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 = 0.0220422, 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, Rules used = {2209} \[ \frac{f^{a+b x^3}}{3 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^3)*x^2,x]

[Out]

f^(a + b*x^3)/(3*b*Log[f])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int f^{a+b x^3} x^2 \, dx &=\frac{f^{a+b x^3}}{3 b \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0024945, 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 = 0.003, size = 19, normalized size = 1. \begin{align*}{\frac{{f}^{b{x}^{3}+a}}{3\,b\ln \left ( f \right ) }} \end{align*}

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 = 1.07176, size = 24, normalized size = 1.2 \begin{align*} \frac{f^{b x^{3} + a}}{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^2,x, algorithm="maxima")

[Out]

1/3*f^(b*x^3 + a)/(b*log(f))

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Fricas [A]  time = 1.89043, size = 41, normalized size = 2.05 \begin{align*} \frac{f^{b x^{3} + a}}{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^2,x, algorithm="fricas")

[Out]

1/3*f^(b*x^3 + a)/(b*log(f))

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Sympy [A]  time = 0.109959, size = 24, normalized size = 1.2 \begin{align*} \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} \end{align*}

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 [A]  time = 1.24623, size = 24, normalized size = 1.2 \begin{align*} \frac{f^{b x^{3} + a}}{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^2,x, algorithm="giac")

[Out]

1/3*f^(b*x^3 + a)/(b*log(f))