Integrand size = 20, antiderivative size = 111 \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=\frac {e^a \operatorname {ExpIntegralEi}(b x)}{c}-\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c} \]
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Time = 0.16 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2303, 2209} \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=-\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {d}}\right )}{2 c}+\frac {e^a \operatorname {ExpIntegralEi}(b x)}{c} \]
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Rule 2209
Rule 2303
Rubi steps \begin{align*} \text {integral}& = \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*}
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=\frac {e^a \left (2 \operatorname {ExpIntegralEi}(b x)-e^{-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )+\operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )\right )\right )}{2 c} \]
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Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )+{\mathrm e}^{-\frac {b \sqrt {-c d}-a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 c}-\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right )}{c}\) | \(112\) |
default | \(\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )+{\mathrm e}^{-\frac {b \sqrt {-c d}-a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 c}-\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right )}{c}\) | \(112\) |
risch | \(-\frac {{\mathrm e}^{a} \operatorname {Ei}_{1}\left (-b x \right )}{c}+\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )}{2 c}+\frac {{\mathrm e}^{\frac {-b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 c}\) | \(113\) |
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none
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.72 \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=-\frac {{\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} + {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )} - 2 \, {\rm Ei}\left (b x\right ) e^{a}}{2 \, c} \]
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\[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=e^{a} \int \frac {e^{b x}}{c x + d x^{3}}\, dx \]
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\[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=\int { \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x} \,d x } \]
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\[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=\int { \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x} \,d x } \]
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Timed out. \[ \int \frac {e^{a+b x}}{x \left (c+d x^2\right )} \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{x\,\left (d\,x^2+c\right )} \,d x \]
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