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 = 0.0109559, 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 \[ -\frac{x f^a \text{Gamma}\left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]

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))

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Rubi in Sympy [A]  time = 1.17719, size = 34, normalized size = 1.06 \[ - \frac{f^{a} x \Gamma{\left (\frac{1}{3},- b x^{3} \log{\left (f \right )} \right )}}{3 \sqrt [3]{- b x^{3} \log{\left (f \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**3+a),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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Mathematica [A]  time = 0.00663037, 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 = 0.018, size = 78, normalized size = 2.4 \[{\frac{{f}^{a}}{3} \left ({\frac{2\,x\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) }\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}}-{x\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }\Gamma \left ({\frac{1}{3}},-b{x}^{3}\ln \left ( f \right ) \right ){\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt [3]{-b}}}{\frac{1}{\sqrt [3]{\ln \left ( f \right ) }}}} \]

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 = 0.833744, size = 35, normalized size = 1.09 \[ -\frac{f^{a} x \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac{1}{3}}} \]

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 = 0.267201, size = 30, normalized size = 0.94 \[ -\frac{f^{a} \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b \log \left (f\right )\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

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)