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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 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]

Rule 6741

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

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*}

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Mathematica [A]  time = 0.02, size = 37, normalized size = 1.00 \[ -\frac {x e^{a+c} \left (-\left ((b+d) x^n\right )\right )^{-1/n} \Gamma \left (\frac {1}{n},-\left ((b+d) x^n\right )\right )}{n} \]

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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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left ({\left (b + d\right )} x^{n} + a + c\right )}, x\right ) \]

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

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)

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maple [C]  time = 0.15, size = 241, normalized size = 6.51 \[ \frac {\left (\frac {\left (\left (-b -d \right ) n \,x^{n}+n +1\right ) n^{2} x^{-n +1} \left (\left (-b -d \right ) x^{n}\right )^{-\frac {n +1}{2 n}} \left (-b -d \right )^{\frac {1}{n}-1} \WhittakerM \left (\frac {1}{n}-\frac {n +1}{2 n}, \frac {n +1}{2 n}+\frac {1}{2}, \left (-b -d \right ) x^{n}\right ) {\mathrm e}^{-\frac {\left (-b -d \right ) x^{n}}{2}}}{\left (n +1\right ) \left (2 n +1\right )}+\frac {\left (n +1\right ) n \,x^{-n +1} \left (\left (-b -d \right ) x^{n}\right )^{-\frac {n +1}{2 n}} \left (-b -d \right )^{\frac {1}{n}-1} \WhittakerM \left (\frac {1}{n}-\frac {n +1}{2 n}+1, \frac {n +1}{2 n}+\frac {1}{2}, \left (-b -d \right ) x^{n}\right ) {\mathrm e}^{-\frac {\left (-b -d \right ) x^{n}}{2}}}{2 n +1}\right ) \left (-b -d \right )^{-\frac {1}{n}} {\mathrm e}^{a +c}}{n} \]

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

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maxima [A]  time = 0.99, size = 36, normalized size = 0.97 \[ -\frac {x e^{\left (a + c\right )} \Gamma \left (\frac {1}{n}, -{\left (b + d\right )} x^{n}\right )}{\left (-{\left (b + d\right )} x^{n}\right )^{\left (\frac {1}{n}\right )} n} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {e}}^{a+c+b\,x^n+d\,x^n} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + c + b*x^n + d*x^n),x)

[Out]

int(exp(a + c + b*x^n + d*x^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} e^{c} \int e^{b x^{n}} e^{d x^{n}}\, dx \]

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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