Optimal. Leaf size=37 \[ -\frac {x e^{a+c} \left (-\left ((b+d) x^n\right )\right )^{-1/n} \Gamma \left (\frac {1}{n},-\left ((b+d) x^n\right )\right )}{n} \]
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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.
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Rule 2208
Rule 6741
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.
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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.
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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.
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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.
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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.
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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.
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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.
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