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

Optimal. Leaf size=57 \[ -\frac{F^a \left (c+d x^n\right )^{b \log (F)+1} \, _2F_1\left (1,b \log (F)+1;b \log (F)+2;\frac{d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]

[Out]

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

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Rubi [A]  time = 0.0638044, antiderivative size = 57, 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, 266, 65} \[ -\frac{F^a \left (c+d x^n\right )^{b \log (F)+1} \, _2F_1\left (1,b \log (F)+1;b \log (F)+2;\frac{d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*Log[c + d*x^n])/x,x]

[Out]

-((F^a*(c + d*x^n)^(1 + b*Log[F])*Hypergeometric2F1[1, 1 + b*Log[F], 2 + b*Log[F], 1 + (d*x^n)/c])/(c*n*(1 + 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

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

Mathematica [A]  time = 0.102058, size = 50, normalized size = 0.88 \[ -\frac{F^{a+b \log \left (c+d x^n\right )} \left (\, _2F_1\left (1,b \log (F);b \log (F)+1;\frac{d x^n}{c}+1\right )-1\right )}{b n \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b\ln \left ( c+d{x}^{n} \right ) }}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(F^(a+b*ln(c+d*x^n))/x,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}\,{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,x, algorithm="maxima")

[Out]

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

[Out]

integral(F^(b*log(d*x^n + c) + a)/x, 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,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}\,{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,x, algorithm="giac")

[Out]

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