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 = 0.25, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 2178
Rule 2271
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*}
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Mathematica [C] time = 0.13, 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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fricas [A] time = 0.42, size = 80, normalized size = 0.72 \[ -\frac {{\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} + {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )} - 2 \, {\rm Ei}\left (b x\right ) e^{a}}{2 \, c} \]
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 \[ \int \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 112, normalized size = 1.01 \[ -\frac {\Ei \left (1, -b x \right ) {\mathrm e}^{a}}{c}+\frac {\Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}+\Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}}{2 c} \]
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 \[ \int \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{x\,\left (d\,x^2+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} \int \frac {e^{b x}}{c x + d x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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