Optimal. Leaf size=25 \[ 2 a b e^x+\frac {1}{2} b^2 e^{2 x}+a^2 x \]
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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 = {2320, 45}
\begin {gather*} a^2 x+2 a b e^x+\frac {1}{2} b^2 e^{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2320
Rubi steps
\begin {align*} \int \left (a+b e^x\right )^2 \, dx &=\text {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,e^x\right )\\ &=\text {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 = 0.02, size = 26, normalized size = 1.04 \begin {gather*} \frac {1}{2} b e^x \left (4 a+b e^x\right )+a^2 \log \left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 24, normalized size = 0.96
method | result | size |
norman | \(2 a b \,{\mathrm e}^{x}+\frac {b^{2} {\mathrm e}^{2 x}}{2}+a^{2} x\) | \(22\) |
risch | \(2 a b \,{\mathrm e}^{x}+\frac {b^{2} {\mathrm e}^{2 x}}{2}+a^{2} x\) | \(22\) |
derivativedivides | \(\frac {b^{2} {\mathrm e}^{2 x}}{2}+2 a b \,{\mathrm e}^{x}+a^{2} \ln \left ({\mathrm e}^{x}\right )\) | \(24\) |
default | \(\frac {b^{2} {\mathrm e}^{2 x}}{2}+2 a b \,{\mathrm e}^{x}+a^{2} \ln \left ({\mathrm e}^{x}\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 21, normalized size = 0.84 \begin {gather*} a^{2} x + \frac {1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 21, normalized size = 0.84 \begin {gather*} a^{2} x + \frac {1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 22, normalized size = 0.88 \begin {gather*} a^{2} x + 2 a b e^{x} + \frac {b^{2} e^{2 x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.92, size = 21, normalized size = 0.84 \begin {gather*} a^{2} x + \frac {1}{2} \, b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} \end {gather*}
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 \begin {gather*} x\,a^2+2\,{\mathrm {e}}^x\,a\,b+\frac {{\mathrm {e}}^{2\,x}\,b^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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