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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 16, normalized size = 1.00 \[ \frac {f^{b x+c x^2}}{\log (f)} \]

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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fricas [A]  time = 0.42, size = 16, normalized size = 1.00 \[ \frac {f^{c x^{2} + b x}}{\log \relax (f)} \]

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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giac [A]  time = 0.30, size = 16, normalized size = 1.00 \[ \frac {f^{c x^{2} + b x}}{\log \relax (f)} \]

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)

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maple [A]  time = 0.01, size = 17, normalized size = 1.06 \[ \frac {f^{c \,x^{2}+b x}}{\ln \relax (f )} \]

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 = 0.85, size = 16, normalized size = 1.00 \[ \frac {f^{c x^{2} + b x}}{\log \relax (f)} \]

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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mupad [B]  time = 3.52, size = 16, normalized size = 1.00 \[ \frac {f^{c\,x^2+b\,x}}{\ln \relax (f)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x + c*x^2)*(b + 2*c*x),x)

[Out]

f^(b*x + c*x^2)/log(f)

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sympy [A]  time = 0.12, size = 22, normalized size = 1.38 \[ \begin {cases} \frac {f^{b x + c x^{2}}}{\log {\relax (f )}} & \text {for}\: \log {\relax (f )} \neq 0 \\b x + c x^{2} & \text {otherwise} \end {cases} \]

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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