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

Optimal. Leaf size=57 \[ -\frac{F^a \left (c+d x^n\right )^{b \log (F)+1} \text{Hypergeometric2F1}\left (1,b \log (F)+1,b \log (F)+2,\frac{d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]

[Out]

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

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Rubi [A]  time = 0.109915, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{F^a \left (c+d x^n\right )^{b \log (F)+1} \text{Hypergeometric2F1}\left (1,b \log (F)+1,b \log (F)+2,\frac{d x^n}{c}+1\right )}{c n (b \log (F)+1)} \]

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])))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{\left (\left (\left (\left (\left (\left (\left (\left (\left (\left (x^{n}\right )^{n}\right )^{n}\right )^{n}\right )^{n}\right )^{n}\right )^{n}\right )^{n}\right )^{n}\right )^{n}\right )^{n}} \frac{F^{a + b \log{\left (c + d x^{n} \right )}}}{x}\, dx}{n^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*ln(c+d*x**n))/x,x)

[Out]

Integral(F**(a + b*log(c + d*x**n))/x, (x, ((((((((((x**n)**n)**n)**n)**n)**n)**
n)**n)**n)**n)**n))/n**11

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Mathematica [A]  time = 0.0889943, size = 50, normalized size = 0.88 \[ -\frac{F^{a+b \log \left (c+d x^n\right )} \left (\text{Hypergeometric2F1}\left (1,b \log (F),b \log (F)+1,\frac{d x^n}{c}+1\right )-1\right )}{b n \log (F)} \]

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]))

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b\ln \left ( c+d{x}^{n} \right ) }}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*ln(c+d*x^n))/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*log(d*x^n + c) + a)/x,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*log(d*x^n + c) + a)/x,x, algorithm="giac")

[Out]

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