Optimal. Leaf size=142 \[ \frac{105}{16} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+\frac{35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{105}{8} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]
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Rubi [A] time = 0.633317, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {6707, 2176, 2180, 2204} \[ \frac{105}{16} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right )+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+\frac{35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{105}{8} e^{a+b x+c x^2} \sqrt{a+b x+c x^2} \]
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 )^{7/2} \, dx &=\operatorname{Subst}\left (\int e^x x^{7/2} \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac{7}{2} \operatorname{Subst}\left (\int e^x x^{5/2} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}+\frac{35}{4} \operatorname{Subst}\left (\int e^x x^{3/2} \, dx,x,a+b x+c x^2\right )\\ &=\frac{35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}-\frac{105}{8} \operatorname{Subst}\left (\int e^x \sqrt{x} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{105}{8} e^{a+b x+c x^2} \sqrt{a+b x+c x^2}+\frac{35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}+\frac{105}{16} \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac{105}{8} e^{a+b x+c x^2} \sqrt{a+b x+c x^2}+\frac{35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}+\frac{105}{8} \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )\\ &=-\frac{105}{8} e^{a+b x+c x^2} \sqrt{a+b x+c x^2}+\frac{35}{4} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{3/2}-\frac{7}{2} e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{5/2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )^{7/2}+\frac{105}{16} \sqrt{\pi } \text{erfi}\left (\sqrt{a+b x+c x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.196046, size = 47, normalized size = 0.33 \[ -\frac{\sqrt{a+x (b+c x)} \text{Gamma}\left (\frac{9}{2},-a-x (b+c x)\right )}{\sqrt{-a-x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 119, normalized size = 0.8 \begin{align*}{\frac{35\,{{\rm e}^{c{x}^{2}+bx+a}}}{4} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{{\rm e}^{c{x}^{2}+bx+a}}}{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{{\rm e}^{c{x}^{2}+bx+a}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}+{\frac{105\,\sqrt{\pi }}{16}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }-{\frac{105\,{{\rm e}^{c{x}^{2}+bx+a}}}{8}\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{7}{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^{4} x^{7} + 7 \, b c^{3} x^{6} + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{5} + 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} x^{4} + a^{3} b +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{3} + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} x^{2} +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} 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{7}{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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