Optimal. Leaf size=138 \[ \frac{2 \left (3 a e^2+c d^2\right ) \left (a e^2+c d^2+2 c d e x\right )}{3 e \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 d}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.0937172, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {792, 613} \[ \frac{2 \left (3 a e^2+c d^2\right ) \left (a e^2+c d^2+2 c d e x\right )}{3 e \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 d}{3 e (d+e x) \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 613
Rubi steps
\begin{align*} \int \frac{x}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac{2 d}{3 e \left (c d^2-a e^2\right ) (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (c d^2+3 a e^2\right ) \int \frac{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 e \left (c d^2-a e^2\right )}\\ &=-\frac{2 d}{3 e \left (c d^2-a e^2\right ) (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{2 \left (c d^2+3 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right )}{3 e \left (c d^2-a e^2\right )^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 0.0373915, size = 100, normalized size = 0.72 \[ \frac{2 \left (a^2 e^3 (2 d+3 e x)+2 a c d e \left (3 d^2+5 d e x+3 e^2 x^2\right )+c^2 d^3 x (3 d+2 e x)\right )}{3 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 149, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 6\,acd{e}^{3}{x}^{2}+2\,{c}^{2}{d}^{3}e{x}^{2}+3\,{a}^{2}{e}^{4}x+10\,ac{d}^{2}{e}^{2}x+3\,{c}^{2}{d}^{4}x+2\,{a}^{2}d{e}^{3}+6\,ac{d}^{3}e \right ) }{3\,{a}^{3}{e}^{6}-9\,{a}^{2}c{d}^{2}{e}^{4}+9\,a{c}^{2}{d}^{4}{e}^{2}-3\,{c}^{3}{d}^{6}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 20.4064, size = 621, normalized size = 4.5 \begin{align*} \frac{2 \,{\left (6 \, a c d^{3} e + 2 \, a^{2} d e^{3} + 2 \,{\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x^{2} +{\left (3 \, c^{2} d^{4} + 10 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{3 \,{\left (a c^{3} d^{8} e - 3 \, a^{2} c^{2} d^{6} e^{3} + 3 \, a^{3} c d^{4} e^{5} - a^{4} d^{2} e^{7} +{\left (c^{4} d^{7} e^{2} - 3 \, a c^{3} d^{5} e^{4} + 3 \, a^{2} c^{2} d^{3} e^{6} - a^{3} c d e^{8}\right )} x^{3} +{\left (2 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} + 3 \, a^{2} c^{2} d^{4} e^{5} + a^{3} c d^{2} e^{7} - a^{4} e^{9}\right )} x^{2} +{\left (c^{4} d^{9} - a c^{3} d^{7} e^{2} - 3 \, a^{2} c^{2} d^{5} e^{4} + 5 \, a^{3} c d^{3} e^{6} - 2 \, a^{4} d e^{8}\right )} x\right )}} \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*} \int \frac{x}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \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*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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