Optimal. Leaf size=123 \[ \frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{5 x+3}}-\frac{19130 \sqrt{1-2 x}}{195657 (5 x+3)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{162 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.320598, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{5 x+3}}-\frac{19130 \sqrt{1-2 x}}{195657 (5 x+3)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{162 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 29.6887, size = 114, normalized size = 0.93 \[ - \frac{162 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} - \frac{400636 \sqrt{5 x + 3}}{2152227 \sqrt{- 2 x + 1}} + \frac{29710}{27951 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{370}{2541 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(5/2)/(2+3*x)/(3+5*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.152722, size = 75, normalized size = 0.61 \[ \frac{20031800 x^3-8854440 x^2-6468522 x+2981164}{2152227 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{81 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.023, size = 250, normalized size = 2. \[{\frac{1}{15065589\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 355776300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+71155260\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-209908017\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+140222600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-21346578\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-61981080\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+32019867\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -45279654\,x\sqrt{-10\,{x}^{2}-x+3}+20868148\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(5/2)/(2+3*x)/(3+5*x)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.50277, size = 117, normalized size = 0.95 \[ \frac{81}{343} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2003180 \, x}{2152227 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1085762}{2152227 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{740 \, x}{2541 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{326}{2541 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(5/2)*(3*x + 2)*(-2*x + 1)^(5/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.234832, size = 147, normalized size = 1.2 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (10015900 \, x^{3} - 4427220 \, x^{2} - 3234261 \, x + 1490582\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3557763 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{15065589 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(5/2)*(3*x + 2)*(-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+5*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.279605, size = 315, normalized size = 2.56 \[ -\frac{25}{702768} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{81}{3430} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{675}{29282} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{32 \,{\left (379 \, \sqrt{5}{\left (5 \, x + 3\right )} - 2277 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{53805675 \,{\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(5/2)*(3*x + 2)*(-2*x + 1)^(5/2)),x, algorithm="giac")
[Out]