3.2288 \(\int \frac{\sqrt{1-2 x}}{(2+3 x) \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=64 \[ -\frac{2}{3} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.103909, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{2}{3} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.66709, size = 60, normalized size = 0.94 \[ - \frac{2 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{15} - \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/15 - 2*sqrt(7)*atan(sqrt(7)*sqrt(-2*
x + 1)/(7*sqrt(5*x + 3)))/3

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Mathematica [A]  time = 0.109432, size = 75, normalized size = 1.17 \[ \frac{1}{15} \left (-5 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - Sqrt[10]*Arc
Tan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/15

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Maple [A]  time = 0.017, size = 69, normalized size = 1.1 \[{\frac{1}{15}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 5\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ) \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))-10^(1/2)*arcsin(20/11*x+1/11))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50698, size = 54, normalized size = 0.84 \[ -\frac{1}{15} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1}{3} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(10)*arcsin(20/11*x + 1/11) + 1/3*sqrt(7)*arcsin(37/11*x/abs(3*x + 2)
+ 20/11/abs(3*x + 2))

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Fricas [A]  time = 0.22881, size = 96, normalized size = 1.5 \[ \frac{1}{15} \, \sqrt{5}{\left (\sqrt{7} \sqrt{5} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - \sqrt{2} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*sqrt(5)*(sqrt(7)*sqrt(5)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqr
t(-2*x + 1))) - sqrt(2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.24191, size = 196, normalized size = 3.06 \[ \frac{1}{30} \, \sqrt{5}{\left (\sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 2 \, \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*sqrt(5)*(sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 2*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)))))