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

Optimal. Leaf size=56 \[ x F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac{1}{n},-b \log (F);1+\frac{1}{n};-\frac{d x^n}{c}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0216632, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2274, 12, 246, 245} \[ x F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac{1}{n},-b \log (F);1+\frac{1}{n};-\frac{d x^n}{c}\right ) \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*Log[c + d*x^n]),x]

[Out]

(F^a*x*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[n^(-1), -(b*Log[F]), 1 + n^(-1), -((d*x^n)/c)])/(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 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 245

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

Rubi steps

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

Mathematica [A]  time = 0.0874138, size = 83, normalized size = 1.48 \[ -\frac{x \left (-\frac{d x^n}{c}\right )^{-1/n} \left (c+d x^n\right ) F^{a+b \log \left (c+d x^n\right )} \, _2F_1\left (\frac{n-1}{n},b \log (F)+1;b \log (F)+2;\frac{d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*Log[c + d*x^n]),x]

[Out]

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

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b\ln \left ( c+d{x}^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

integral(F^(b*log(d*x^n + c) + a), 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)

[Out]

Timed out

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

[Out]

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