\(\int F^{c (a+b x)} (d+e x)^m \, dx\) [23]

Optimal result

Integrand size = 17, antiderivative size = 67 \[ \int F^{c (a+b x)} (d+e x)^m \, dx=\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^m \Gamma \left (1+m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{-m}}{b c \log (F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2212} \[ \int F^{c (a+b x)} (d+e x)^m \, dx=\frac {(d+e x)^m F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{-m} \Gamma \left (m+1,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]

[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)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^m \Gamma \left (1+m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{-m}}{b c \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} (d+e x)^m \, dx=\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^m \Gamma \left (1+m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{-m}}{b c \log (F)} \]

[In]

Integrate[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)

Maple [F]

\[\int F^{c \left (b x +a \right )} \left (e x +d \right )^{m}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int F^{c (a+b x)} (d+e x)^m \, dx=\frac {e^{\left (-\frac {e m \log \left (-\frac {b c \log \left (F\right )}{e}\right ) + {\left (b c d - a c e\right )} \log \left (F\right )}{e}\right )} \Gamma \left (m + 1, -\frac {{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right )}{b c \log \left (F\right )} \]

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} (d+e x)^m \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (d+e\,x\right )}^m \,d x \]

[In]

int(F^(c*(a + b*x))*(d + e*x)^m,x)

[Out]

int(F^(c*(a + b*x))*(d + e*x)^m, x)