Optimal. Leaf size=151 \[ \frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{4 (3 x+2)^4}+\frac{55 (5 x+3)^{3/2} (1-2 x)^{3/2}}{24 (3 x+2)^3}+\frac{605 (5 x+3)^{3/2} \sqrt{1-2 x}}{32 (3 x+2)^2}-\frac{6655 \sqrt{5 x+3} \sqrt{1-2 x}}{448 (3 x+2)}-\frac{73205 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.215887, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{4 (3 x+2)^4}+\frac{55 (5 x+3)^{3/2} (1-2 x)^{3/2}}{24 (3 x+2)^3}+\frac{605 (5 x+3)^{3/2} \sqrt{1-2 x}}{32 (3 x+2)^2}-\frac{6655 \sqrt{5 x+3} \sqrt{1-2 x}}{448 (3 x+2)}-\frac{73205 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^5,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.0809, size = 136, normalized size = 0.9 \[ - \frac{55 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{168 \left (3 x + 2\right )^{3}} + \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{4 \left (3 x + 2\right )^{4}} + \frac{605 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{672 \left (3 x + 2\right )^{2}} + \frac{6655 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{448 \left (3 x + 2\right )} - \frac{73205 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3136} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**5,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0876901, size = 82, normalized size = 0.54 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (518715 x^3+1059032 x^2+723428 x+164688\right )}{1344 (3 x+2)^4}-\frac{73205 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{896 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^5,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.017, size = 250, normalized size = 1.7 \[{\frac{1}{18816\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 17788815\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+47436840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+47436840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+7262010\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+21083040\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+14826448\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3513840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +10127992\,x\sqrt{-10\,{x}^{2}-x+3}+2305632\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^5,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.55112, size = 212, normalized size = 1.4 \[ \frac{73205}{6272} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3025}{336} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{12 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{17 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1815 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{224 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{22385 \, \sqrt{-10 \, x^{2} - x + 3}}{1344 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^5,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.22421, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (518715 \, x^{3} + 1059032 \, x^{2} + 723428 \, x + 164688\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 219615 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{18816 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^5,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((1-2*x)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**5,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.415004, size = 512, normalized size = 3.39 \[ \frac{14641}{12544} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{73205 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 4088 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 862400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 65856000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{672 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^5,x, algorithm="giac")
[Out]