Optimal. Leaf size=93 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac{11 \sqrt{1-2 x} \sqrt{5 x+3}}{28 (3 x+2)}-\frac{121 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.127195, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}-\frac{11 \sqrt{1-2 x} \sqrt{5 x+3}}{28 (3 x+2)}-\frac{121 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 10.3472, size = 82, normalized size = 0.88 \[ - \frac{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{28 \left (3 x + 2\right )} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{2 \left (3 x + 2\right )^{2}} - \frac{121 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{196} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(1/2)*(3+5*x)**(1/2)/(2+3*x)**3,x)
[Out]
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Mathematica [A] time = 0.0638836, size = 72, normalized size = 0.77 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} (37 x+20)}{28 (3 x+2)^2}-\frac{121 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^3,x]
[Out]
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Maple [B] time = 0.016, size = 154, normalized size = 1.7 \[{\frac{1}{392\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1089\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1452\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+484\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +518\,x\sqrt{-10\,{x}^{2}-x+3}+280\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3,x)
[Out]
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Maxima [A] time = 1.52067, size = 122, normalized size = 1.31 \[ \frac{121}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{5}{21} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{37 \, \sqrt{-10 \, x^{2} - x + 3}}{84 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232829, size = 107, normalized size = 1.15 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 121 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{392 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(1/2)*(3+5*x)**(1/2)/(2+3*x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.301972, size = 343, normalized size = 3.69 \[ \frac{121}{3920} \, \sqrt{5}{\left (\sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{280 \, \sqrt{2}{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - \frac{280 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{1120 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2)^3,x, algorithm="giac")
[Out]