Optimal. Leaf size=144 \[ -\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{5 x+3}}+\frac{44475 \sqrt{1-2 x}}{2744 (3 x+2) \sqrt{5 x+3}}+\frac{255 \sqrt{1-2 x}}{196 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{4616025 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
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Rubi [A] time = 0.322053, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{5 x+3}}+\frac{44475 \sqrt{1-2 x}}{2744 (3 x+2) \sqrt{5 x+3}}+\frac{255 \sqrt{1-2 x}}{196 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{4616025 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]
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Rubi in Sympy [A] time = 28.078, size = 131, normalized size = 0.91 \[ - \frac{7396875 \sqrt{- 2 x + 1}}{30184 \sqrt{5 x + 3}} + \frac{44475 \sqrt{- 2 x + 1}}{2744 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{255 \sqrt{- 2 x + 1}}{196 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{\sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} + \frac{4616025 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2+3*x)**4/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)
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Mathematica [A] time = 0.106817, size = 82, normalized size = 0.57 \[ \frac{50776275 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{14 \sqrt{1-2 x} \left (199715625 x^3+395028225 x^2+260298990 x+57135248\right )}{(3 x+2)^3 \sqrt{5 x+3}}}{422576} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.025, size = 250, normalized size = 1.7 \[ -{\frac{1}{422576\, \left ( 2+3\,x \right ) ^{3}} \left ( 6854797125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+17822472525\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+17365486050\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+2796018750\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7514888700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+5530395150\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1218630600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3644185860\,x\sqrt{-10\,{x}^{2}-x+3}+799893472\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2+3*x)^4/(3+5*x)^(3/2)/(1-2*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{4} \sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1)),x, algorithm="maxima")
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Fricas [A] time = 0.237866, size = 147, normalized size = 1.02 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (199715625 \, x^{3} + 395028225 \, x^{2} + 260298990 \, x + 57135248\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 50776275 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{422576 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1)),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2+3*x)**4/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.344673, size = 509, normalized size = 3.53 \[ -\frac{923205}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{125}{22} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{7425 \,{\left (487 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 217280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 25693248 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1372 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1)),x, algorithm="giac")
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