Optimal. Leaf size=93 \[ \frac{2 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3}}{49 (3 x+2)}+\frac{33 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.129385, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3}}{49 (3 x+2)}+\frac{33 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 10.6797, size = 76, normalized size = 0.82 \[ \frac{33 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} + \frac{33 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1}} - \frac{\left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )} \]
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)**2,x)
[Out]
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Mathematica [A] time = 0.0803327, size = 75, normalized size = 0.81 \[ \frac{33 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{98 \sqrt{7}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (64 x+45)}{49 \left (6 x^2+x-2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^2),x]
[Out]
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Maple [B] time = 0.019, size = 161, normalized size = 1.7 \[ -{\frac{1}{ \left ( 1372+2058\,x \right ) \left ( -1+2\,x \right ) } \left ( 198\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+33\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-66\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +896\,x\sqrt{-10\,{x}^{2}-x+3}+630\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
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)^2,x)
[Out]
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Maxima [A] time = 1.50456, size = 124, normalized size = 1.33 \[ -\frac{33}{686} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{320 \, x}{147 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{611}{441 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1}{63 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^2*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224877, size = 101, normalized size = 1.09 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (64 \, x + 45\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 33 \,{\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{686 \,{\left (6 \, x^{2} + x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^2*(-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((3+5*x)**(3/2)/(1-2*x)**(3/2)/(2+3*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.311052, size = 296, normalized size = 3.18 \[ -\frac{33}{6860} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{22 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{245 \,{\left (2 \, x - 1\right )}} + \frac{22 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{49 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^2*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]