3.458 \(\int f^{b x+c x^2} (b+2 c x) \, dx\)

Optimal. Leaf size=16 \[ \frac{f^{b x+c x^2}}{\log (f)} \]

[Out]

f^(b*x + c*x^2)/Log[f]

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Rubi [A]  time = 0.0141611, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2236} \[ \frac{f^{b x+c x^2}}{\log (f)} \]

Antiderivative was successfully verified.

[In]

Int[f^(b*x + c*x^2)*(b + 2*c*x),x]

[Out]

f^(b*x + c*x^2)/Log[f]

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0272758, size = 16, normalized size = 1. \[ \frac{f^{b x+c x^2}}{\log (f)} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(b*x + c*x^2)*(b + 2*c*x),x]

[Out]

f^(b*x + c*x^2)/Log[f]

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Maple [A]  time = 0.002, size = 17, normalized size = 1.1 \begin{align*}{\frac{{f}^{c{x}^{2}+bx}}{\ln \left ( f \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^(c*x^2+b*x)/ln(f)

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Maxima [A]  time = 0.994273, size = 22, normalized size = 1.38 \begin{align*} \frac{f^{c x^{2} + b x}}{\log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^(c*x^2 + b*x)/log(f)

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Fricas [A]  time = 1.52006, size = 32, normalized size = 2. \begin{align*} \frac{f^{c x^{2} + b x}}{\log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^(c*x^2 + b*x)/log(f)

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Sympy [A]  time = 0.113387, size = 22, normalized size = 1.38 \begin{align*} \begin{cases} \frac{f^{b x + c x^{2}}}{\log{\left (f \right )}} & \text{for}\: \log{\left (f \right )} \neq 0 \\b x + c x^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26103, size = 22, normalized size = 1.38 \begin{align*} \frac{f^{c x^{2} + b x}}{\log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f^(c*x^2 + b*x)/log(f)