Optimal. Leaf size=160 \[ \frac{2}{21} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{131}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{131}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{9013 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1890} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.336018, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{21} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1}{7} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{131}{189} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{131}{945} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{9013 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1890} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/Sqrt[2 + 3*x],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 31.0322, size = 141, normalized size = 0.88 \[ \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{5}{2}}}{21} - \frac{\sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{7} - \frac{131 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{189} - \frac{9013 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5670} - \frac{1441 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{33075} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.354525, size = 97, normalized size = 0.61 \[ \frac{75 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (90 x^2+81 x-10\right )-4690 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+9013 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2835 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/Sqrt[2 + 3*x],x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.017, size = 174, normalized size = 1.1 \[{\frac{1}{170100\,{x}^{3}+130410\,{x}^{2}-39690\,x-34020}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 4690\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9013\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +405000\,{x}^{5}+675000\,{x}^{4}+139950\,{x}^{3}-200550\,{x}^{2}-62400\,x+9000 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/sqrt(3*x + 2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/sqrt(3*x + 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)**(1/2)/(2+3*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{\sqrt{3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/sqrt(3*x + 2),x, algorithm="giac")
[Out]