3.287 \(\int \frac{F^{a+b (c+d x)^3}}{c+d x} \, dx\)

Optimal. Leaf size=22 \[ \frac{F^a \text{Ei}\left (b (c+d x)^3 \log (F)\right )}{3 d} \]

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]])/(3*d)

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Rubi [A]  time = 0.0658895, antiderivative size = 22, 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 = {2210} \[ \frac{F^a \text{Ei}\left (b (c+d x)^3 \log (F)\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]])/(3*d)

Rule 2210

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

Rubi steps

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

Mathematica [A]  time = 0.0064917, size = 22, normalized size = 1. \[ \frac{F^a \text{Ei}\left (b (c+d x)^3 \log (F)\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]])/(3*d)

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{d x + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c), x)

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Fricas [B]  time = 1.6311, size = 97, normalized size = 4.41 \begin{align*} \frac{F^{a}{\rm Ei}\left ({\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*F^a*Ei((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + b \left (c + d x\right )^{3}}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(F**(a + b*(c + d*x)**3)/(c + d*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{d x + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c), x)