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 = 0.36, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 2176
Rule 2180
Rule 2204
Rule 6707
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*}
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Mathematica [A] time = 0.11, size = 47, normalized size = 0.57 \[ -\frac {\sqrt {a+x (b+c x)} \Gamma \left (\frac {5}{2},-a-x (b+c x)\right )}{\sqrt {-a-x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + a b + {\left (b^{2} + 2 \, a c\right )} x\right )} \sqrt {c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 65, normalized size = 0.79 \[ \frac {3}{4} \, \sqrt {\pi } i \operatorname {erf}\left (-\sqrt {c x^{2} + b x + a} i\right ) + \frac {1}{2} \, {\left (2 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {c x^{2} + b x + a}\right )} e^{\left (c x^{2} + b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 69, normalized size = 0.84 \[ \frac {3 \sqrt {\pi }\, \erfi \left (\sqrt {c \,x^{2}+b x +a}\right )}{4}+\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} {\mathrm e}^{c \,x^{2}+b x +a}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, {\mathrm e}^{c \,x^{2}+b x +a}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.77, size = 102, normalized size = 1.24 \[ \frac {3\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,{\left (-c\,x^2-b\,x-a\right )}^{3/2}}-\frac {3\,{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}\,\sqrt {c\,x^2+b\,x+a}}{2}+{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 86.53, size = 94, normalized size = 1.15 \[ \frac {\left (- \sqrt {- a - b x - c x^{2}} \left (a + b x + c x^{2} - \frac {3}{2}\right ) e^{a + b x + c x^{2}} + \frac {3 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )}}{4}\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (- a - b x - c x^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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