac{1}{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, algorithm="fricas")

[Out]

1/3*(-b*log(f))^(2/3)*f^a*gamma(1/3, -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}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^3 + a), x)