Optimal. Leaf size=111 \[ \frac {e^a \text {Ei}(b x)}{c}-\frac {e^{a+\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (-\frac {b \left (\sqrt {-c}-\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c}-\frac {e^{a-\frac {b \sqrt {-c}}{\sqrt {d}}} \text {Ei}\left (\frac {b \left (\sqrt {-c}+\sqrt {d} x\right )}{\sqrt {d}}\right )}{2 c} \]
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Rubi [A]
time = 0.18, 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 = {2303, 2209}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2303
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] Result contains complex when optimal does not.
time = 0.18, size = 93, normalized size = 0.84 \begin {gather*} \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 (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )+\text {Ei}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )\right )\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 112, normalized size = 1.01
method | result | size |
derivativedivides | \(\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \expIntegral \left (1, \frac {b \sqrt {-c d}+a d -\left (b x +a \right ) d}{d}\right )+{\mathrm e}^{-\frac {b \sqrt {-c d}-a d}{d}} \expIntegral \left (1, -\frac {b \sqrt {-c d}-a d +\left (b x +a \right ) d}{d}\right )}{2 c}-\frac {{\mathrm e}^{a} \expIntegral \left (1, -b x \right )}{c}\) | \(112\) |
default | \(\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \expIntegral \left (1, \frac {b \sqrt {-c d}+a d -\left (b x +a \right ) d}{d}\right )+{\mathrm e}^{-\frac {b \sqrt {-c d}-a d}{d}} \expIntegral \left (1, -\frac {b \sqrt {-c d}-a d +\left (b x +a \right ) d}{d}\right )}{2 c}-\frac {{\mathrm e}^{a} \expIntegral \left (1, -b x \right )}{c}\) | \(112\) |
risch | \(\frac {{\mathrm e}^{\frac {-b \sqrt {-c d}+a d}{d}} \expIntegral \left (1, -\frac {b \sqrt {-c d}-a d +\left (b x +a \right ) d}{d}\right )}{2 c}+\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+a d}{d}} \expIntegral \left (1, \frac {b \sqrt {-c d}+a d -\left (b x +a \right ) d}{d}\right )}{2 c}-\frac {{\mathrm e}^{a} \expIntegral \left (1, -b x \right )}{c}\) | \(113\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 80, normalized size = 0.72 \begin {gather*} -\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 {gather*}
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 \begin {gather*} e^{a} \int \frac {e^{b x}}{c x + d x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {e}}^{a+b\,x}}{x\,\left (d\,x^2+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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