3.3.87 \(\int \frac {F^{a+b (c+d x)^3}}{c+d x} \, dx\) [287]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {2241} \begin {gather*} \frac {F^a \text {Ei}\left (b (c+d x)^3 \log (F)\right )}{3 d} \end {gather*}

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 2241

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*}

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Mathematica [A]
time = 0.19, size = 22, normalized size = 1.00 \begin {gather*} \frac {F^a \text {Ei}\left (b (c+d x)^3 \log (F)\right )}{3 d} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{d x +c}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
time = 0.37, size = 44, normalized size = 2.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{a + b \left (c + d x\right )^{3}}}{c + d x}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 3.58, size = 20, normalized size = 0.91 \begin {gather*} \frac {F^a\,\mathrm {ei}\left (b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}{3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^a*ei(b*log(F)*(c + d*x)^3))/(3*d)

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