Optimal. Leaf size=180 \[ \frac{121 \sqrt{1-2 x} (5 x+3)^{5/2}}{16 (3 x+2)^3}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{5/2}}{8 (3 x+2)^4}+\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{3/2}}{448 (3 x+2)^2}-\frac{43923 \sqrt{1-2 x} \sqrt{5 x+3}}{6272 (3 x+2)}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.268611, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{121 \sqrt{1-2 x} (5 x+3)^{5/2}}{16 (3 x+2)^3}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{5/2}}{8 (3 x+2)^4}+\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{3/2}}{448 (3 x+2)^2}-\frac{43923 \sqrt{1-2 x} \sqrt{5 x+3}}{6272 (3 x+2)}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 20.6133, size = 163, normalized size = 0.91 \[ - \frac{121 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{784 \left (3 x + 2\right )^{3}} - \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{56 \left (3 x + 2\right )^{4}} + \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{5 \left (3 x + 2\right )^{5}} + \frac{1331 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3136 \left (3 x + 2\right )^{2}} + \frac{43923 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6272 \left (3 x + 2\right )} - \frac{483153 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{43904} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**6,x)
[Out]
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Mathematica [A] time = 0.11699, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (17153435 x^4+46327530 x^3+47166452 x^2+21361768 x+3620448\right )}{(3 x+2)^5}-2415765 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{439040} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]
[Out]
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Maple [B] time = 0.017, size = 298, normalized size = 1.7 \[{\frac{1}{439040\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 587030895\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+1956769650\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+2609026200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+240148090\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+1739350800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+648585420\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+579783600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+660330328\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+77304480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +299064752\,x\sqrt{-10\,{x}^{2}-x+3}+50686272\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^6,x)
[Out]
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Maxima [A] time = 1.52821, size = 306, normalized size = 1.7 \[ \frac{90695}{32928} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{56 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1221 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{784 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{54417 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{738705}{21952} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{483153}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{650859}{43904} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{215303 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{131712 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225231, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (17153435 \, x^{4} + 46327530 \, x^{3} + 47166452 \, x^{2} + 21361768 \, x + 3620448\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 2415765 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{439040 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^6,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)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.52418, size = 594, normalized size = 3.3 \[ \frac{483153}{878080} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{161051 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 3920 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 2007040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 307328000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 18439680000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{3136 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="giac")
[Out]