3.365 \(\int \frac{F^{a+b (c+d x)^n}}{(c+d x)^2} \, dx\)

Optimal. Leaf size=52 \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.0600595, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^n)/(c + d*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.86974, size = 48, normalized size = 0.92 \[ - \frac{F^{a} \left (- b \left (c + d x\right )^{n} \log{\left (F \right )}\right )^{\frac{1}{n}} \Gamma{\left (- \frac{1}{n},- b \left (c + d x\right )^{n} \log{\left (F \right )} \right )}}{d n \left (c + d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**n)/(d*x+c)**2,x)

[Out]

-F**a*(-b*(c + d*x)**n*log(F))**(1/n)*Gamma(-1/n, -b*(c + d*x)**n*log(F))/(d*n*(
c + d*x))

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Mathematica [A]  time = 0.0298042, size = 52, normalized size = 1. \[ -\frac{F^a \left (-b \log (F) (c+d x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^n)/(c + d*x)^2,x]

[Out]

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

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{n}}}{ \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^n)/(d*x+c)^2,x)

[Out]

int(F^(a+b*(d*x+c)^n)/(d*x+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^n*b + a)/(d*x + c)^2,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^n*b + a)/(d*x + c)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{F^{{\left (d x + c\right )}^{n} b + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^n*b + a)/(d*x + c)^2,x, algorithm="fricas")

[Out]

integral(F^((d*x + c)^n*b + a)/(d^2*x^2 + 2*c*d*x + c^2), 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*(d*x+c)**n)/(d*x+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((d*x + c)^n*b + a)/(d*x + c)^2,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^n*b + a)/(d*x + c)^2, x)