Optimal. Leaf size=92 \[ \frac {b (c+d x)^2 \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2}-\frac {(c+d x) (b c-a d) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^2} \]
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Rubi [A] time = 0.06, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 2208
Rule 2218
Rule 2226
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*}
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Mathematica [A] time = 0.09, size = 85, normalized size = 0.92 \[ \frac {(c+d x) \left ((a d-b c) \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )+b (c+d x) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )\right )}{3 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 169, normalized size = 1.84 \[ -\frac {b d^{2} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 2 \, {\left (b c d - a d^{2}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (b d^{2} x^{2} + 2 \, a d^{2} x - b c^{2} + 2 \, a c d\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} + \int \frac {3 \, {\left (b d e x^{2} + 2 \, a d e x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{2 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}\,\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b x\right ) e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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