Optimal. Leaf size=72 \[ -\frac {96 \sqrt {e^{a+b x}}}{b^4}+\frac {48 x \sqrt {e^{a+b x}}}{b^3}-\frac {12 x^2 \sqrt {e^{a+b x}}}{b^2}+\frac {2 x^3 \sqrt {e^{a+b x}}}{b} \]
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Rubi [A] time = 0.10, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2176, 2194} \[ -\frac {12 x^2 \sqrt {e^{a+b x}}}{b^2}+\frac {48 x \sqrt {e^{a+b x}}}{b^3}-\frac {96 \sqrt {e^{a+b x}}}{b^4}+\frac {2 x^3 \sqrt {e^{a+b x}}}{b} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int \sqrt {e^{a+b x}} x^3 \, dx &=\frac {2 \sqrt {e^{a+b x}} x^3}{b}-\frac {6 \int \sqrt {e^{a+b x}} x^2 \, dx}{b}\\ &=-\frac {12 \sqrt {e^{a+b x}} x^2}{b^2}+\frac {2 \sqrt {e^{a+b x}} x^3}{b}+\frac {24 \int \sqrt {e^{a+b x}} x \, dx}{b^2}\\ &=\frac {48 \sqrt {e^{a+b x}} x}{b^3}-\frac {12 \sqrt {e^{a+b x}} x^2}{b^2}+\frac {2 \sqrt {e^{a+b x}} x^3}{b}-\frac {48 \int \sqrt {e^{a+b x}} \, dx}{b^3}\\ &=-\frac {96 \sqrt {e^{a+b x}}}{b^4}+\frac {48 \sqrt {e^{a+b x}} x}{b^3}-\frac {12 \sqrt {e^{a+b x}} x^2}{b^2}+\frac {2 \sqrt {e^{a+b x}} x^3}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.51 \[ \frac {2 \left (b^3 x^3-6 b^2 x^2+24 b x-48\right ) \sqrt {e^{a+b x}}}{b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 35, normalized size = 0.49 \[ \frac {2 \, {\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 24 \, b x - 48\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 35, normalized size = 0.49 \[ \frac {2 \, {\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 24 \, b x - 48\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.49 \[ \frac {2 \left (b^{3} x^{3}-6 b^{2} x^{2}+24 b x -48\right ) \sqrt {{\mathrm e}^{b x +a}}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 48, normalized size = 0.67 \[ \frac {2 \, {\left (b^{3} x^{3} e^{\left (\frac {1}{2} \, a\right )} - 6 \, b^{2} x^{2} e^{\left (\frac {1}{2} \, a\right )} + 24 \, b x e^{\left (\frac {1}{2} \, a\right )} - 48 \, e^{\left (\frac {1}{2} \, a\right )}\right )} e^{\left (\frac {1}{2} \, b x\right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 37, normalized size = 0.51 \[ \sqrt {{\mathrm {e}}^{a+b\,x}}\,\left (\frac {48\,x}{b^3}-\frac {96}{b^4}+\frac {2\,x^3}{b}-\frac {12\,x^2}{b^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 42, normalized size = 0.58 \[ \begin {cases} \frac {\left (2 b^{3} x^{3} - 12 b^{2} x^{2} + 48 b x - 96\right ) \sqrt {e^{a + b x}}}{b^{4}} & \text {for}\: b^{4} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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