Optimal. Leaf size=20 \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]
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Rubi [A] time = 0.597482, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {6742, 2262} \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2262
Rubi steps
\begin{align*} \int \left (\frac{x^2 \left (5 e^x+3 x^2\right )}{5 \sqrt{5 e^x+x^3}}+\frac{4}{5} x \sqrt{5 e^x+x^3}\right ) \, dx &=\frac{1}{5} \int \frac{x^2 \left (5 e^x+3 x^2\right )}{\sqrt{5 e^x+x^3}} \, dx+\frac{4}{5} \int x \sqrt{5 e^x+x^3} \, dx\\ &=\frac{1}{5} \int \left (\frac{5 e^x x^2}{\sqrt{5 e^x+x^3}}+\frac{3 x^4}{\sqrt{5 e^x+x^3}}\right ) \, dx+\frac{4}{5} \int x \sqrt{5 e^x+x^3} \, dx\\ &=\frac{3}{5} \int \frac{x^4}{\sqrt{5 e^x+x^3}} \, dx+\frac{4}{5} \int x \sqrt{5 e^x+x^3} \, dx+\int \frac{e^x x^2}{\sqrt{5 e^x+x^3}} \, dx\\ &=\frac{2}{5} x^2 \sqrt{5 e^x+x^3}\\ \end{align*}
Mathematica [A] time = 0.161067, size = 20, normalized size = 1. \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 16, normalized size = 0.8 \begin{align*}{\frac{2\,{x}^{2}}{5}\sqrt{5\,{{\rm e}^{x}}+{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13466, size = 31, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (x^{5} + 5 \, x^{2} e^{x}\right )}}{5 \, \sqrt{x^{3} + 5 \, e^{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{7 x^{4}}{\sqrt{x^{3} + 5 e^{x}}}\, dx + \int \frac{20 x e^{x}}{\sqrt{x^{3} + 5 e^{x}}}\, dx + \int \frac{5 x^{2} e^{x}}{\sqrt{x^{3} + 5 e^{x}}}\, dx}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x^{2} + 5 \, e^{x}\right )} x^{2}}{5 \, \sqrt{x^{3} + 5 \, e^{x}}} + \frac{4}{5} \, \sqrt{x^{3} + 5 \, e^{x}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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