Optimal. Leaf size=128 \[ -\frac {e^{a c+(b c+a d) x+b d x^2}}{x}+\sqrt {b} \sqrt {d} e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )+(b c+a d) \text {Int}\left (\frac {e^{a c+(b c+a d) x+b d x^2}}{x},x\right ) \]
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Rubi [A]
time = 0.19, 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 = {}
\begin {gather*} \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx \end {gather*}
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 = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{(a+b x) (c+d x)}}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.01, 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 \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 = 0.00, 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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{a c} \int \frac {e^{a d x} e^{b c x} e^{b d x^{2}}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {e}}^{\left (a+b\,x\right )\,\left (c+d\,x\right )}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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