Optimal. Leaf size=151 \[ -\frac {2 b (c+d x)^2 (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {(c+d x) (b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}-\frac {b^2 e \text {Ei}\left (\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{(c+d x)^3}}}{3 d^3} \]
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Rubi [A] time = 0.13, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2226, 2208, 2218, 2214, 2210} \[ -\frac {2 b (c+d x)^2 (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {(c+d x) (b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}-\frac {b^2 e \text {Ei}\left (\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{(c+d x)^3}}}{3 d^3} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 2210
Rule 2214
Rule 2218
Rule 2226
Rubi steps
\begin {align*} \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx &=\int \left (\frac {(-b c+a d)^2 e^{\frac {e}{(c+d x)^3}}}{d^2}-\frac {2 b (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)}{d^2}+\frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^2}{d^2}\right ) \, dx\\ &=\frac {b^2 \int e^{\frac {e}{(c+d x)^3}} (c+d x)^2 \, dx}{d^2}-\frac {(2 b (b c-a d)) \int e^{\frac {e}{(c+d x)^3}} (c+d x) \, dx}{d^2}+\frac {(b c-a d)^2 \int e^{\frac {e}{(c+d x)^3}} \, dx}{d^2}\\ &=\frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{3 d^3}-\frac {2 b (b c-a d) \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^3}+\frac {(b c-a d)^2 \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^3}+\frac {\left (b^2 e\right ) \int \frac {e^{\frac {e}{(c+d x)^3}}}{c+d x} \, dx}{d^2}\\ &=\frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{3 d^3}-\frac {b^2 e \text {Ei}\left (\frac {e}{(c+d x)^3}\right )}{3 d^3}-\frac {2 b (b c-a d) \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^3}+\frac {(b c-a d)^2 \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^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 136, normalized size = 0.90 \[ \frac {-2 b (c+d x)^2 (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )+(c+d x) (b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )-b^2 e \text {Ei}\left (\frac {e}{(c+d x)^3}\right )+b^2 (c+d x)^3 e^{\frac {e}{(c+d x)^3}}}{3 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 259, normalized size = 1.72 \[ -\frac {b^{2} e {\rm Ei}\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 3 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + 3 \, a^{2} d^{3} x + b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{3 \, d^{3}} \]
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 \[ \int {\left (b x + a\right )}^{2} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}}\, dx \]
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 \[ \frac {1}{3} \, {\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )} + \int \frac {{\left (b^{2} d e x^{3} + 3 \, a b d e x^{2} + 3 \, a^{2} d e x\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} \]
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 \[ \int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}\,{\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int \left (a + b x\right )^{2} e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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