Optimal. Leaf size=111 \[ -\frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}-\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 c}+\frac{e^a \text{Ei}(b x)}{c} \]
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Rubi [A] time = 0.246697, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2271, 2178} \[ -\frac{e^{a+\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (-\frac{b \left (\sqrt{-c}-\sqrt{d} x\right )}{\sqrt{d}}\right )}{2 c}-\frac{e^{a-\frac{b \sqrt{-c}}{\sqrt{d}}} \text{Ei}\left (\frac{b \left (\sqrt{d} x+\sqrt{-c}\right )}{\sqrt{d}}\right )}{2 c}+\frac{e^a \text{Ei}(b x)}{c} \]
Antiderivative was successfully verified.
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Rule 2271
Rule 2178
Rubi steps
\begin{align*} \int \frac{e^{a+b x}}{x \left (c+d x^2\right )} \, dx &=\int \left (\frac{e^{a+b x}}{c x}-\frac{d e^{a+b x} x}{c \left (c+d x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{e^{a+b x}}{x} \, dx}{c}-\frac{d \int \frac{e^{a+b x} x}{c+d x^2} \, dx}{c}\\ &=\frac{e^a \text{Ei}(b x)}{c}-\frac{d \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}{c}\\ &=\frac{e^a \text{Ei}(b x)}{c}+\frac{\sqrt{d} \int \frac{e^{a+b x}}{\sqrt{-c}-\sqrt{d} x} \, dx}{2 c}-\frac{\sqrt{d} \int \frac{e^{a+b x}}{\sqrt{-c}+\sqrt{d} x} \, dx}{2 c}\\ &=\frac{e^a \text{Ei}(b x)}{c}-\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 c}-\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 c}\\ \end{align*}
Mathematica [C] time = 0.120942, size = 93, normalized size = 0.84 \[ \frac{e^a \left (2 \text{Ei}(b x)-e^{-\frac{i b \sqrt{c}}{\sqrt{d}}} \left (e^{\frac{2 i b \sqrt{c}}{\sqrt{d}}} \text{Ei}\left (b \left (x-\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )+\text{Ei}\left (b \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )\right )\right )\right )}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 112, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{c}}+{\frac{1}{2\,c} \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 ) } \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*} \int \frac{e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55933, size = 167, normalized size = 1.5 \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 \,{\rm Ei}\left (b x\right ) e^{a}}{2 \, c} \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{e^{b x}}{c x + d x^{3}}\, 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{e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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