Integrand size = 33, antiderivative size = 52 \[ \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx=e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6839, 2207, 2211, 2235} \[ \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx=e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
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Rule 2207
Rule 2211
Rule 2235
Rule 6839
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int e^x \sqrt {x} \, dx,x,a+b x+c x^2\right ) \\ & = e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,a+b x+c x^2\right ) \\ & = e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {a+b x+c x^2}\right ) \\ & = e^{a+b x+c x^2} \sqrt {a+b x+c x^2}-\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \\ \end{align*}
Time = 1.68 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {a+x (b+c x)} \Gamma \left (\frac {3}{2},-a-x (b+c x)\right )}{\sqrt {-a-x (b+c x)}} \]
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Time = 0.36 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {\operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }}{2}+{\mathrm e}^{c \,x^{2}+b x +a} \sqrt {c \,x^{2}+b x +a}\) | \(44\) |
default | \(-\frac {\operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }}{2}+{\mathrm e}^{c \,x^{2}+b x +a} \sqrt {c \,x^{2}+b x +a}\) | \(44\) |
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\[ \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )} \,d x } \]
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Time = 1.85 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.50 \[ \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx=\frac {\left (\sqrt {- a - b x - c x^{2}} e^{a + b x + c x^{2}} + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )}}{2}\right ) \sqrt {a + b x + c x^{2}}}{\sqrt {- a - b x - c x^{2}}} \]
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\[ \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx=-\frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {c x^{2} + b x + a}\right ) + \sqrt {c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )} \]
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Time = 0.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.46 \[ \int e^{a+b x+c x^2} (b+2 c x) \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,\sqrt {c\,x^2+b\,x+a}}{2\,\sqrt {-c\,x^2-b\,x-a}}+{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}\,\sqrt {c\,x^2+b\,x+a} \]
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