Optimal. Leaf size=93 \[ \frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}+\frac{33 \sqrt{5 x+3} \sqrt{1-2 x}}{4 (3 x+2)}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.124558, 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 \[ \frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{2 (3 x+2)^2}+\frac{33 \sqrt{5 x+3} \sqrt{1-2 x}}{4 (3 x+2)}-\frac{363 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(3/2)/((2 + 3*x)^3*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 10.6408, size = 82, normalized size = 0.88 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2 \left (3 x + 2\right )^{2}} + \frac{33 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4 \left (3 x + 2\right )} - \frac{363 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{28} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)/(2+3*x)**3/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0714672, size = 72, normalized size = 0.77 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} (95 x+68)}{4 (3 x+2)^2}-\frac{363 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^3*Sqrt[3 + 5*x]),x]
[Out]
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Maple [B] time = 0.019, size = 154, normalized size = 1.7 \[{\frac{1}{56\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3267\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+4356\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1452\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1330\,x\sqrt{-10\,{x}^{2}-x+3}+952\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50012, size = 103, normalized size = 1.11 \[ \frac{363}{56} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{6 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{95 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/(sqrt(5*x + 3)*(3*x + 2)^3),x, algorithm="maxima")
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Fricas [A] time = 0.227855, size = 107, normalized size = 1.15 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (95 \, x + 68\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 363 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{56 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/(sqrt(5*x + 3)*(3*x + 2)^3),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)**(3/2)/(2+3*x)**3/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.287708, size = 346, normalized size = 3.72 \[ \frac{363}{560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{605 \,{\left (\sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 168 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{2 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(3/2)/(sqrt(5*x + 3)*(3*x + 2)^3),x, algorithm="giac")
[Out]