Optimal. Leaf size=180 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{245529161 \sqrt{5 x+3} \sqrt{1-2 x}}{169344 (3 x+2)}+\frac{2347559 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^2}+\frac{67187 \sqrt{5 x+3} \sqrt{1-2 x}}{2160 (3 x+2)^3}+\frac{2023 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{104040277 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.369315, antiderivative size = 180, 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 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{245529161 \sqrt{5 x+3} \sqrt{1-2 x}}{169344 (3 x+2)}+\frac{2347559 \sqrt{5 x+3} \sqrt{1-2 x}}{12096 (3 x+2)^2}+\frac{67187 \sqrt{5 x+3} \sqrt{1-2 x}}{2160 (3 x+2)^3}+\frac{2023 \sqrt{5 x+3} \sqrt{1-2 x}}{360 (3 x+2)^4}-\frac{104040277 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 35.2568, size = 165, normalized size = 0.92 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{5}} + \frac{245529161 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{169344 \left (3 x + 2\right )} + \frac{2347559 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12096 \left (3 x + 2\right )^{2}} + \frac{67187 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2160 \left (3 x + 2\right )^{3}} + \frac{2023 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{360 \left (3 x + 2\right )^{4}} - \frac{104040277 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{43904} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)/(2+3*x)**6/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.113237, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (11048812245 x^4+29956486710 x^3+30475811404 x^2+13788819736 x+2341358496\right )}{(3 x+2)^5}-1560604155 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{1317120} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]
[Out]
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Maple [B] time = 0.023, size = 298, normalized size = 1.7 \[{\frac{1}{1317120\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 379226809665\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+1264089365550\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+1685452487400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+154683371430\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+1123634991600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+419390813940\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+374544997200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+426661359656\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+49939332960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +193043476304\,x\sqrt{-10\,{x}^{2}-x+3}+32779018944\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50292, size = 248, normalized size = 1.38 \[ \frac{104040277}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{45 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{637 \, \sqrt{-10 \, x^{2} - x + 3}}{120 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{67187 \, \sqrt{-10 \, x^{2} - x + 3}}{2160 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{2347559 \, \sqrt{-10 \, x^{2} - x + 3}}{12096 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{245529161 \, \sqrt{-10 \, x^{2} - x + 3}}{169344 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230638, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (11048812245 \, x^{4} + 29956486710 \, x^{3} + 30475811404 \, x^{2} + 13788819736 \, x + 2341358496\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1560604155 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1317120 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(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)/(2+3*x)**6/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.494343, size = 594, normalized size = 3.3 \[ \frac{104040277}{878080} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1331 \,{\left (706299 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 493892560 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 156884295680 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 24022907776000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 1441374466560000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9408 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^6),x, algorithm="giac")
[Out]