3.617 \(\int F^{a+b x+c x^3} (b+3 c x^2) \, dx\)

Optimal. Leaf size=17 \[ \frac{F^{a+b x+c x^3}}{\log (F)} \]

[Out]

F^(a + b*x + c*x^3)/Log[F]

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Rubi [A]  time = 0.0520096, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {6706} \[ \frac{F^{a+b x+c x^3}}{\log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*x + c*x^3)*(b + 3*c*x^2),x]

[Out]

F^(a + b*x + c*x^3)/Log[F]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

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

Mathematica [A]  time = 0.0519045, size = 17, normalized size = 1. \[ \frac{F^{a+b x+c x^3}}{\log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*x + c*x^3)*(b + 3*c*x^2),x]

[Out]

F^(a + b*x + c*x^3)/Log[F]

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Maple [A]  time = 0.043, size = 18, normalized size = 1.1 \begin{align*}{\frac{{F}^{c{x}^{3}+bx+a}}{\ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*x^3+b*x+a)*(3*c*x^2+b),x)

[Out]

F^(c*x^3+b*x+a)/ln(F)

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Maxima [A]  time = 0.959697, size = 23, normalized size = 1.35 \begin{align*} \frac{F^{c x^{3} + b x + a}}{\log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*x^3+b*x+a)*(3*c*x^2+b),x, algorithm="maxima")

[Out]

F^(c*x^3 + b*x + a)/log(F)

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Fricas [A]  time = 0.924443, size = 38, normalized size = 2.24 \begin{align*} \frac{F^{c x^{3} + b x + a}}{\log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*x^3+b*x+a)*(3*c*x^2+b),x, algorithm="fricas")

[Out]

F^(c*x^3 + b*x + a)/log(F)

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Sympy [A]  time = 0.12406, size = 24, normalized size = 1.41 \begin{align*} \begin{cases} \frac{F^{a + b x + c x^{3}}}{\log{\left (F \right )}} & \text{for}\: \log{\left (F \right )} \neq 0 \\b x + c x^{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*x**3+b*x+a)*(3*c*x**2+b),x)

[Out]

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

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Giac [A]  time = 1.33893, size = 23, normalized size = 1.35 \begin{align*} \frac{F^{c x^{3} + b x + a}}{\log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*x^3+b*x+a)*(3*c*x^2+b),x, algorithm="giac")

[Out]

F^(c*x^3 + b*x + a)/log(F)