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

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

[Out]

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

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Rubi [A]  time = 0.037701, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{x^4 f^a \text{Gamma}\left (\frac{4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]

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))

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Rubi in Sympy [A]  time = 3.11918, size = 36, normalized size = 1.06 \[ - \frac{f^{a} x^{4} \Gamma{\left (\frac{4}{3},- b x^{3} \log{\left (f \right )} \right )}}{3 \left (- b x^{3} \log{\left (f \right )}\right )^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**3+a)*x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0307404, size = 56, normalized size = 1.65 \[ -\frac{x^4 f^a \left (\text{Gamma}\left (\frac{1}{3},-b x^3 \log (f)\right )+3 f^{b x^3} \sqrt [3]{-b x^3 \log (f)}\right )}{9 \left (-b x^3 \log (f)\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^3)*x^3,x]

[Out]

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

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Maple [B]  time = 0.033, size = 109, normalized size = 3.2 \[ -{\frac{{f}^{a}}{3\,b} \left ( -{\frac{2\,x\pi \,\sqrt{3}}{9\,b\Gamma \left ( 2/3 \right ) } \left ( -b \right ) ^{{\frac{4}{3}}}\sqrt [3]{\ln \left ( f \right ) }{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}}+{\frac{x{{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{b} \left ( -b \right ) ^{{\frac{4}{3}}}\sqrt [3]{\ln \left ( f \right ) }}+{\frac{x}{3\,b} \left ( -b \right ) ^{{\frac{4}{3}}}\sqrt [3]{\ln \left ( f \right ) }\Gamma \left ({\frac{1}{3}},-b{x}^{3}\ln \left ( f \right ) \right ){\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}} \right ) \left ( \ln \left ( f \right ) \right ) ^{-{\frac{4}{3}}}{\frac{1}{\sqrt [3]{-b}}}} \]

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)/GA
MMA(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 = 0.868942, size = 73, normalized size = 2.15 \[ \frac{f^{b x^{3}} f^{a} x}{3 \, b \log \left (f\right )} + \frac{f^{a} x \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right )}{9 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac{1}{3}} b \log \left (f\right )} \]

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

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Fricas [A]  time = 0.270085, size = 68, normalized size = 2. \[ \frac{3 \, \left (-b \log \left (f\right )\right )^{\frac{1}{3}} f^{b x^{3} + a} x + f^{a} \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right )}{9 \, \left (-b \log \left (f\right )\right )^{\frac{1}{3}} b \log \left (f\right )} \]

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

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

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

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)