Optimal. Leaf size=20 \[ \frac {2}{5} x^2 \sqrt {x^3+5 e^x} \]
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Rubi [A] time = 0.60, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6742, 2262} \[ \frac {2}{5} x^2 \sqrt {x^3+5 e^x} \]
Antiderivative was successfully verified.
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Rule 2262
Rule 6742
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*}
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Mathematica [A] time = 0.17, size = 20, normalized size = 1.00 \[ \frac {2}{5} x^2 \sqrt {x^3+5 e^x} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {{\left (3 \, x^{2} + 5 \, e^{x}\right )} x^{2}}{5 \, \sqrt {x^{3} + 5 \, e^{x}}} + \frac {4}{5} \, \sqrt {x^{3} + 5 \, e^{x}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 16, normalized size = 0.80 \[ \frac {2 \sqrt {x^{3}+5 \,{\mathrm e}^{x}}\, x^{2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 23, normalized size = 1.15 \[ \frac {2 \, {\left (x^{5} + 5 \, x^{2} e^{x}\right )}}{5 \, \sqrt {x^{3} + 5 \, e^{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 15, normalized size = 0.75 \[ \frac {2\,x^2\,\sqrt {5\,{\mathrm {e}}^x+x^3}}{5} \]
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 \[ \frac {\int \frac {7 x^{4}}{\sqrt {x^{3} + 5 e^{x}}}\, dx + \int \frac {20 x e^{x}}{\sqrt {x^{3} + 5 e^{x}}}\, dx + \int \frac {5 x^{2} e^{x}}{\sqrt {x^{3} + 5 e^{x}}}\, dx}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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