3.23 \(\int F^{c (a+b x)} (d+e x)^m \, dx\)

Optimal. Leaf size=67 \[ \frac{(d+e x)^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.0455326, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{(d+e x)^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.74382, size = 60, normalized size = 0.9 \[ \frac{F^{\frac{c \left (a e - b d\right )}{e}} \left (\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e}\right )^{- m} \left (d + e x\right )^{m} \Gamma{\left (m + 1,\frac{b c \left (- d - e x\right ) \log{\left (F \right )}}{e} \right )}}{b c \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0179293, size = 66, normalized size = 0.99 \[ -\frac{(d+e x)^{m+1} F^{a c-\frac{b c d}{e}} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m-1} \text{Gamma}\left (m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*(e*x+d)^m,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.268747, size = 88, normalized size = 1.31 \[ \frac{e^{\left (-\frac{e m \log \left (-\frac{b c \log \left (F\right )}{e}\right ) +{\left (b c d - a c e\right )} \log \left (F\right )}{e}\right )} \Gamma \left (m + 1, -\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right )}{b c \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*x+d)**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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