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

Optimal. Leaf size=77 \[ \frac{F^a (d x)^{m+1} \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \text{Hypergeometric2F1}\left (\frac{m+1}{n},-b \log (F),\frac{m+n+1}{n},-\frac{d x^n}{c}\right )}{d (m+1)} \]

[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]))

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Rubi [A]  time = 0.10707, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{F^a (d x)^{m+1} \left (c+d x^n\right )^{b \log (F)} \left (\frac{d x^n}{c}+1\right )^{-b \log (F)} \text{Hypergeometric2F1}\left (\frac{m+1}{n},-b \log (F),\frac{m+n+1}{n},-\frac{d x^n}{c}\right )}{d (m+1)} \]

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]))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.286473, size = 0, normalized size = 0. \[ \int F^{a+b \log \left (c+d x^n\right )} (d x)^m \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \[ \int{F}^{a+b\ln \left ( c+d{x}^{n} \right ) } \left ( dx \right ) ^{m}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m*F^(b*log(d*x^n + c) + a),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., size = 0, normalized size = 0. \[{\rm integral}\left (\left (d x\right )^{m} F^{b \log \left (d x^{n} + c\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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