Optimal. Leaf size=115 \[ -\frac{3895 \sqrt{5 x+3}}{7546 \sqrt{1-2 x}}+\frac{345 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)}+\frac{3 \sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}-\frac{12465 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.246803, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3895 \sqrt{5 x+3}}{7546 \sqrt{1-2 x}}+\frac{345 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)}+\frac{3 \sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}-\frac{12465 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^3*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 21.7745, size = 105, normalized size = 0.91 \[ - \frac{12465 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} - \frac{3895 \sqrt{5 x + 3}}{7546 \sqrt{- 2 x + 1}} + \frac{345 \sqrt{5 x + 3}}{196 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{3 \sqrt{5 x + 3}}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**3/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0996097, size = 77, normalized size = 0.67 \[ \frac{-\frac{14 \sqrt{5 x+3} \left (70110 x^2+13785 x-25204\right )}{\sqrt{1-2 x} (3 x+2)^2}-137115 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{211288} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^3*Sqrt[3 + 5*x]),x]
[Out]
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Maple [B] time = 0.023, size = 209, normalized size = 1.8 \[{\frac{1}{211288\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) } \left ( 2468070\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+2056725\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-548460\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+981540\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-548460\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +192990\,x\sqrt{-10\,{x}^{2}-x+3}-352856\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229318, size = 127, normalized size = 1.1 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (70110 \, x^{2} + 13785 \, x - 25204\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 137115 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{211288 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*(-2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
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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/(1-2*x)**(3/2)/(2+3*x)**3/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.341157, size = 382, normalized size = 3.32 \[ \frac{2493}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{16 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{18865 \,{\left (2 \, x - 1\right )}} + \frac{297 \,{\left (9 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 1640 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]