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

Optimal. Leaf size=20 \[ \frac{f^{a+b x^2}}{2 b \log (f)} \]

[Out]

f^(a + b*x^2)/(2*b*Log[f])

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Rubi [A]  time = 0.0129456, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2209} \[ \frac{f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)*x,x]

[Out]

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

Mathematica [A]  time = 0.0023447, size = 20, normalized size = 1. \[ \frac{f^{a+b x^2}}{2 b \log (f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)*x,x]

[Out]

f^(a + b*x^2)/(2*b*Log[f])

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \begin{align*}{\frac{{f}^{b{x}^{2}+a}}{2\,b\ln \left ( f \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)*x,x)

[Out]

1/2*f^(b*x^2+a)/b/ln(f)

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Maxima [A]  time = 1.04665, size = 24, normalized size = 1.2 \begin{align*} \frac{f^{b x^{2} + a}}{2 \, b \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f^(b*x^2 + a)/(b*log(f))

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Fricas [A]  time = 1.52332, size = 41, normalized size = 2.05 \begin{align*} \frac{f^{b x^{2} + a}}{2 \, b \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f^(b*x^2 + a)/(b*log(f))

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Sympy [A]  time = 0.121413, size = 24, normalized size = 1.2 \begin{align*} \begin{cases} \frac{f^{a + b x^{2}}}{2 b \log{\left (f \right )}} & \text{for}\: 2 b \log{\left (f \right )} \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)*x,x)

[Out]

Piecewise((f**(a + b*x**2)/(2*b*log(f)), Ne(2*b*log(f), 0)), (x**2/2, True))

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Giac [A]  time = 1.17054, size = 24, normalized size = 1.2 \begin{align*} \frac{f^{b x^{2} + a}}{2 \, b \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*f^(b*x^2 + a)/(b*log(f))