Optimal. Leaf size=51 \[ 2 \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}} \]
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Rubi [A] time = 0.31, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6707, 2177, 2180, 2204} \[ 2 \sqrt {\pi } \text {Erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {2 e^{a+b x+c x^2}}{\sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2180
Rule 2204
Rule 6707
Rubi steps
\begin {align*} \int \frac {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 \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 \operatorname {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 \operatorname {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*}
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Mathematica [A] time = 0.10, size = 62, normalized size = 1.22 \[ \frac {2 \sqrt {-a-x (b+c x)} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )-2 e^{a+x (b+c x)}}{\sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 0.88 \[ 2 \sqrt {\pi }\, \erfi \left (\sqrt {c \,x^{2}+b x +a}\right )-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{\sqrt {c \,x^{2}+b x +a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.00, size = 79, normalized size = 1.55 \[ -\frac {{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (2\,c\,x^2+2\,b\,x+2\,a\right )+2\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (-c\,x^2-b\,x-a\right )}^{3/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.49, size = 80, normalized size = 1.57 \[ \frac {\left (- 2 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )} + \frac {2 e^{a + b x + c x^{2}}}{\sqrt {- a - b x - c x^{2}}}\right ) \left (- a - b x - c x^{2}\right )^{\frac {3}{2}}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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