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

Optimal. Leaf size=72 \[ \frac{\left ((d+e x)^n\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m n} \text{Gamma}\left (m n+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0947758, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\left ((d+e x)^n\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m n} \text{Gamma}\left (m n+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.9849, size = 65, normalized size = 0.9 \[ \frac{F^{\frac{c \left (a e - b d\right )}{e}} \left (\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e}\right )^{- m n} \left (\left (d + e x\right )^{n}\right )^{m} \Gamma{\left (m n + 1,\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e} \right )}}{b c \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0484099, size = 74, normalized size = 1.03 \[ -\frac{(d+e x) \left ((d+e x)^n\right )^m F^{a c-\frac{b c d}{e}} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m n-1} \text{Gamma}\left (m n+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{F}^{c \left ( bx+a \right ) } \left ( \left ( ex+d \right ) ^{n} \right ) ^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(F^(c*(b*x+a))*((e*x+d)^n)^m,x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.253858, size = 92, normalized size = 1.28 \[ \frac{e^{\left (-\frac{e m n \log \left (-\frac{b c \log \left (F\right )}{e}\right ) +{\left (b c d - a c e\right )} \log \left (F\right )}{e}\right )} \Gamma \left (m n + 1, -\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right )}{b c \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(-(e*m*n*log(-b*c*log(F)/e) + (b*c*d - a*c*e)*log(F))/e)*gamma(m*n + 1, -(b*c*
e*x + b*c*d)*log(F)/e)/(b*c*log(F))

_______________________________________________________________________________________

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*x+d)**n)**m,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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