Optimal. Leaf size=180 \[ \frac{16985 \sqrt{1-2 x} \sqrt{5 x+3}}{153664 (3 x+2)}-\frac{745 \sqrt{1-2 x} \sqrt{5 x+3}}{10976 (3 x+2)^2}-\frac{89 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)^3}-\frac{131 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^4}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^4}-\frac{279015 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.375722, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{16985 \sqrt{1-2 x} \sqrt{5 x+3}}{153664 (3 x+2)}-\frac{745 \sqrt{1-2 x} \sqrt{5 x+3}}{10976 (3 x+2)^2}-\frac{89 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)^3}-\frac{131 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^4}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^4}-\frac{279015 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^5),x]
[Out]
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Rubi in Sympy [A] time = 36.6654, size = 165, normalized size = 0.92 \[ \frac{16985 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{153664 \left (3 x + 2\right )} - \frac{745 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{10976 \left (3 x + 2\right )^{2}} - \frac{89 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{392 \left (3 x + 2\right )^{3}} - \frac{131 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{196 \left (3 x + 2\right )^{4}} - \frac{279015 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1075648} + \frac{11 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(3/2)/(1-2*x)**(3/2)/(2+3*x)**5,x)
[Out]
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Mathematica [A] time = 0.133796, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{5 x+3} \left (-917190 x^4-1188045 x^3+60048 x^2+538276 x+163152\right )}{\sqrt{1-2 x} (3 x+2)^4}-279015 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2151296} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^5),x]
[Out]
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Maple [B] time = 0.023, size = 305, normalized size = 1.7 \[{\frac{1}{2151296\, \left ( 2+3\,x \right ) ^{4} \left ( -1+2\,x \right ) } \left ( 45200430\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+97934265\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+60267240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+12840660\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-6696360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+16632630\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-17856960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-840672\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-4464240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -7535864\,x\sqrt{-10\,{x}^{2}-x+3}-2284128\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^5,x)
[Out]
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Maxima [A] time = 1.51599, size = 400, normalized size = 2.22 \[ \frac{279015}{2151296} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{84925 \, x}{230496 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{131015}{460992 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1}{252 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{169}{3528 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{649}{4704 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{2475}{21952 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^5*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239198, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (917190 \, x^{4} + 1188045 \, x^{3} - 60048 \, x^{2} - 538276 \, x - 163152\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 279015 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2151296 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^5*(-2*x + 1)^(3/2)),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((3+5*x)**(3/2)/(1-2*x)**(3/2)/(2+3*x)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.607672, size = 547, normalized size = 3.04 \[ \frac{55803}{4302592} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{176 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{84035 \,{\left (2 \, x - 1\right )}} - \frac{11 \,{\left (178579 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 183436680 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 17824632000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 2829942080000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{537824 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^5*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]