3.2250 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{2+3 x} \, dx\)

Optimal. Leaf size=84 \[ \frac{1}{3} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{37 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9 \sqrt{10}}+\frac{2}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3 + (37*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(9*Sqrt[
10]) + (2*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi [A]  time = 0.169727, antiderivative size = 84, 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 \[ \frac{1}{3} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{37 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9 \sqrt{10}}+\frac{2}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x),x]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3 + (37*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(9*Sqrt[
10]) + (2*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi in Sympy [A]  time = 16.0836, size = 76, normalized size = 0.9 \[ \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3} + \frac{37 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{90} + \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(1/2)*(3+5*x)**(1/2)/(2+3*x),x)

[Out]

sqrt(-2*x + 1)*sqrt(5*x + 3)/3 + 37*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/90
+ 2*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/9

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Mathematica [A]  time = 0.137265, size = 95, normalized size = 1.13 \[ \frac{1}{180} \left (60 \sqrt{1-2 x} \sqrt{5 x+3}+20 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+37 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x),x]

[Out]

(60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] + 20*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*
x]*Sqrt[3 + 5*x])] + 37*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50
*x])])/180

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Maple [A]  time = 0.015, size = 83, normalized size = 1. \[{\frac{1}{180}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 37\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -20\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +60\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x),x)

[Out]

1/180*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(37*10^(1/2)*arcsin(20/11*x+1/11)-20*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+60*(-10*x^2-x+3)^(1/2))/(-10*x
^2-x+3)^(1/2)

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Maxima [A]  time = 1.51106, size = 73, normalized size = 0.87 \[ \frac{37}{180} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{9} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{1}{3} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2),x, algorithm="maxima")

[Out]

37/180*sqrt(10)*arcsin(20/11*x + 1/11) - 1/9*sqrt(7)*arcsin(37/11*x/abs(3*x + 2)
+ 20/11/abs(3*x + 2)) + 1/3*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.230264, size = 115, normalized size = 1.37 \[ -\frac{1}{180} \, \sqrt{10}{\left (2 \, \sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{10} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 37 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2),x, algorithm="fricas")

[Out]

-1/180*sqrt(10)*(2*sqrt(10)*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x +
3)*sqrt(-2*x + 1))) - 6*sqrt(10)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 37*arctan(1/20*s
qrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(1/2)*(3+5*x)**(1/2)/(2+3*x),x)

[Out]

Integral(sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*x + 2), x)

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GIAC/XCAS [A]  time = 0.260388, size = 219, normalized size = 2.61 \[ -\frac{1}{180} \, \sqrt{5}{\left (2 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 37 \, \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 12 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*sqrt(-2*x + 1)/(3*x + 2),x, algorithm="giac")

[Out]

-1/180*sqrt(5)*(2*sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*
((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22)))) - 37*sqrt(2)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*
x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 12*
sqrt(5*x + 3)*sqrt(-10*x + 5))