Optimal. Leaf size=152 \[ \frac{3 d e (b c-a d)^2 \text{Unintegrable}\left (\frac{e^{e (c+d x)^3}}{a+b x},x\right )}{b^3}-\frac{d e (c+d x) (b c-a d) \text{Gamma}\left (\frac{1}{3},-e (c+d x)^3\right )}{b^3 \sqrt [3]{-e (c+d x)^3}}-\frac{d e (c+d x)^2 \text{Gamma}\left (\frac{2}{3},-e (c+d x)^3\right )}{b^2 \left (-e (c+d x)^3\right )^{2/3}}-\frac{e^{e (c+d x)^3}}{b (a+b x)} \]
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Rubi [A] time = 0.350316, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{e (c+d x)^3}}{(a+b x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{e^{e (c+d x)^3}}{(a+b x)^2} \, dx &=-\frac{e^{e (c+d x)^3}}{b (a+b x)}+\frac{(3 d e) \int \frac{e^{e (c+d x)^3} (c+d x)^2}{a+b x} \, dx}{b}\\ &=-\frac{e^{e (c+d x)^3}}{b (a+b x)}+\frac{(3 d e) \int \left (\frac{d (b c-a d) e^{e (c+d x)^3}}{b^2}+\frac{(b c-a d)^2 e^{e (c+d x)^3}}{b^2 (a+b x)}+\frac{d e^{e (c+d x)^3} (c+d x)}{b}\right ) \, dx}{b}\\ &=-\frac{e^{e (c+d x)^3}}{b (a+b x)}+\frac{\left (3 d^2 e\right ) \int e^{e (c+d x)^3} (c+d x) \, dx}{b^2}+\frac{\left (3 d^2 (b c-a d) e\right ) \int e^{e (c+d x)^3} \, dx}{b^3}+\frac{\left (3 d (b c-a d)^2 e\right ) \int \frac{e^{e (c+d x)^3}}{a+b x} \, dx}{b^3}\\ &=-\frac{e^{e (c+d x)^3}}{b (a+b x)}-\frac{d (b c-a d) e (c+d x) \Gamma \left (\frac{1}{3},-e (c+d x)^3\right )}{b^3 \sqrt [3]{-e (c+d x)^3}}-\frac{d e (c+d x)^2 \Gamma \left (\frac{2}{3},-e (c+d x)^3\right )}{b^2 \left (-e (c+d x)^3\right )^{2/3}}+\frac{\left (3 d (b c-a d)^2 e\right ) \int \frac{e^{e (c+d x)^3}}{a+b x} \, dx}{b^3}\\ \end{align*}
Mathematica [A] time = 2.05486, size = 0, normalized size = 0. \[ \int \frac{e^{e (c+d x)^3}}{(a+b x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{e \left ( dx+c \right ) ^{3}}}}{ \left ( bx+a \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left ({\left (d x + c\right )}^{3} e\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 = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + 3 \, c^{2} d e x + c^{3} e\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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