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

Optimal. Leaf size=47 \[ \frac{F^a (c+d x) \sqrt [3]{-\frac{b \log (F)}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[Out]

(F^a*(c + d*x)*Gamma[-1/3, -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(1/3))/(3*d)

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Rubi [A]  time = 0.0068177, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2208} \[ \frac{F^a (c+d x) \sqrt [3]{-\frac{b \log (F)}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3),x]

[Out]

(F^a*(c + d*x)*Gamma[-1/3, -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(1/3))/(3*d)

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin{align*} \int F^{a+\frac{b}{(c+d x)^3}} \, dx &=\frac{F^a (c+d x) \Gamma \left (-\frac{1}{3},-\frac{b \log (F)}{(c+d x)^3}\right ) \sqrt [3]{-\frac{b \log (F)}{(c+d x)^3}}}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0119261, size = 47, normalized size = 1. \[ \frac{F^a (c+d x) \sqrt [3]{-\frac{b \log (F)}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3),x]

[Out]

(F^a*(c + d*x)*Gamma[-1/3, -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(1/3))/(3*d)

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 3 \, F^{a} b d \int \frac{F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} x}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} \log \left (F\right ) + F^{a} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*F^a*b*d*integrate(F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*x/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 +
4*c^3*d*x + c^4), x)*log(F) + F^a*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*x

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Fricas [B]  time = 1.62543, size = 284, normalized size = 6.04 \begin{align*} -\frac{F^{a} d \left (-\frac{b \log \left (F\right )}{d^{3}}\right )^{\frac{1}{3}} \Gamma \left (\frac{2}{3}, -\frac{b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) -{\left (d x + c\right )} F^{\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^(a + b/(d*x + c)^3), x)