Optimal. Leaf size=108 \[ -\frac{4390 \sqrt{5 x+3}}{124509 \sqrt{1-2 x}}+\frac{3 \sqrt{5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190 \sqrt{5 x+3}}{1617 (1-2 x)^{3/2}}-\frac{405 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]
[Out]
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Rubi [A] time = 0.24952, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{4390 \sqrt{5 x+3}}{124509 \sqrt{1-2 x}}+\frac{3 \sqrt{5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190 \sqrt{5 x+3}}{1617 (1-2 x)^{3/2}}-\frac{405 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 21.9494, size = 99, normalized size = 0.92 \[ - \frac{405 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2401} - \frac{4390 \sqrt{5 x + 3}}{124509 \sqrt{- 2 x + 1}} - \frac{190 \sqrt{5 x + 3}}{1617 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{3 \sqrt{5 x + 3}}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(5/2)/(2+3*x)**2/(3+5*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.105248, size = 77, normalized size = 0.71 \[ \frac{\sqrt{5 x+3} \left (26340 x^2-39500 x+15321\right )}{124509 (1-2 x)^{3/2} (3 x+2)}-\frac{405 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{686 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x]),x]
[Out]
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Maple [B] time = 0.023, size = 209, normalized size = 1.9 \[{\frac{1}{ \left ( 3486252+5229378\,x \right ) \left ( -1+2\,x \right ) ^{2}} \left ( 1764180\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-588060\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-735075\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+368760\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+294030\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -553000\,x\sqrt{-10\,{x}^{2}-x+3}+214494\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(5/2)/(2+3*x)^2/(3+5*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225232, size = 127, normalized size = 1.18 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (26340 \, x^{2} - 39500 \, x + 15321\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 147015 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1743126 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(5/2)),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)**(5/2)/(2+3*x)**2/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.323293, size = 313, normalized size = 2.9 \[ \frac{81}{9604} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{594 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{343 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} - \frac{8 \,{\left (536 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3333 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{3112725 \,{\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(5/2)),x, algorithm="giac")
[Out]