Optimal. Leaf size=38 \[ \text {Ei}\left (c x^2+b x+a\right )-\frac {e^{a+b x+c x^2}}{a+b x+c x^2} \]
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Rubi [A] time = 0.20, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6707, 2177, 2178} \[ \text {Ei}\left (c x^2+b x+a\right )-\frac {e^{a+b x+c x^2}}{a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
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 )^2} \, dx &=\operatorname {Subst}\left (\int \frac {e^x}{x^2} \, dx,x,a+b x+c x^2\right )\\ &=-\frac {e^{a+b x+c x^2}}{a+b x+c x^2}+\operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,a+b x+c x^2\right )\\ &=-\frac {e^{a+b x+c x^2}}{a+b x+c x^2}+\text {Ei}\left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 35, normalized size = 0.92 \[ \text {Ei}(a+x (b+c x))-\frac {e^{a+x (b+c x)}}{a+x (b+c x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 49, normalized size = 1.29 \[ \frac {{\left (c x^{2} + b x + a\right )} {\rm Ei}\left (c x^{2} + b x + a\right ) - e^{\left (c x^{2} + b x + a\right )}}{c x^{2} + b x + a} \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 45, normalized size = 1.18 \[ -\Ei \left (1, -c \,x^{2}-b x -a \right )-\frac {{\mathrm e}^{c \,x^{2}+b x +a}}{c \,x^{2}+b x +a} \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.99, size = 44, normalized size = 1.16 \[ -\mathrm {expint}\left (-c\,x^2-b\,x-a\right )-\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,{\mathrm {e}}^{c\,x^2}}{c\,x^2+b\,x+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 159.33, size = 24, normalized size = 0.63 \[ - \frac {\operatorname {E}_{2}\left (- a - b x - c x^{2}\right )}{a + b x + c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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