Optimal. Leaf size=21 \[ \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
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Rubi [A] time = 0.26, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6707, 2180, 2204} \[ \sqrt {\pi } \text {Erfi}\left (\sqrt {a+b x+c x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 6707
Rubi steps
\begin {align*} \int \frac {e^{a+b x+c x^2} (b+2 c x)}{\sqrt {a+b x+c x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,a+b x+c x^2\right )\\ &=2 \operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )\\ &=\sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )\\ \end {align*}
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Mathematica [B] time = 0.06, size = 46, normalized size = 2.19 \[ \frac {\sqrt {-a-x (b+c x)} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )}{\sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 21, normalized size = 1.00 \[ \sqrt {\pi } i \operatorname {erf}\left (-\sqrt {c x^{2} + b x + a} i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 0.86 \[ \sqrt {\pi }\, \erfi \left (\sqrt {c \,x^{2}+b x +a}\right ) \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 49, normalized size = 2.33 \[ \frac {\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,\sqrt {-c\,x^2-b\,x-a}}{\sqrt {c\,x^2+b\,x+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.43, size = 49, normalized size = 2.33 \[ \frac {\sqrt {\pi } \sqrt {- a - b x - c x^{2}} \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )}}{\sqrt {a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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