Optimal. Leaf size=240 \[ \frac{b d^2 e^2 e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4}+\frac{d^2 e e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac{d^2 e e^{\frac{e}{c+d x}}}{2 (b c-a d)^3}+\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2} \]
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Rubi [A] time = 1.03656, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ \frac{b d^2 e^2 e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4}+\frac{d^2 e e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac{d^2 e e^{\frac{e}{c+d x}}}{2 (b c-a d)^3}+\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2} \]
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)^3} \, dx &=-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}-\frac{(d e) \int \frac{e^{\frac{e}{c+d x}}}{(a+b x)^2 (c+d x)^2} \, dx}{2 b}\\ &=-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}-\frac{(d e) \int \left (\frac{b^2 e^{\frac{e}{c+d x}}}{(b c-a d)^2 (a+b x)^2}-\frac{2 b^2 d e^{\frac{e}{c+d x}}}{(b c-a d)^3 (a+b x)}+\frac{d^2 e^{\frac{e}{c+d x}}}{(b c-a d)^2 (c+d x)^2}+\frac{2 b d^2 e^{\frac{e}{c+d x}}}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b}\\ &=-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}+\frac{\left (b d^2 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{a+b x} \, dx}{(b c-a d)^3}-\frac{\left (d^3 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{(b c-a d)^3}-\frac{(b d e) \int \frac{e^{\frac{e}{c+d x}}}{(a+b x)^2} \, dx}{2 (b c-a d)^2}-\frac{\left (d^3 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{(c+d x)^2} \, dx}{2 b (b c-a d)^2}\\ &=\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac{d^2 e \text{Ei}\left (\frac{e}{c+d x}\right )}{(b c-a d)^3}+\frac{\left (d^3 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{(b c-a d)^3}+\frac{\left (d^2 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{(b c-a d)^2}+\frac{\left (d^2 e^2\right ) \int \frac{e^{\frac{e}{c+d x}}}{(a+b x) (c+d x)^2} \, dx}{2 (b c-a d)^2}\\ &=\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac{\left (d^2 e\right ) \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)^3}+\frac{\left (d^2 e^2\right ) \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}{2 (b c-a d)^2}\\ &=\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac{d^2 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)^3}+\frac{\left (b^2 d^2 e^2\right ) \int \frac{e^{\frac{e}{c+d x}}}{a+b x} \, dx}{2 (b c-a d)^4}-\frac{\left (b d^3 e^2\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{2 (b c-a d)^4}-\frac{\left (d^3 e^2\right ) \int \frac{e^{\frac{e}{c+d x}}}{(c+d x)^2} \, dx}{2 (b c-a d)^3}\\ &=\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}+\frac{d^2 e e^{\frac{e}{c+d x}}}{2 (b c-a d)^3}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac{b d^2 e^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{2 (b c-a d)^4}+\frac{d^2 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)^3}+\frac{\left (b d^3 e^2\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{2 (b c-a d)^4}+\frac{\left (b d^2 e^2\right ) \int \frac{e^{\frac{e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{2 (b c-a d)^3}\\ &=\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}+\frac{d^2 e e^{\frac{e}{c+d x}}}{2 (b c-a d)^3}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac{d^2 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)^3}+\frac{\left (b d^2 e^2\right ) \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 )}{2 (b c-a d)^4}\\ &=\frac{d^2 e^{\frac{e}{c+d x}}}{2 b (b c-a d)^2}+\frac{d^2 e e^{\frac{e}{c+d x}}}{2 (b c-a d)^3}-\frac{e^{\frac{e}{c+d x}}}{2 b (a+b x)^2}+\frac{d e e^{\frac{e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac{d^2 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)^3}+\frac{b d^2 e^2 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 )}{2 (b c-a d)^4}\\ \end{align*}
Mathematica [F] time = 0.401324, size = 0, normalized size = 0. \[ \int \frac{e^{\frac{e}{c+d x}}}{(a+b x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.01, size = 240, normalized size = 1. \begin{align*} -{\frac{e}{d} \left ( -{\frac{be{d}^{3}}{ \left ( ad-bc \right ) ^{4}} \left ( -{\frac{1}{2}{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-bc}} \right ) ^{-2}}-{\frac{1}{2}{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-bc}} \right ) ^{-1}}-{\frac{1}{2}{{\rm e}^{-{\frac{be}{ad-bc}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-bc}} \right ) } \right ) }+{\frac{{d}^{3}}{ \left ( ad-bc \right ) ^{3}} \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 ) } \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6449, size = 1007, normalized size = 4.2 \begin{align*} \frac{{\left (a^{2} b d^{2} e^{2} +{\left (b^{3} d^{2} e^{2} + 2 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \,{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} e + 2 \,{\left (a b^{2} d^{2} e^{2} + 2 \,{\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} e\right )} x\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^{3} c^{4} - 4 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3} -{\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4} +{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} -{\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e -{\left (2 \, a b^{2} c^{2} d^{2} - 4 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4} +{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{2 \,{\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4} +{\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} x\right )}} \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 )^{3}}\, 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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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