3.2251 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=91 \[ -\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)}-\frac{2}{9} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{37 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9 \sqrt{7}} \]

[Out]

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

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Rubi [A]  time = 0.1726, antiderivative size = 91, 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 \[ -\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)}-\frac{2}{9} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{37 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.0242, size = 82, normalized size = 0.9 \[ - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )} - \frac{2 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{9} - \frac{37 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{63} \]

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)**2,x)

[Out]

-sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*(3*x + 2)) - 2*sqrt(10)*asin(sqrt(22)*sqrt(5*x
+ 3)/11)/9 - 37*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/63

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Mathematica [A]  time = 0.169653, size = 102, normalized size = 1.12 \[ -\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{9 x+6}-\frac{37 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{18 \sqrt{7}}-\frac{1}{9} \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6 + 9*x)) - (37*ArcTan[(-20 - 37*x)/(2*Sqrt[7 -
14*x]*Sqrt[3 + 5*x])])/(18*Sqrt[7]) - (Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2
*x]*Sqrt[30 + 50*x])])/9

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Maple [A]  time = 0.016, size = 131, normalized size = 1.4 \[ -{\frac{1}{252+378\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 42\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-111\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+28\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -74\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +42\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^2,x)

[Out]

-1/126*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(42*10^(1/2)*arcsin(20/11*x+1/11)*x-111*7^(1/
2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+28*10^(1/2)*arcsin(20/11
*x+1/11)-74*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+42*(-10*x
^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.5002, size = 82, normalized size = 0.9 \[ -\frac{1}{9} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{37}{126} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{3 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(10)*arcsin(20/11*x + 1/11) + 37/126*sqrt(7)*arcsin(37/11*x/abs(3*x + 2
) + 20/11/abs(3*x + 2)) - 1/3*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.234118, size = 138, normalized size = 1.52 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{10} \sqrt{7}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 37 \,{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{126 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/126*sqrt(7)*(2*sqrt(10)*sqrt(7)*(3*x + 2)*arctan(1/20*sqrt(10)*(20*x + 1)/(sq
rt(5*x + 3)*sqrt(-2*x + 1))) - 37*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqr
t(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

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

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)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.296462, size = 358, normalized size = 3.93 \[ \frac{1}{1260} \, \sqrt{5}{\left (37 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 140 \, \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{9240 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1260*sqrt(5)*(37*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)))) - 140*sqrt(2)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 9
240*sqrt(2)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3
)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280))