3.101 \(\int f^{a+b x^3} x^2 \, dx\)

Optimal. Leaf size=20 \[ \frac{f^{a+b x^3}}{3 b \log (f)} \]

[Out]

f^(a + b*x^3)/(3*b*Log[f])

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Rubi [A]  time = 0.0357309, antiderivative size = 20, 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 \[ \frac{f^{a+b x^3}}{3 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^3)*x^2,x]

[Out]

f^(a + b*x^3)/(3*b*Log[f])

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Rubi in Sympy [A]  time = 3.46217, size = 14, normalized size = 0.7 \[ \frac{f^{a + b x^{3}}}{3 b \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**3+a)*x**2,x)

[Out]

f**(a + b*x**3)/(3*b*log(f))

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Mathematica [A]  time = 0.00502405, size = 20, normalized size = 1. \[ \frac{f^{a+b x^3}}{3 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^3)*x^2,x]

[Out]

f^(a + b*x^3)/(3*b*Log[f])

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Maple [A]  time = 0.004, size = 19, normalized size = 1. \[{\frac{{f}^{b{x}^{3}+a}}{3\,b\ln \left ( f \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^3+a)*x^2,x)

[Out]

1/3*f^(b*x^3+a)/b/ln(f)

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Maxima [A]  time = 0.813683, size = 24, normalized size = 1.2 \[ \frac{f^{b x^{3} + a}}{3 \, b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^2,x, algorithm="maxima")

[Out]

1/3*f^(b*x^3 + a)/(b*log(f))

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Fricas [A]  time = 0.308497, size = 24, normalized size = 1.2 \[ \frac{f^{b x^{3} + a}}{3 \, b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^2,x, algorithm="fricas")

[Out]

1/3*f^(b*x^3 + a)/(b*log(f))

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Sympy [A]  time = 0.19023, size = 24, normalized size = 1.2 \[ \begin{cases} \frac{f^{a + b x^{3}}}{3 b \log{\left (f \right )}} & \text{for}\: 3 b \log{\left (f \right )} \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**3+a)*x**2,x)

[Out]

Piecewise((f**(a + b*x**3)/(3*b*log(f)), Ne(3*b*log(f), 0)), (x**3/3, True))

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GIAC/XCAS [A]  time = 0.223749, size = 24, normalized size = 1.2 \[ \frac{f^{b x^{3} + a}}{3 \, b{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^2,x, algorithm="giac")

[Out]

1/3*f^(b*x^3 + a)/(b*ln(f))