Optimal. Leaf size=125 \[ -\frac{68 \sqrt{3 x+2} \sqrt{5 x+3}}{231 \sqrt{1-2 x}}+\frac{\sqrt{3 x+2} \sqrt{5 x+3}}{3 (1-2 x)^{3/2}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{34 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.258712, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{68 \sqrt{3 x+2} \sqrt{5 x+3}}{231 \sqrt{1-2 x}}+\frac{\sqrt{3 x+2} \sqrt{5 x+3}}{3 (1-2 x)^{3/2}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{34 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(1 - 2*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 23.841, size = 110, normalized size = 0.88 \[ - \frac{34 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{231} - \frac{\sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{231} - \frac{68 \sqrt{3 x + 2} \sqrt{5 x + 3}}{231 \sqrt{- 2 x + 1}} + \frac{\sqrt{3 x + 2} \sqrt{5 x + 3}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(1/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.241114, size = 115, normalized size = 0.92 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (136 x+9)+35 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-68 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{462 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(1 - 2*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.024, size = 276, normalized size = 2.2 \[{\frac{1}{462\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 70\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-136\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +68\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4080\,{x}^{3}+5438\,{x}^{2}+1974\,x+108 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(3*x + 2)/(-2*x + 1)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(3*x + 2)/(-2*x + 1)^(5/2),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((2+3*x)**(1/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*sqrt(3*x + 2)/(-2*x + 1)^(5/2),x, algorithm="giac")
[Out]