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

Optimal. Leaf size=37 \[ -\frac{x e^{a+c} \left (-(b+d) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-(b+d) x^n\right )}{n} \]

[Out]

-((E^(a + c)*x*Gamma[n^(-1), -((b + d)*x^n)])/(n*(-((b + d)*x^n))^n^(-1)))

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Rubi [A]  time = 0.0351137, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6741, 2208} \[ -\frac{x e^{a+c} \left (-(b+d) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-(b+d) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + c + b*x^n + d*x^n),x]

[Out]

-((E^(a + c)*x*Gamma[n^(-1), -((b + d)*x^n)])/(n*(-((b + d)*x^n))^n^(-1)))

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

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

Mathematica [A]  time = 0.01572, size = 37, normalized size = 1. \[ -\frac{x e^{a+c} \left (-(b+d) x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-(b+d) x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a + c + b*x^n + d*x^n),x]

[Out]

-((E^(a + c)*x*Gamma[n^(-1), -((b + d)*x^n)])/(n*(-((b + d)*x^n))^n^(-1)))

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Maple [C]  time = 0.24, size = 241, normalized size = 6.5 \begin{align*}{\frac{{{\rm e}^{a+c}}}{n} \left ( -b-d \right ) ^{-{n}^{-1}} \left ({\frac{{n}^{2}{x}^{1-n} \left ( -b-d \right ) ^{{n}^{-1}-1} \left ( n{x}^{n} \left ( -b-d \right ) +n+1 \right ) }{ \left ( 1+n \right ) \left ( 1+2\,n \right ) } \left ({x}^{n} \left ( -b-d \right ) \right ) ^{-{\frac{1+n}{2\,n}}}{{\rm e}^{-{\frac{{x}^{n} \left ( -b-d \right ) }{2}}}}{{\sl M}_{{n}^{-1}-{\frac{1+n}{2\,n}},\,{\frac{1+n}{2\,n}}+{\frac{1}{2}}}\left ({x}^{n} \left ( -b-d \right ) \right )}}+{\frac{n{x}^{1-n} \left ( -b-d \right ) ^{{n}^{-1}-1} \left ( 1+n \right ) }{1+2\,n} \left ({x}^{n} \left ( -b-d \right ) \right ) ^{-{\frac{1+n}{2\,n}}}{{\rm e}^{-{\frac{{x}^{n} \left ( -b-d \right ) }{2}}}}{{\sl M}_{{n}^{-1}-{\frac{1+n}{2\,n}}+1,\,{\frac{1+n}{2\,n}}+{\frac{1}{2}}}\left ({x}^{n} \left ( -b-d \right ) \right )}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a+c+b*x^n+d*x^n),x)

[Out]

exp(a+c)/n*(-b-d)^(-1/n)*(n^2*x^(1-n)*(-b-d)^(1/n-1)*(n*x^n*(-b-d)+n+1)/(1+n)/(1+2*n)*(x^n*(-b-d))^(-1/2*(1+n)
/n)*exp(-1/2*x^n*(-b-d))*WhittakerM(1/n-1/2*(1+n)/n,1/2*(1+n)/n+1/2,x^n*(-b-d))+n*x^(1-n)*(-b-d)^(1/n-1)*(1+n)
/(1+2*n)*(x^n*(-b-d))^(-1/2*(1+n)/n)*exp(-1/2*x^n*(-b-d))*WhittakerM(1/n-1/2*(1+n)/n+1,1/2*(1+n)/n+1/2,x^n*(-b
-d)))

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Maxima [A]  time = 1.09025, size = 49, normalized size = 1.32 \begin{align*} -\frac{x e^{\left (a + c\right )} \Gamma \left (\frac{1}{n}, -{\left (b + d\right )} x^{n}\right )}{\left (-{\left (b + d\right )} x^{n}\right )^{\left (\frac{1}{n}\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a+c+b*x^n+d*x^n),x, algorithm="fricas")

[Out]

integral(e^((b + d)*x^n + a + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} e^{c} \int e^{b x^{n}} e^{d x^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a+c+b*x**n+d*x**n),x)

[Out]

exp(a)*exp(c)*Integral(exp(b*x**n)*exp(d*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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