Optimal. Leaf size=100 \[ \frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d}+\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 d} \]
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Rubi [A] time = 0.128842, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2271, 2178} \[ \frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d}+\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2271
Rule 2178
Rubi steps
\begin{align*} \int \frac{e^{a+b x} x}{c+d x^2} \, dx &=\int \left (-\frac{e^{a+b x}}{2 \sqrt{d} \left (\sqrt{-c}-\sqrt{d} x\right )}+\frac{e^{a+b x}}{2 \sqrt{d} \left (\sqrt{-c}+\sqrt{d} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{e^{a+b x}}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 \sqrt{d}}+\frac{\int \frac{e^{a+b x}}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 \sqrt{d}}\\ &=\frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d}+\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{-c}+\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 d}\\ \end{align*}
Mathematica [C] time = 0.0542735, size = 83, normalized size = 0.83 \[ \frac{e^{a-\frac{i b \sqrt{c}}{\sqrt{d}}} \left (e^{\frac{2 i b \sqrt{c}}{\sqrt{d}}} \text{Ei}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\text{Ei}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 323, normalized size = 3.2 \begin{align*}{\frac{1}{{b}^{2}} \left ( -{\frac{b}{2\,d} \left ( \sqrt{-cd}{{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) b+\sqrt{-cd}{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) b+{{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) ad-{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) ad \right ){\frac{1}{\sqrt{-cd}}}}+{\frac{ab}{2} \left ({{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+ad \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}-d \left ( bx+a \right ) +ad \right ) } \right ) -{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-ad \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+d \left ( bx+a \right ) -ad \right ) } \right ) \right ){\frac{1}{\sqrt{-cd}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x e^{\left (b x + a\right )}}{b d x^{2} + b c} + \int \frac{{\left (d x^{2} e^{a} - c e^{a}\right )} e^{\left (b x\right )}}{b d^{2} x^{4} + 2 \, b c d x^{2} + b c^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57449, size = 144, normalized size = 1.44 \begin{align*} \frac{{\rm Ei}\left (b x - \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a + \sqrt{-\frac{b^{2} c}{d}}\right )} +{\rm Ei}\left (b x + \sqrt{-\frac{b^{2} c}{d}}\right ) e^{\left (a - \sqrt{-\frac{b^{2} c}{d}}\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int \frac{x e^{b x}}{c + d x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{\left (b x + a\right )}}{d x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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