3.6.85 \(\int F^{a+b \log (c+d x^n)} (d x)^m \, dx\) [585]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2306, 12, 372, 371} \begin {gather*} \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)} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.15, size = 94, normalized size = 1.22 \begin {gather*} -\frac {F^{a+b \log \left (c+d x^n\right )} x (d x)^m \left (-\frac {d x^n}{c}\right )^{-\frac {1+m}{n}} \left (c+d x^n\right ) \, _2F_1\left (1-\frac {1+m}{n},1+b \log (F);2+b \log (F);1+\frac {d x^n}{c}\right )}{c n (1+b \log (F))} \end {gather*}

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])))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int F^{a +b \ln \left (c +d \,x^{n}\right )} \left (d x \right )^{m}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int F^{a+b\,\ln \left (c+d\,x^n\right )}\,{\left (d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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