3.358 \(\int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^5} \, dx\)

Optimal. Leaf size=49 \[ \frac{F^a \text{Gamma}\left (\frac{4}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x)^4 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{4/3}} \]

[Out]

(F^a*Gamma[4/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)^4*(-((b*Log[F])/(c + d*x)^3))^(4/3))

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Rubi [A]  time = 0.0435577, antiderivative size = 49, 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 = {2218} \[ \frac{F^a \text{Gamma}\left (\frac{4}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x)^4 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)/(c + d*x)^5,x]

[Out]

(F^a*Gamma[4/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)^4*(-((b*Log[F])/(c + d*x)^3))^(4/3))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^5} \, dx &=\frac{F^a \Gamma \left (\frac{4}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x)^4 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{4/3}}\\ \end{align*}

Mathematica [A]  time = 0.0358109, size = 49, normalized size = 1. \[ \frac{F^a \text{Gamma}\left (\frac{4}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d (c+d x)^4 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^5,x]

[Out]

(F^a*Gamma[4/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)^4*(-((b*Log[F])/(c + d*x)^3))^(4/3))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{5}}{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)/(d*x+c)^5,x)

[Out]

int(F^(a+b/(d*x+c)^3)/(d*x+c)^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{5}}\,{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)^5,x, algorithm="maxima")

[Out]

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^5, x)

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Fricas [B]  time = 1.68297, size = 350, normalized size = 7.14 \begin{align*} \frac{{\left (d^{3} x + c d^{2}\right )} F^{a} \left (-\frac{b \log \left (F\right )}{d^{3}}\right )^{\frac{2}{3}} \Gamma \left (\frac{1}{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 ) - 3 \, 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}}} b \log \left (F\right )}{9 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^5,x, algorithm="fricas")

[Out]

1/9*((d^3*x + c*d^2)*F^a*(-b*log(F)/d^3)^(2/3)*gamma(1/3, -b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))
- 3*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))*b*log
(F))/((b^2*d^2*x + b^2*c*d)*log(F)^2)

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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)/(d*x+c)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{5}}\,{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)^5,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^5, x)