3.26 \(\int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right )^{-m} \, dx\)

Optimal. Leaf size=73 \[ \frac{\left ((d+e x)^3\right )^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3 m} \text{Gamma}\left (1-3 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]

[Out]

(F^(c*(a - (b*d)/e))*Gamma[1 - 3*m, -((b*c*(d + e*x)*Log[F])/e)]*(-((b*c*(d + e*
x)*Log[F])/e))^(3*m))/(b*c*((d + e*x)^3)^m*Log[F])

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Rubi [A]  time = 0.103617, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049 \[ \frac{\left ((d+e x)^3\right )^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3 m} \text{Gamma}\left (1-3 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))/(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + e^3*x^3)^m,x]

[Out]

(F^(c*(a - (b*d)/e))*Gamma[1 - 3*m, -((b*c*(d + e*x)*Log[F])/e)]*(-((b*c*(d + e*
x)*Log[F])/e))^(3*m))/(b*c*((d + e*x)^3)^m*Log[F])

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Rubi in Sympy [A]  time = 26.2484, size = 65, normalized size = 0.89 \[ \frac{F^{\frac{c \left (a e - b d\right )}{e}} \left (\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e}\right )^{3 m} \left (\left (d + e x\right )^{3}\right )^{- m} \Gamma{\left (- 3 m + 1,\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e} \right )}}{b c \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))/((e**3*x**3+3*d*e**2*x**2+3*d**2*e*x+d**3)**m),x)

[Out]

F**(c*(a*e - b*d)/e)*(b*c*(-d - e*x)*log(F)/e)**(3*m)*((d + e*x)**3)**(-m)*Gamma
(-3*m + 1, b*c*(-d - e*x)*log(F)/e)/(b*c*log(F))

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Mathematica [A]  time = 0.0466042, size = 75, normalized size = 1.03 \[ -\frac{(d+e x) \left ((d+e x)^3\right )^{-m} F^{a c-\frac{b c d}{e}} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3 m-1} \text{Gamma}\left (1-3 m,-\frac{b c \log (F) (d+e x)}{e}\right )}{e} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))/(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + e^3*x^3)^m,x]

[Out]

-((F^(a*c - (b*c*d)/e)*(d + e*x)*Gamma[1 - 3*m, -((b*c*(d + e*x)*Log[F])/e)]*(-(
(b*c*(d + e*x)*Log[F])/e))^(-1 + 3*m))/(e*((d + e*x)^3)^m))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{{F}^{c \left ( bx+a \right ) }}{ \left ({e}^{3}{x}^{3}+3\,d{e}^{2}{x}^{2}+3\,{d}^{2}ex+{d}^{3} \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))/((e^3*x^3+3*d*e^2*x^2+3*d^2*e*x+d^3)^m),x)

[Out]

int(F^(c*(b*x+a))/((e^3*x^3+3*d*e^2*x^2+3*d^2*e*x+d^3)^m),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)^m,x, algorithm="maxima")

[Out]

integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{F^{b c x + a c}}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)^m,x, algorithm="fricas")

[Out]

integral(F^(b*c*x + a*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)^m, 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**(c*(b*x+a))/((e**3*x**3+3*d*e**2*x**2+3*d**2*e*x+d**3)**m),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)^m,x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)^m, x)