Optimal. Leaf size=180 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^4}-\frac{167155 \sqrt{1-2 x} \sqrt{5 x+3}}{1382976 (3 x+2)}-\frac{38365 \sqrt{1-2 x} \sqrt{5 x+3}}{98784 (3 x+2)^2}-\frac{3653 \sqrt{1-2 x} \sqrt{5 x+3}}{3528 (3 x+2)^3}+\frac{131 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^4}-\frac{168795 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.370498, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^4}-\frac{167155 \sqrt{1-2 x} \sqrt{5 x+3}}{1382976 (3 x+2)}-\frac{38365 \sqrt{1-2 x} \sqrt{5 x+3}}{98784 (3 x+2)^2}-\frac{3653 \sqrt{1-2 x} \sqrt{5 x+3}}{3528 (3 x+2)^3}+\frac{131 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^4}-\frac{168795 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^5),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 37.6869, size = 165, normalized size = 0.92 \[ - \frac{167155 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1382976 \left (3 x + 2\right )} - \frac{38365 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{98784 \left (3 x + 2\right )^{2}} - \frac{3653 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3528 \left (3 x + 2\right )^{3}} + \frac{131 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{588 \left (3 x + 2\right )^{4}} - \frac{168795 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1075648} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**5,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.127359, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{5 x+3} \left (1002930 x^4+2578615 x^3+2184144 x^2+687828 x+53136\right )}{\sqrt{1-2 x} (3 x+2)^4}-168795 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2151296} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^5),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.022, size = 305, normalized size = 1.7 \[{\frac{1}{2151296\, \left ( 2+3\,x \right ) ^{4} \left ( -1+2\,x \right ) } \left ( 27344790\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+59247045\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+36459720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-14041020\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4051080\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-36100610\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-10802880\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-30578016\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-2700720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -9629592\,x\sqrt{-10\,{x}^{2}-x+3}-743904\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^5,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.51882, size = 400, normalized size = 2.22 \[ \frac{168795}{2151296} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{835775 \, x}{2074464 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{843155}{4148928 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{756 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{787}{31752 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{20681}{127008 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{69575}{197568 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^5*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.241732, size = 167, normalized size = 0.93 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1002930 \, x^{4} + 2578615 \, x^{3} + 2184144 \, x^{2} + 687828 \, x + 53136\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 168795 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2151296 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^5*(-2*x + 1)^(3/2)),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((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**5,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.624503, size = 547, normalized size = 3.04 \[ \frac{33759}{4302592} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{968 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{84035 \,{\left (2 \, x - 1\right )}} - \frac{121 \,{\left (10277 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 10598840 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 3966648000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 122821440000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{537824 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)/((3*x + 2)^5*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]