Optimal. Leaf size=92 \[ \frac{b (c+d x)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^2}-\frac{(c+d x) (b c-a d) \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^2} \]
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Rubi [A] time = 0.0567096, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2226, 2208, 2218} \[ \frac{b (c+d x)^2 \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^2}-\frac{(c+d x) (b c-a d) \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 2226
Rule 2208
Rule 2218
Rubi steps
\begin{align*} \int e^{\frac{e}{(c+d x)^3}} (a+b x) \, dx &=\int \left (\frac{(-b c+a d) e^{\frac{e}{(c+d x)^3}}}{d}+\frac{b e^{\frac{e}{(c+d x)^3}} (c+d x)}{d}\right ) \, dx\\ &=\frac{b \int e^{\frac{e}{(c+d x)^3}} (c+d x) \, dx}{d}+\frac{(-b c+a d) \int e^{\frac{e}{(c+d x)^3}} \, dx}{d}\\ &=\frac{b \left (-\frac{e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^2}-\frac{(b c-a d) \sqrt [3]{-\frac{e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )}{3 d^2}\\ \end{align*}
Mathematica [A] time = 0.0847917, size = 85, normalized size = 0.92 \[ \frac{(c+d x) \left ((a d-b c) \sqrt [3]{-\frac{e}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{e}{(c+d x)^3}\right )+b (c+d x) \left (-\frac{e}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{e}{(c+d x)^3}\right )\right )}{3 d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{3}}}}} \left ( bx+a \right ) \, dx \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*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} + \int \frac{3 \,{\left (b d e x^{2} + 2 \, a d e x\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{2 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53108, size = 370, normalized size = 4.02 \begin{align*} -\frac{b d^{2} \left (-\frac{e}{d^{3}}\right )^{\frac{2}{3}} \Gamma \left (\frac{1}{3}, -\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 2 \,{\left (b c d - a d^{2}\right )} \left (-\frac{e}{d^{3}}\right )^{\frac{1}{3}} \Gamma \left (\frac{2}{3}, -\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) -{\left (b d^{2} x^{2} + 2 \, a d^{2} x - b c^{2} + 2 \, a c d\right )} e^{\left (\frac{e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{2 \, d^{2}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )} e^{\left (\frac{e}{{\left (d x + c\right )}^{3}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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