Optimal. Leaf size=135 \[ -\frac{2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{27}{100} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2-\frac{63 (35-8 x) (1-2 x)^{3/2} \sqrt{5 x+3}}{16000}+\frac{35511 \sqrt{1-2 x} \sqrt{5 x+3}}{160000}+\frac{390621 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{160000 \sqrt{10}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.20442, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{27}{100} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2-\frac{63 (35-8 x) (1-2 x)^{3/2} \sqrt{5 x+3}}{16000}+\frac{35511 \sqrt{1-2 x} \sqrt{5 x+3}}{160000}+\frac{390621 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{160000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(3 + 5*x)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 21.3479, size = 124, normalized size = 0.92 \[ - \frac{\left (- 1890 x + \frac{33075}{4}\right ) \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{60000} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{5 \sqrt{5 x + 3}} + \frac{27 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{100} + \frac{35511 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{160000} + \frac{390621 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1600000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(2+3*x)**3/(3+5*x)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.195285, size = 70, normalized size = 0.52 \[ \frac{\frac{10 \sqrt{1-2 x} \left (-432000 x^4-439200 x^3+287460 x^2+317125 x+46783\right )}{\sqrt{5 x+3}}-390621 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1600000} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(3 + 5*x)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.017, size = 133, normalized size = 1. \[{\frac{1}{3200000} \left ( -8640000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-8784000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1953105\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+5749200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1171863\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +6342500\,x\sqrt{-10\,{x}^{2}-x+3}+935660\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(2+3*x)^3/(3+5*x)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.51027, size = 248, normalized size = 1.84 \[ \frac{27}{500} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{35937}{1000000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{1378113}{16000000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{171}{10000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{297}{2500} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{9801}{40000} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{6831}{50000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{28809}{800000} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1250 \,{\left (5 \, x + 3\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{3125 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-2*x + 1)^(3/2)/(5*x + 3)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.220607, size = 113, normalized size = 0.84 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (432000 \, x^{4} + 439200 \, x^{3} - 287460 \, x^{2} - 317125 \, x - 46783\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 390621 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3200000 \,{\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-2*x + 1)^(3/2)/(5*x + 3)^(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((1-2*x)**(3/2)*(2+3*x)**3/(3+5*x)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.303854, size = 185, normalized size = 1.37 \[ -\frac{1}{4000000} \,{\left (36 \,{\left (8 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 83 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 805 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 128915 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{390621}{1600000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{31250 \, \sqrt{5 \, x + 3}} + \frac{22 \, \sqrt{10} \sqrt{5 \, x + 3}}{15625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^3*(-2*x + 1)^(3/2)/(5*x + 3)^(3/2),x, algorithm="giac")
[Out]