Optimal. Leaf size=25 \[ a^2 x+2 a b e^x+\frac {1}{2} b^2 e^{2 x} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2282, 43} \[ a^2 x+2 a b e^x+\frac {1}{2} b^2 e^{2 x} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2282
Rubi steps
\begin {align*} \int \left (a+b e^x\right )^2 \, dx &=\operatorname {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,e^x\right )\\ &=2 a b e^x+\frac {1}{2} b^2 e^{2 x}+a^2 x\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ a^2 x+2 a b e^x+\frac {1}{2} b^2 e^{2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 21, normalized size = 0.84 \[ a^{2} x + \frac {1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 21, normalized size = 0.84 \[ a^{2} x + \frac {1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 24, normalized size = 0.96 \[ a^{2} \ln \left ({\mathrm e}^{x}\right )+2 a b \,{\mathrm e}^{x}+\frac {b^{2} {\mathrm e}^{2 x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 21, normalized size = 0.84 \[ a^{2} x + \frac {1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 21, normalized size = 0.84 \[ x\,a^2+2\,{\mathrm {e}}^x\,a\,b+\frac {{\mathrm {e}}^{2\,x}\,b^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 22, normalized size = 0.88 \[ a^{2} x + 2 a b e^{x} + \frac {b^{2} e^{2 x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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