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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 12

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

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])

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 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]

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 = 0.11, 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]))

________________________________________________________________________________________

fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \ln \left (d \,x^{n}+c \right )+a}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \log \left (d x^{n} + c\right ) + a}}{x}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {F^{a+b\,\ln \left (c+d\,x^n\right )}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*log(c + d*x^n))/x,x)

[Out]

int(F^(a + b*log(c + d*x^n))/x, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + b \log {\left (c + d x^{n} \right )}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________