3.771 \(\int e^{(a+b x)^n} (a+b x)^m \, dx\)

Optimal. Leaf size=52 \[ -\frac{(a+b x)^{m+1} \left (-(a+b x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-(a+b x)^n\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0263118, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2218} \[ -\frac{(a+b x)^{m+1} \left (-(a+b x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-(a+b x)^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[E^(a + b*x)^n*(a + b*x)^m,x]

[Out]

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

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.0139617, size = 52, normalized size = 1. \[ -\frac{(a+b x)^{m+1} \left (-(a+b x)^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-(a+b x)^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.294, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{ \left ( bx+a \right ) ^{n}}} \left ( bx+a \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^m*e^((b*x + a)^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((b*x+a)**n)*(b*x+a)**m,x)

[Out]

Integral((a + b*x)**m*exp((a + b*x)**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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