Optimal. Leaf size=137 \[ -\frac{57595 \sqrt{5 x+3}}{249018 \sqrt{1-2 x}}+\frac{51 \sqrt{5 x+3}}{28 (1-2 x)^{3/2} (3 x+2)}-\frac{1735 \sqrt{5 x+3}}{3234 (1-2 x)^{3/2}}+\frac{3 \sqrt{5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}-\frac{5805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.318894, antiderivative size = 137, 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 \[ -\frac{57595 \sqrt{5 x+3}}{249018 \sqrt{1-2 x}}+\frac{51 \sqrt{5 x+3}}{28 (1-2 x)^{3/2} (3 x+2)}-\frac{1735 \sqrt{5 x+3}}{3234 (1-2 x)^{3/2}}+\frac{3 \sqrt{5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}-\frac{5805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^3*Sqrt[3 + 5*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 28.8236, size = 126, normalized size = 0.92 \[ - \frac{5805 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} - \frac{57595 \sqrt{5 x + 3}}{249018 \sqrt{- 2 x + 1}} - \frac{1735 \sqrt{5 x + 3}}{3234 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{51 \sqrt{5 x + 3}}{28 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} + \frac{3 \sqrt{5 x + 3}}{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(5/2)/(2+3*x)**3/(3+5*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.116479, size = 85, normalized size = 0.62 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (2073420 x^3-676860 x^2-945629 x+391476\right )}{498036 \left (6 x^2+x-2\right )^2}-\frac{5805 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^3*Sqrt[3 + 5*x]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.025, size = 257, normalized size = 1.9 \[{\frac{1}{6972504\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) ^{2}} \left ( 75859740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+25286580\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-48465945\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+29027880\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-8428860\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-9476040\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+8428860\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -13238806\,x\sqrt{-10\,{x}^{2}-x+3}+5480664\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(5/2)/(2+3*x)^3/(3+5*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*(-2*x + 1)^(5/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.226095, size = 147, normalized size = 1.07 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (2073420 \, x^{3} - 676860 \, x^{2} - 945629 \, x + 391476\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 2107215 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{6972504 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*(-2*x + 1)^(5/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-2*x)**(5/2)/(2+3*x)**3/(3+5*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.389163, size = 400, normalized size = 2.92 \[ \frac{1161}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{32 \,{\left (367 \, \sqrt{5}{\left (5 \, x + 3\right )} - 2211 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{21789075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{297 \,{\left (197 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 36680 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4802 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^3*(-2*x + 1)^(5/2)),x, algorithm="giac")
[Out]