Integrand size = 33, antiderivative size = 51 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}+2 \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 51, 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, 2208, 2211, 2235} \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=2 \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}} \]
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Rule 2208
Rule 2211
Rule 2235
Rule 6839
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {e^x}{x^{3/2}} \, dx,x,a+b x+c x^2\right ) \\ & = -\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}+2 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,a+b x+c x^2\right ) \\ & = -\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}+4 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {a+b x+c x^2}\right ) \\ & = -\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}}+2 \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \\ \end{align*}
Time = 1.83 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.22 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-2 e^{a+x (b+c x)}+2 \sqrt {-a-x (b+c x)} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )}{\sqrt {a+x (b+c x)}} \]
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Time = 0.48 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(2 \,\operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{\sqrt {c \,x^{2}+b x +a}}\) | \(45\) |
default | \(2 \,\operatorname {erfi}\left (\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {\pi }-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{\sqrt {c \,x^{2}+b x +a}}\) | \(45\) |
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\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 2.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.57 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\left (- 2 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )} + \frac {2 e^{a + b x + c x^{2}}}{\sqrt {- a - b x - c x^{2}}}\right ) \left (- a - b x - c x^{2}\right )^{\frac {3}{2}}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}} \]
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\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.75 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.55 \[ \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (2\,c\,x^2+2\,b\,x+2\,a\right )+2\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (-c\,x^2-b\,x-a\right )}^{3/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
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