Optimal. Leaf size=72 \[ -\frac{12 x^2 \sqrt{e^{a+b x}}}{b^2}+\frac{48 x \sqrt{e^{a+b x}}}{b^3}-\frac{96 \sqrt{e^{a+b x}}}{b^4}+\frac{2 x^3 \sqrt{e^{a+b x}}}{b} \]
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Rubi [A] time = 0.101391, antiderivative size = 72, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0134152, 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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Maple [A] time = 0.003, size = 35, normalized size = 0.5 \begin{align*} 2\,{\frac{ \left ({x}^{3}{b}^{3}-6\,{x}^{2}{b}^{2}+24\,bx-48 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09332, size = 65, normalized size = 0.9 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45923, size = 85, normalized size = 1.18 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.108289, size = 42, normalized size = 0.58 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25374, size = 47, normalized size = 0.65 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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