Optimal. Leaf size=84 \[ \frac{3 (1-2 x)^{3/2}}{7 (3 x+2) \sqrt{5 x+3}}-\frac{103 \sqrt{1-2 x}}{7 \sqrt{5 x+3}}+\frac{103 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]
[Out]
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Rubi [A] time = 0.122437, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 (1-2 x)^{3/2}}{7 (3 x+2) \sqrt{5 x+3}}-\frac{103 \sqrt{1-2 x}}{7 \sqrt{5 x+3}}+\frac{103 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - 2*x]/((2 + 3*x)^2*(3 + 5*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 10.1469, size = 82, normalized size = 0.98 \[ - \frac{10 \left (- 2 x + 1\right )^{\frac{3}{2}}}{11 \left (3 x + 2\right ) \sqrt{5 x + 3}} - \frac{103 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{11 \left (3 x + 2\right )} + \frac{103 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(1/2)/(2+3*x)**2/(3+5*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0940225, size = 70, normalized size = 0.83 \[ \frac{103 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2 \sqrt{7}}-\frac{\sqrt{1-2 x} (45 x+29)}{(3 x+2) \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^2*(3 + 5*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.021, size = 154, normalized size = 1.8 \[ -{\frac{1}{28+42\,x} \left ( 1545\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1957\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+618\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +630\,x\sqrt{-10\,{x}^{2}-x+3}+406\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.50582, size = 124, normalized size = 1.48 \[ -\frac{103}{14} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{30 \, x}{\sqrt{-10 \, x^{2} - x + 3}} - \frac{47}{3 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{7}{3 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/((5*x + 3)^(3/2)*(3*x + 2)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222111, size = 107, normalized size = 1.27 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (45 \, x + 29\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 103 \,{\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{14 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/((5*x + 3)^(3/2)*(3*x + 2)^2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(1/2)/(2+3*x)**2/(3+5*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.253921, size = 347, normalized size = 4.13 \[ -\frac{1}{140} \, \sqrt{5}{\left (103 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + 70 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{9240 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*x + 1)/((5*x + 3)^(3/2)*(3*x + 2)^2),x, algorithm="giac")
[Out]