\(\int F^{a+b \log (c+d x^n)} x \, dx\) [580]

Optimal result

Integrand size = 16, antiderivative size = 65 \[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx=\frac {1}{2} F^a x^2 \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (\frac {2}{n},-b \log (F),\frac {2+n}{n},-\frac {d x^n}{c}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2306, 12, 372, 371} \[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx=\frac {1}{2} x^2 F^a \left (c+d x^n\right )^{b \log (F)} \left (\frac {d x^n}{c}+1\right )^{-b \log (F)} \operatorname {Hypergeometric2F1}\left (\frac {2}{n},-b \log (F),\frac {n+2}{n},-\frac {d x^n}{c}\right ) \]

[In]

Int[F^(a + b*Log[c + d*x^n])*x,x]

[Out]

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

Rule 12

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 2306

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]

Rubi steps \begin{align*} \text {integral}& = \int F^a x \left (c+d x^n\right )^{b \log (F)} \, dx \\ & = F^a \int x \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 x \left (1+\frac {d x^n}{c}\right )^{b \log (F)} \, dx \\ & = \frac {1}{2} F^a x^2 \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.31 \[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx=-\frac {F^{a+b \log \left (c+d x^n\right )} x^2 \left (-\frac {d x^n}{c}\right )^{-2/n} \left (c+d x^n\right ) \operatorname {Hypergeometric2F1}\left (\frac {-2+n}{n},1+b \log (F),2+b \log (F),1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))} \]

[In]

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

[Out]

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

Maple [F]

\[\int F^{a +b \ln \left (c +d \,x^{n}\right )} x d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx=\int { F^{b \log \left (d x^{n} + c\right ) + a} x \,d x } \]

[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)

Sympy [F]

\[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx=\int F^{a + b \log {\left (c + d x^{n} \right )}} x\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx=\int { F^{b \log \left (d x^{n} + c\right ) + a} x \,d x } \]

[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)

Giac [F]

\[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx=\int { F^{b \log \left (d x^{n} + c\right ) + a} x \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int F^{a+b \log \left (c+d x^n\right )} x \, dx=\int F^{a+b\,\ln \left (c+d\,x^n\right )}\,x \,d x \]

[In]

int(F^(a + b*log(c + d*x^n))*x,x)

[Out]

int(F^(a + b*log(c + d*x^n))*x, x)