Optimal. Leaf size=138 \[ -\frac{1}{10} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{33}{160} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{121 \sqrt{5 x+3} (1-2 x)^{5/2}}{1600}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{3/2}}{6400}+\frac{43923 \sqrt{5 x+3} \sqrt{1-2 x}}{64000}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000 \sqrt{10}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.130503, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{10} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{33}{160} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{121 \sqrt{5 x+3} (1-2 x)^{5/2}}{1600}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{3/2}}{6400}+\frac{43923 \sqrt{5 x+3} \sqrt{1-2 x}}{64000}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.031, size = 124, normalized size = 0.9 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{25} + \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{200} + \frac{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{2000} - \frac{1331 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{16000} - \frac{43923 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{64000} + \frac{483153 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{640000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.106187, size = 70, normalized size = 0.51 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (256000 x^4-124800 x^3-177440 x^2+116420 x+29673\right )-483153 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{640000} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 120, normalized size = 0.9 \[{\frac{1}{25} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}}+{\frac{11}{200} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}}+{\frac{121}{2000} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{16000} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{43923}{64000}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{483153\,\sqrt{10}}{1280000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.49862, size = 113, normalized size = 0.82 \[ \frac{1}{25} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{11}{40} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{11}{800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3993}{3200} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{483153}{1280000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3993}{64000} \, \sqrt{-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.216363, size = 97, normalized size = 0.7 \[ \frac{1}{1280000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (256000 \, x^{4} - 124800 \, x^{3} - 177440 \, x^{2} + 116420 \, x + 29673\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 483153 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 146.246, size = 311, normalized size = 2.25 \[ \begin{cases} \frac{40 i \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{\sqrt{10 x - 5}} - \frac{319 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{10 x - 5}} + \frac{8833 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{40 \sqrt{10 x - 5}} - \frac{171699 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{1600 \sqrt{10 x - 5}} - \frac{14641 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{6400 \sqrt{10 x - 5}} + \frac{483153 i \sqrt{x + \frac{3}{5}}}{64000 \sqrt{10 x - 5}} - \frac{483153 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{640000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{483153 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{640000} - \frac{40 \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{\sqrt{- 10 x + 5}} + \frac{319 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{- 10 x + 5}} - \frac{8833 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{40 \sqrt{- 10 x + 5}} + \frac{171699 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{1600 \sqrt{- 10 x + 5}} + \frac{14641 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{6400 \sqrt{- 10 x + 5}} - \frac{483153 \sqrt{x + \frac{3}{5}}}{64000 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(5/2)*(3+5*x)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.258564, size = 317, normalized size = 2.3 \[ \frac{1}{9600000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{240000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{7}{24000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2),x, algorithm="giac")
[Out]