Optimal. Leaf size=93 \[ \frac{333 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3}}{14 (3 x+2)^2}-\frac{3827 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]
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Rubi [A] time = 0.160907, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{333 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3}}{14 (3 x+2)^2}-\frac{3827 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{196 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x]),x]
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Rubi in Sympy [A] time = 15.0055, size = 83, normalized size = 0.89 \[ \frac{333 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{196 \left (3 x + 2\right )} + \frac{3 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{14 \left (3 x + 2\right )^{2}} - \frac{3827 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1372} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2+3*x)**3/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0775722, size = 72, normalized size = 0.77 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} (333 x+236)}{(3 x+2)^2}-3827 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2744} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x]),x]
[Out]
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Maple [B] time = 0.02, size = 154, normalized size = 1.7 \[{\frac{1}{2744\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 34443\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+45924\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+15308\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +13986\,x\sqrt{-10\,{x}^{2}-x+3}+9912\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2+3*x)^3/(1-2*x)^(1/2)/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.51391, size = 103, normalized size = 1.11 \[ \frac{3827}{2744} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3 \, \sqrt{-10 \, x^{2} - x + 3}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{333 \, \sqrt{-10 \, x^{2} - x + 3}}{196 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*sqrt(-2*x + 1)),x, algorithm="maxima")
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Fricas [A] time = 0.237485, size = 107, normalized size = 1.15 \[ \frac{\sqrt{7}{\left (6 \, \sqrt{7}{\left (333 \, x + 236\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3827 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2744 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*sqrt(-2*x + 1)),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2+3*x)**3/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270251, size = 347, normalized size = 3.73 \[ \frac{3827}{27440} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{33 \,{\left (181 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 32200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*sqrt(-2*x + 1)),x, algorithm="giac")
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