Optimal. Leaf size=38 \[ -e^{a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6839, 2207,
2225} \begin {gather*} e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 6839
Rubi steps
\begin {align*} \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx &=\text {Subst}\left (\int e^x x \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )-\text {Subst}\left (\int e^x \, dx,x,a+b x+c x^2\right )\\ &=-e^{a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 23, normalized size = 0.61 \begin {gather*} e^{a+x (b+c x)} \left (-1+a+b x+c x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 37, normalized size = 0.97
method | result | size |
gosper | \(\left (c \,x^{2}+b x +a -1\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(24\) |
risch | \(\left (c \,x^{2}+b x +a -1\right ) {\mathrm e}^{c \,x^{2}+b x +a}\) | \(24\) |
derivativedivides | \(-{\mathrm e}^{c \,x^{2}+b x +a}+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )\) | \(37\) |
default | \(-{\mathrm e}^{c \,x^{2}+b x +a}+{\mathrm e}^{c \,x^{2}+b x +a} \left (c \,x^{2}+b x +a \right )\) | \(37\) |
norman | \(\left (a -1\right ) {\mathrm e}^{c \,x^{2}+b x +a}+b x \,{\mathrm e}^{c \,x^{2}+b x +a}+c \,x^{2} {\mathrm e}^{c \,x^{2}+b x +a}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.43, size = 501, normalized size = 13.18 \begin {gather*} \frac {\sqrt {\pi } a b \operatorname {erf}\left (\sqrt {-c} x - \frac {b}{2 \, \sqrt {-c}}\right ) e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{2 \, \sqrt {-c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} b^{2} e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} - \frac {1}{2} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} a \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} + \frac {3}{8} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {5}{2}}} - \frac {4 \, b e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} b \sqrt {c} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} - \frac {1}{8} \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {7}{2}}} - \frac {6 \, b^{2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {5}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {7}{2}}} + \frac {8 \, \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{c^{\frac {3}{2}}}\right )} c^{\frac {3}{2}} e^{\left (a - \frac {b^{2}}{4 \, c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 23, normalized size = 0.61 \begin {gather*} {\left (c x^{2} + b x + a - 1\right )} e^{\left (c x^{2} + b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 22, normalized size = 0.58 \begin {gather*} \left (a + b x + c x^{2} - 1\right ) e^{a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.62, size = 23, normalized size = 0.61 \begin {gather*} {\left (c x^{2} + b x + a - 1\right )} e^{\left (c x^{2} + b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 23, normalized size = 0.61 \begin {gather*} {\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (c\,x^2+b\,x+a-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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