Optimal. Leaf size=145 \[ \frac {\sqrt {d} e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {e^a b \text {Ei}(b x)}{c}-\frac {e^{a+b x}}{c x} \]
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Rubi [A] time = 0.35, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2271, 2177, 2178, 2269} \[ \frac {\sqrt {d} e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (\frac {b \left (\sqrt {d} x+\sqrt {-c}\right )}{\sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {e^a b \text {Ei}(b x)}{c}-\frac {e^{a+b x}}{c x} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2269
Rule 2271
Rubi steps
\begin {align*} \int \frac {e^{a+b x}}{x^2 \left (c+d x^2\right )} \, dx &=\int \left (\frac {e^{a+b x}}{c x^2}-\frac {d e^{a+b x}}{c \left (c+d x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {e^{a+b x}}{x^2} \, dx}{c}-\frac {d \int \frac {e^{a+b x}}{c+d x^2} \, dx}{c}\\ &=-\frac {e^{a+b x}}{c x}+\frac {b \int \frac {e^{a+b x}}{x} \, dx}{c}-\frac {d \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}{c}\\ &=-\frac {e^{a+b x}}{c x}+\frac {b e^a \text {Ei}(b x)}{c}-\frac {d \int \frac {e^{a+b x}}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 (-c)^{3/2}}-\frac {d \int \frac {e^{a+b x}}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 (-c)^{3/2}}\\ &=-\frac {e^{a+b x}}{c x}+\frac {b e^a \text {Ei}(b x)}{c}+\frac {\sqrt {d} 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)^{3/2}}-\frac {\sqrt {d} 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)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 133, normalized size = 0.92 \[ \frac {e^a \left (i \sqrt {d} x e^{\frac {i b \sqrt {c}}{\sqrt {d}}} \text {Ei}\left (b \left (x-\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )-i \sqrt {d} x e^{-\frac {i b \sqrt {c}}{\sqrt {d}}} \text {Ei}\left (b \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )\right )+2 b \sqrt {c} x \text {Ei}(b x)-2 \sqrt {c} e^{b x}\right )}{2 c^{3/2} x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 128, normalized size = 0.88 \[ \frac {2 \, b^{2} c x {\rm Ei}\left (b x\right ) e^{a} + \sqrt {-\frac {b^{2} c}{d}} d x {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a + \sqrt {-\frac {b^{2} c}{d}}\right )} - \sqrt {-\frac {b^{2} c}{d}} d x {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) e^{\left (a - \sqrt {-\frac {b^{2} c}{d}}\right )} - 2 \, b c e^{\left (b x + a\right )}}{2 \, b c^{2} x} \]
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 \[ \int \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 142, normalized size = 0.98 \[ \left (-\frac {\Ei \left (1, -b x \right ) {\mathrm e}^{a}}{c}+\frac {\left (\Ei \left (1, \frac {a d +\sqrt {-c d}\, b -\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{\frac {a d +\sqrt {-c d}\, b}{d}}-\Ei \left (1, -\frac {-a d +\sqrt {-c d}\, b +\left (b x +a \right ) d}{d}\right ) {\mathrm e}^{-\frac {-a d +\sqrt {-c d}\, b}{d}}\right ) d}{2 \sqrt {-c d}\, b c}-\frac {{\mathrm e}^{b x +a}}{b c x}\right ) b \]
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 \[ \int \frac {e^{\left (b x + a\right )}}{{\left (d x^{2} + c\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{x^2\,\left (d\,x^2+c\right )} \,d x \]
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 \[ e^{a} \int \frac {e^{b x}}{c x^{2} + d x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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