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

Optimal. Leaf size=68 \[ -\frac{F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (-\frac{2}{n},-b \log (F);-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 x^2} \]

[Out]

-(F^a*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[-2/n, -(b*Log[F]), -((2 - n)/n), -((d*x^n)/c)])/(2*x^2*(1 + (d*
x^n)/c)^(b*Log[F]))

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Rubi [A]  time = 0.0449979, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2274, 12, 365, 364} \[ -\frac{F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (-\frac{2}{n},-b \log (F);-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*Log[c + d*x^n])/x^3,x]

[Out]

-(F^a*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[-2/n, -(b*Log[F]), -((2 - n)/n), -((d*x^n)/c)])/(2*x^2*(1 + (d*
x^n)/c)^(b*Log[F]))

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{F^{a+b \log \left (c+d x^n\right )}}{x^3} \, dx &=\int \frac{F^a \left (c+d x^n\right )^{b \log (F)}}{x^3} \, dx\\ &=F^a \int \frac{\left (c+d x^n\right )^{b \log (F)}}{x^3} \, dx\\ &=\left (F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac{d x^n}{c}\right )^{-b \log (F)}\right ) \int \frac{\left (1+\frac{d x^n}{c}\right )^{b \log (F)}}{x^3} \, dx\\ &=-\frac{F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac{d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (-\frac{2}{n},-b \log (F);-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.117691, size = 85, normalized size = 1.25 \[ -\frac{\left (-\frac{d x^n}{c}\right )^{2/n} \left (c+d x^n\right ) F^{a+b \log \left (c+d x^n\right )} \, _2F_1\left (\frac{n+2}{n},b \log (F)+1;b \log (F)+2;\frac{d x^n}{c}+1\right )}{c n x^2 (b \log (F)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((F^(a + b*Log[c + d*x^n])*(-((d*x^n)/c))^(2/n)*(c + d*x^n)*Hypergeometric2F1[(2 + n)/n, 1 + b*Log[F], 2 + b*
Log[F], 1 + (d*x^n)/c])/(c*n*x^2*(1 + b*Log[F])))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b\ln \left ( c+d{x}^{n} \right ) }}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(F^(b*log(d*x^n + c) + a)/x^3, x)

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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*ln(c+d*x**n))/x**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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