3.6.84 \(\int \frac {F^{a+b \log (c+d x^n)}}{x^3} \, dx\) [584]

Optimal. Leaf size=68 \[ -\frac {F^a \left (c+d x^n\right )^{b \log (F)} \left (1+\frac {d x^n}{c}\right )^{-b \log (F)} \, _2F_1\left (-\frac {2}{n},-b \log (F);-\frac {2-n}{n};-\frac {d x^n}{c}\right )}{2 x^2} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 68, 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 = {2306, 12, 372, 371} \begin {gather*} -\frac {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 {2}{n},-b \log (F);-\frac {2-n}{n};-\frac {d x^n}{c}\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b*Log[c + d*x^n])/x^3,x]

[Out]

-1/2*(F^a*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[-2/n, -(b*Log[F]), -((2 - n)/n), -((d*x^n)/c)])/(x^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*} \int \frac {F^{a+b \log \left (c+d x^n\right )}}{x^3} \, dx &=\int \frac {F^a \left (c+d x^n\right )^{b \log (F)}}{x^3} \, dx\\ &=F^a \int \frac {\left (c+d x^n\right )^{b \log (F)}}{x^3} \, 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 \frac {\left (1+\frac {d x^n}{c}\right )^{b \log (F)}}{x^3} \, dx\\ &=-\frac {F^a \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 )}{2 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 85, normalized size = 1.25 \begin {gather*} -\frac {F^{a+b \log \left (c+d x^n\right )} \left (-\frac {d x^n}{c}\right )^{2/n} \left (c+d x^n\right ) \, _2F_1\left (\frac {2+n}{n},1+b \log (F);2+b \log (F);1+\frac {d x^n}{c}\right )}{c n x^2 (1+b \log (F))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*Log[c + d*x^n])/x^3,x]

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {F^{a +b \ln \left (c +d \,x^{n}\right )}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*ln(c+d*x^n))/x^3,x)

[Out]

int(F^(a+b*ln(c+d*x^n))/x^3,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))/x^3,x, algorithm="maxima")

[Out]

integrate(F^(b*log(d*x^n + c) + a)/x^3, 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))/x^3,x, algorithm="fricas")

[Out]

integral(F^(b*log(d*x^n + c) + a)/x^3, 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))/x**3,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))/x^3,x, algorithm="giac")

[Out]

integrate(F^(b*log(d*x^n + c) + a)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {F^{a+b\,\ln \left (c+d\,x^n\right )}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*log(c + d*x^n))/x^3,x)

[Out]

int(F^(a + b*log(c + d*x^n))/x^3, x)

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