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

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

[Out]

-(b*x+a)^(1+m)*GAMMA((1+m)/n,-(b*x+a)^n)/b/n/((-(b*x+a)^n)^((1+m)/n))

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Rubi [A]  time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 52, normalized size = 1.00 \[ -\frac {(a+b x)^{m+1} \left (-(a+b x)^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-(a+b x)^n\right )}{b n} \]

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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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{m} e^{\left ({\left (b x + a\right )}^{n}\right )}, x\right ) \]

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

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)

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{m} {\mathrm e}^{\left (b x +a \right )^{n}}\, dx \]

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

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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mupad [B]  time = 3.89, size = 77, normalized size = 1.48 \[ \frac {{\mathrm {e}}^{\frac {{\left (a+b\,x\right )}^n}{2}}\,{\left (a+b\,x\right )}^{m+1}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left ({\left (a+b\,x\right )}^n\right )}{b\,{\left ({\left (a+b\,x\right )}^n\right )}^{\frac {m+n+1}{2\,n}}\,\left (m+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((a + b*x)^n)*(a + b*x)^m,x)

[Out]

(exp((a + b*x)^n/2)*(a + b*x)^(m + 1)*whittakerM(1 - (m + n + 1)/(2*n), (m + n + 1)/(2*n) - 1/2, (a + b*x)^n))
/(b*((a + b*x)^n)^((m + n + 1)/(2*n))*(m + 1))

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

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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