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

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))

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Rubi [A]  time = 0.0536632, 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, Rules used = {2188, 2181} \[ \frac{\left ((d+e x)^n\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m n} \text{Gamma}\left (m n+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]

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))

Rule 2188

Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*(u_)^(m_.), x_Symbol] :> Module[{uu = NormalizePowerOfLine
ar[u, x], z}, Simp[z = If[PowerQ[uu] && FreeQ[uu[[2]], x], uu[[1]]^(m*uu[[2]]), uu^m]; (uu^m*Int[z*(a + b*(F^(
g*ExpandToSum[v, x]))^n)^p, x])/z, x]] /; FreeQ[{F, a, b, g, m, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ[u
, x] &&  !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) &&  !IntegerQ[m]

Rule 2181

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

Rubi steps

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

Mathematica [A]  time = 0.0130439, size = 72, normalized size = 1. \[ \frac{\left ((d+e x)^n\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m n} \text{Gamma}\left (m n+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[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))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( \left ( ex+d \right ) ^{n} \right ) ^{m}\, dx \end{align*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.77738, size = 158, normalized size = 2.19 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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