Optimal. Leaf size=128 \[ (a d+b c) \text {Int}\left (\frac {e^{x (a d+b c)+a c+b d x^2}}{x},x\right )+\sqrt {\pi } \sqrt {b} \sqrt {d} e^{-\frac {(b c-a d)^2}{4 b d}} \text {erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )-\frac {e^{x (a d+b c)+a c+b d x^2}}{x} \]
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Rubi [A] time = 0.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx &=\int \frac {e^{a c+(b c+a d) x+b d x^2}}{x^2} \, dx\\ &=-\frac {e^{a c+(b c+a d) x+b d x^2}}{x}+(2 b d) \int e^{a c+(b c+a d) x+b d x^2} \, dx-(-b c-a d) \int \frac {e^{a c+(b c+a d) x+b d x^2}}{x} \, dx\\ &=-\frac {e^{a c+(b c+a d) x+b d x^2}}{x}-(-b c-a d) \int \frac {e^{a c+(b c+a d) x+b d x^2}}{x} \, dx+\left (2 b d e^{-\frac {(b c-a d)^2}{4 b d}}\right ) \int e^{\frac {(b c+a d+2 b d x)^2}{4 b d}} \, dx\\ &=-\frac {e^{a c+(b c+a d) x+b d x^2}}{x}+\sqrt {b} \sqrt {d} e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )-(-b c-a d) \int \frac {e^{a c+(b c+a d) x+b d x^2}}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left ({\left (b x + a\right )} {\left (d x + c\right )}\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{\left (b x +a \right ) \left (d x +c \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left ({\left (b x + a\right )} {\left (d x + c\right )}\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{\left (a+b\,x\right )\,\left (c+d\,x\right )}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a c} \int \frac {e^{a d x} e^{b c x} e^{b d x^{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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