Optimal. Leaf size=151 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^{5/2}}{8 (3 x+2)^3}+\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}-\frac{121 \sqrt{1-2 x} (5 x+3)^{3/2}}{224 (3 x+2)^2}-\frac{3993 \sqrt{1-2 x} \sqrt{5 x+3}}{3136 (3 x+2)}-\frac{43923 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.216138, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^{5/2}}{8 (3 x+2)^3}+\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}-\frac{121 \sqrt{1-2 x} (5 x+3)^{3/2}}{224 (3 x+2)^2}-\frac{3993 \sqrt{1-2 x} \sqrt{5 x+3}}{3136 (3 x+2)}-\frac{43923 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 17.1334, size = 136, normalized size = 0.9 \[ - \frac{363 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1568 \left (3 x + 2\right )^{2}} - \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{56 \left (3 x + 2\right )^{3}} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{4 \left (3 x + 2\right )^{4}} + \frac{3993 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3136 \left (3 x + 2\right )} - \frac{43923 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{21952} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**5,x)
[Out]
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Mathematica [A] time = 0.101496, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{43904} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^5,x]
[Out]
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Maple [B] time = 0.018, size = 250, normalized size = 1.7 \[{\frac{1}{43904\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3557763\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+9487368\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+9487368\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1402226\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4216608\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2985360\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+702768\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +2043160\,x\sqrt{-10\,{x}^{2}-x+3}+453600\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^5,x)
[Out]
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Maxima [A] time = 1.50695, size = 251, normalized size = 1.66 \[ \frac{8245}{16464} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{111 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{392 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{4947 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{67155}{10976} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{43923}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{59169}{21952} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{19573 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{65856 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222612, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (100159 \, x^{3} + 213240 \, x^{2} + 145940 \, x + 32400\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 43923 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{43904 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^5,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)*(3+5*x)**(3/2)/(2+3*x)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.41175, size = 512, normalized size = 3.39 \[ \frac{43923}{439040} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 3080 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 862400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 65856000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1568 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^5,x, algorithm="giac")
[Out]