Optimal. Leaf size=86 \[ -\frac{58 \sqrt{5 x+3}}{539 \sqrt{1-2 x}}+\frac{3 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)}-\frac{123 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.173424, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{58 \sqrt{5 x+3}}{539 \sqrt{1-2 x}}+\frac{3 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)}-\frac{123 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.8876, size = 78, normalized size = 0.91 \[ - \frac{123 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} - \frac{58 \sqrt{5 x + 3}}{539 \sqrt{- 2 x + 1}} + \frac{3 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**2/(3+5*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0773751, size = 75, normalized size = 0.87 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} (174 x-115)}{539 \left (6 x^2+x-2\right )}-\frac{123 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{98 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.022, size = 161, normalized size = 1.9 \[{\frac{1}{ \left ( 15092+22638\,x \right ) \left ( -1+2\,x \right ) } \left ( 8118\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1353\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-2706\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +2436\,x\sqrt{-10\,{x}^{2}-x+3}-1610\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.221099, size = 101, normalized size = 1.17 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (174 \, x - 115\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1353 \,{\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7546 \,{\left (6 \, x^{2} + x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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)**2/(3+5*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.298698, size = 296, normalized size = 3.44 \[ \frac{123}{6860} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{8 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{2695 \,{\left (2 \, x - 1\right )}} + \frac{198 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{49 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^2*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]