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

Optimal. Leaf size=77 \[ \frac{F^a (d x)^{m+1} \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac{m+1}{n},-b \log (F);\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{d (m+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0547465, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2274, 12, 365, 364} \[ \frac{F^a (d x)^{m+1} \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \, _2F_1\left (\frac{m+1}{n},-b \log (F);\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(F^a*(d*x)^(1 + m)*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[(1 + m)/n, -(b*Log[F]), (1 + m + n)/n, -((d*x^n)/c
)])/(d*(1 + m)*(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 F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx &=\int F^a (d x)^m \left (c+d x^n\right )^{b \log (F)} \, dx\\ &=F^a \int (d x)^m \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 (d x)^m \left (1+\frac{d x^n}{c}\right )^{b \log (F)} \, dx\\ &=\frac{F^a (d x)^{1+m} \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+m}{n},-b \log (F);\frac{1+m+n}{n};-\frac{d x^n}{c}\right )}{d (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.159498, size = 94, normalized size = 1.22 \[ -\frac{x (d x)^m \left (c+d x^n\right ) \left (-\frac{d x^n}{c}\right )^{-\frac{m+1}{n}} F^{a+b \log \left (c+d x^n\right )} \, _2F_1\left (1-\frac{m+1}{n},b \log (F)+1;b \log (F)+2;\frac{d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b\ln \left ( c+d{x}^{n} \right ) } \left ( dx \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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))*(d*x)**m,x)

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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