Optimal. Leaf size=173 \[ -\frac{7986105 \sqrt{5 x+3}}{845152 \sqrt{1-2 x}}+\frac{698295 \sqrt{5 x+3}}{21952 \sqrt{1-2 x} (3 x+2)}+\frac{6621 \sqrt{5 x+3}}{1568 \sqrt{1-2 x} (3 x+2)^2}+\frac{263 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)^3}+\frac{3 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^4}-\frac{24922335 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]
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Rubi [A] time = 0.412221, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{7986105 \sqrt{5 x+3}}{845152 \sqrt{1-2 x}}+\frac{698295 \sqrt{5 x+3}}{21952 \sqrt{1-2 x} (3 x+2)}+\frac{6621 \sqrt{5 x+3}}{1568 \sqrt{1-2 x} (3 x+2)^2}+\frac{263 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)^3}+\frac{3 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^4}-\frac{24922335 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^5*Sqrt[3 + 5*x]),x]
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Rubi in Sympy [A] time = 36.9032, size = 160, normalized size = 0.92 \[ - \frac{24922335 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1075648} - \frac{7986105 \sqrt{5 x + 3}}{845152 \sqrt{- 2 x + 1}} + \frac{698295 \sqrt{5 x + 3}}{21952 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{6621 \sqrt{5 x + 3}}{1568 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{263 \sqrt{5 x + 3}}{392 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{3 \sqrt{5 x + 3}}{28 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**5/(3+5*x)**(1/2),x)
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Mathematica [A] time = 0.143821, size = 87, normalized size = 0.5 \[ \frac{-\frac{14 \sqrt{5 x+3} \left (1293749010 x^4+1998242055 x^3+482249808 x^2-491393004 x-205593328\right )}{\sqrt{1-2 x} (3 x+2)^4}-274145685 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{23664256} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^5*Sqrt[3 + 5*x]),x]
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Maple [B] time = 0.027, size = 305, normalized size = 1.8 \[{\frac{1}{23664256\, \left ( 2+3\,x \right ) ^{4} \left ( -1+2\,x \right ) } \left ( 44411600970\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+96225135435\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+59215467960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+18112486140\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-6579496440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+27975388770\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-17545323840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+6751497312\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-4386330960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -6879502056\,x\sqrt{-10\,{x}^{2}-x+3}-2878306592\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(3/2)/(2+3*x)^5/(3+5*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{5}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^5*(-2*x + 1)^(3/2)),x, algorithm="maxima")
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Fricas [A] time = 0.237871, size = 167, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1293749010 \, x^{4} + 1998242055 \, x^{3} + 482249808 \, x^{2} - 491393004 \, x - 205593328\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 274145685 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{23664256 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^5*(-2*x + 1)^(3/2)),x, algorithm="fricas")
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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/(1-2*x)**(3/2)/(2+3*x)**5/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.569646, size = 547, normalized size = 3.16 \[ \frac{4984467}{4302592} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{64 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{924385 \,{\left (2 \, x - 1\right )}} + \frac{99 \,{\left (4411181 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 2388710520 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 506212728000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 38676680000000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{537824 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^5*(-2*x + 1)^(3/2)),x, algorithm="giac")
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