3.1.19 \(\int F^{c (a+b x)} ((d+e x)^n)^m \, dx\) [19]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 = {1973, 2212} \begin {gather*} \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)} \end {gather*}

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 1973

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/
a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]

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*} \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*}

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Mathematica [A]
time = 0.03, size = 72, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (\left (e x +d \right )^{n}\right )^{m}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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(((x*e + d)^n)^m*F^((b*x + a)*c), x)

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Fricas [A]
time = 0.10, size = 68, normalized size = 0.94 \begin {gather*} \frac {e^{\left (-{\left (m n e \log \left (-b c e^{\left (-1\right )} \log \left (F\right )\right ) + {\left (b c d - a c e\right )} \log \left (F\right )\right )} e^{\left (-1\right )}\right )} \Gamma \left (m n + 1, -{\left (b c x e + b c d\right )} e^{\left (-1\right )} \log \left (F\right )\right )}{b c \log \left (F\right )} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{c \left (a + b x\right )} \left (\left (d + e x\right )^{n}\right )^{m}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(((x*e + d)^n)^m*F^((b*x + a)*c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int F^{c\,\left (a+b\,x\right )}\,{\left ({\left (d+e\,x\right )}^n\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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