Integrand size = 17, antiderivative size = 118 \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2301, 2209} \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Rule 2209
Rule 2301
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-c} e^{a+b x}}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} e^{a+b x}}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {e^{a+b x}}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {e^{a+b x}}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-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 \sqrt {-c} \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 \sqrt {-c} \sqrt {d}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=-\frac {i e^{a-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )\right )}{2 \sqrt {c} \sqrt {d}} \]
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Time = 0.43 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )-{\mathrm e}^{-\frac {b \sqrt {-c d}-a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 \sqrt {-c d}}\) | \(102\) |
default | \(-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )-{\mathrm e}^{-\frac {b \sqrt {-c d}-a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 \sqrt {-c d}}\) | \(102\) |
risch | \(-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (\frac {b \sqrt {-c d}+a d -d \left (b x +a \right )}{d}\right )}{2 \sqrt {-c d}}+\frac {{\mathrm e}^{\frac {-b \sqrt {-c d}+a d}{d}} \operatorname {Ei}_{1}\left (-\frac {b \sqrt {-c d}-a d +d \left (b x +a \right )}{d}\right )}{2 \sqrt {-c d}}\) | \(106\) |
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none
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83 \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=-\frac {\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )}}{2 \, b c} \]
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\[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=e^{a} \int \frac {e^{b x}}{c + d x^{2}}\, dx \]
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\[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=\int { \frac {e^{\left (b x + a\right )}}{d x^{2} + c} \,d x } \]
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\[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=\int { \frac {e^{\left (b x + a\right )}}{d x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {e^{a+b x}}{c+d x^2} \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{d\,x^2+c} \,d x \]
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