Optimal. Leaf size=80 \[ \frac {a (a+b x) \Gamma \left (\frac {1}{3},-(a+b x)^3\right )}{3 b^2 \sqrt [3]{-(a+b x)^3}}-\frac {(a+b x)^2 \Gamma \left (\frac {2}{3},-(a+b x)^3\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2258, 2239,
2250} \begin {gather*} \frac {a (a+b x) \text {Gamma}\left (\frac {1}{3},-(a+b x)^3\right )}{3 b^2 \sqrt [3]{-(a+b x)^3}}-\frac {(a+b x)^2 \text {Gamma}\left (\frac {2}{3},-(a+b x)^3\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 2250
Rule 2258
Rubi steps
\begin {align*} \int e^{x^n} x^m \, dx &=-\frac {x^{1+m} \left (-x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-x^n\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 74, normalized size = 0.92 \begin {gather*} \frac {(a+b x) \left (a \sqrt [3]{-(a+b x)^3} \Gamma \left (\frac {1}{3},-(a+b x)^3\right )-(a+b x) \Gamma \left (\frac {2}{3},-(a+b x)^3\right )\right )}{3 b^2 \left (-(a+b x)^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\left (b x +a \right )^{3}} x\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 89, normalized size = 1.11 \begin {gather*} -\frac {\left (-b^{3}\right )^{\frac {2}{3}} a \Gamma \left (\frac {1}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right ) - \left (-b^{3}\right )^{\frac {1}{3}} b \Gamma \left (\frac {2}{3}, -b^{3} x^{3} - 3 \, a b^{2} x^{2} - 3 \, a^{2} b x - a^{3}\right )}{3 \, b^{4}} \end {gather*}
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 \begin {gather*} e^{a^{3}} \int x e^{b^{3} x^{3}} e^{3 a b^{2} x^{2}} e^{3 a^{2} b x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x\,{\mathrm {e}}^{{\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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