3.2446 \(\int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx\)

Optimal. Leaf size=62 \[ \frac{1}{3} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3 \sqrt{7}} \]

[Out]

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

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Rubi [A]  time = 0.107283, antiderivative size = 62, 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 \[ \frac{1}{3} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.86168, size = 56, normalized size = 0.9 \[ \frac{\sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{3} + \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/3 + 2*sqrt(7)*atan(sqrt(7)*sqrt(-2*x +
1)/(7*sqrt(5*x + 3)))/21

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Mathematica [A]  time = 0.113928, size = 75, normalized size = 1.21 \[ \frac{1}{42} \left (2 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+7 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 7*Sqrt[10]*Ar
cTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/42

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Maple [A]  time = 0.016, size = 69, normalized size = 1.1 \[ -{\frac{1}{42}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -7\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/42*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))-7*10^(1/2)*arcsin(20/11*x+1/11))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50029, size = 54, normalized size = 0.87 \[ \frac{1}{6} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{21} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(10)*arcsin(20/11*x + 1/11) - 1/21*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) +
20/11/abs(3*x + 2))

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Fricas [A]  time = 0.23036, size = 88, normalized size = 1.42 \[ \frac{1}{42} \, \sqrt{7}{\left (\sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 2 \, \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*sqrt(7)*(sqrt(10)*sqrt(7)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))) - 2*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1)
)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.267202, size = 190, normalized size = 3.06 \[ -\frac{1}{210} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{6} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/210*sqrt(70)*sqrt(10)*(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)
))) + 1/6*sqrt(10)*(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))))