Optimal. Leaf size=107 \[ -\frac{d e e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^2}-\frac{d e^{\frac{e}{c+d x}}}{b (b c-a d)}-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)} \]
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Rubi [A] time = 0.544528, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {2223, 6742, 2222, 2210, 2228, 2178, 2209} \[ -\frac{d e e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^2}-\frac{d e^{\frac{e}{c+d x}}}{b (b c-a d)}-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 2223
Rule 6742
Rule 2222
Rule 2210
Rule 2228
Rule 2178
Rule 2209
Rubi steps
\begin{align*} \int \frac{e^{\frac{e}{c+d x}}}{(a+b x)^2} \, dx &=-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)}-\frac{(d e) \int \frac{e^{\frac{e}{c+d x}}}{(a+b x) (c+d x)^2} \, dx}{b}\\ &=-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)}-\frac{(d e) \int \left (\frac{b^2 e^{\frac{e}{c+d x}}}{(b c-a d)^2 (a+b x)}-\frac{d e^{\frac{e}{c+d x}}}{(b c-a d) (c+d x)^2}-\frac{b d e^{\frac{e}{c+d x}}}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b}\\ &=-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)}-\frac{(b d e) \int \frac{e^{\frac{e}{c+d x}}}{a+b x} \, dx}{(b c-a d)^2}+\frac{\left (d^2 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{(b c-a d)^2}+\frac{\left (d^2 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{(c+d x)^2} \, dx}{b (b c-a d)}\\ &=-\frac{d e^{\frac{e}{c+d x}}}{b (b c-a d)}-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)}-\frac{d e \text{Ei}\left (\frac{e}{c+d x}\right )}{(b c-a d)^2}-\frac{\left (d^2 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{(b c-a d)^2}-\frac{(d e) \int \frac{e^{\frac{e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{b c-a d}\\ &=-\frac{d e^{\frac{e}{c+d x}}}{b (b c-a d)}-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)}-\frac{(d e) \operatorname{Subst}\left (\int \frac{\exp \left (-\frac{b e}{-b c+a d}+\frac{d e x}{-b c+a d}\right )}{x} \, dx,x,\frac{a+b x}{c+d x}\right )}{(b c-a d)^2}\\ &=-\frac{d e^{\frac{e}{c+d x}}}{b (b c-a d)}-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)}-\frac{d e e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^2}\\ \end{align*}
Mathematica [A] time = 0.162542, size = 105, normalized size = 0.98 \[ -\frac{d e e^{\frac{b e}{b c-a d}} \text{Ei}\left (\frac{e}{c+d x}-\frac{b e}{b c-a d}\right )}{(a d-b c)^2}-\frac{d e^{\frac{e}{c+d x}}}{b (b c-a d)}-\frac{e^{\frac{e}{c+d x}}}{b (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 97, normalized size = 0.9 \begin{align*} -{\frac{de}{ \left ( ad-bc \right ) ^{2}} \left ( -{{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-bc}} \right ) ^{-1}}-{{\rm e}^{-{\frac{be}{ad-bc}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-bc}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\frac{e}{d x + c}\right )}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59035, size = 312, normalized size = 2.92 \begin{align*} -\frac{{\left (b d e x + a d e\right )}{\rm Ei}\left (-\frac{b d e x + a d e}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac{b e}{b c - a d}\right )} +{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\frac{e}{c + d x}}}{\left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\frac{e}{d x + c}\right )}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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