Optimal. Leaf size=82 \[ \frac{3}{4} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{3}{2} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]
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Rubi [A] time = 0.362943, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {6707, 2176, 2180, 2204} \[ \frac{3}{4} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{3}{2} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 6707
Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int e^x x^{3/2} \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{3}{2} \operatorname{Subst}\left (\int e^x \sqrt{x} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{3}{2} e^{a+b x+c x^2} \sqrt{a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{3}{2} e^{a+b x+c x^2} \sqrt{a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+\frac{3}{2} \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )\\ &=-\frac{3}{2} e^{a+b x+c x^2} \sqrt{a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}+\frac{3}{4} \sqrt{\pi } \text{erfi}\left (\sqrt{a+b x+c x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.109704, size = 47, normalized size = 0.57 \[ -\frac{\sqrt{a+x (b+c x)} \text{Gamma}\left (\frac{5}{2},-a-x (b+c x)\right )}{\sqrt{-a-x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 69, normalized size = 0.8 \begin{align*}{{\rm e}^{c{x}^{2}+bx+a}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}+{\frac{3\,\sqrt{\pi }}{4}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }-{\frac{3\,{{\rm e}^{c{x}^{2}+bx+a}}}{2}\sqrt{c{x}^{2}+bx+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + a b +{\left (b^{2} + 2 \, a c\right )} x\right )} \sqrt{c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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