Optimal. Leaf size=85 \[ \frac {4}{3} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {4 e^{a+b x+c x^2}}{3 \sqrt {a+b x+c x^2}}-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.35, antiderivative size = 85, 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, 2177, 2180, 2204} \[ \frac {4}{3} \sqrt {\pi } \text {Erfi}\left (\sqrt {a+b x+c x^2}\right )-\frac {4 e^{a+b x+c x^2}}{3 \sqrt {a+b x+c x^2}}-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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 )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {e^x}{x^{5/2}} \, dx,x,a+b x+c x^2\right )\\ &=-\frac {2 e^{a+b x+c x^2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {2}{3} \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}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e^{a+b x+c x^2}}{3 \sqrt {a+b x+c x^2}}+\frac {4}{3} \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}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e^{a+b x+c x^2}}{3 \sqrt {a+b x+c x^2}}+\frac {8}{3} \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}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 e^{a+b x+c x^2}}{3 \sqrt {a+b x+c x^2}}+\frac {4}{3} \sqrt {\pi } \text {erfi}\left (\sqrt {a+b x+c x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 77, normalized size = 0.91 \[ -\frac {2 \left (e^{a+x (b+c x)} (2 (a+x (b+c x))+1)+2 (-a-x (b+c x))^{3/2} \Gamma \left (\frac {1}{2},-a-x (b+c x)\right )\right )}{3 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 70, normalized size = 0.82 \[ \frac {4 \sqrt {\pi }\, \erfi \left (\sqrt {c \,x^{2}+b x +a}\right )}{3}-\frac {2 \,{\mathrm e}^{c \,x^{2}+b x +a}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {4 \,{\mathrm e}^{c \,x^{2}+b x +a}}{3 \sqrt {c \,x^{2}+b x +a}} \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 104, normalized size = 1.22 \[ -\frac {{\mathrm {e}}^{c\,x^2+b\,x+a}\,\left (2\,c\,x^2+2\,b\,x+2\,a\right )+4\,{\mathrm {e}}^{c\,x^2+b\,x+a}\,{\left (c\,x^2+b\,x+a\right )}^2-4\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-c\,x^2-b\,x-a}\right )\,{\left (-c\,x^2-b\,x-a\right )}^{5/2}}{3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.58, size = 105, normalized size = 1.24 \[ \frac {\left (\frac {4 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- a - b x - c x^{2}} \right )}}{3} - \frac {\left (- \frac {4 a}{3} - \frac {4 b x}{3} - \frac {4 c x^{2}}{3} - \frac {2}{3}\right ) e^{a + b x + c x^{2}}}{\left (- a - b x - c x^{2}\right )^{\frac {3}{2}}}\right ) \left (- a - b x - c x^{2}\right )^{\frac {5}{2}}}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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