3.2376 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=135 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac{1}{3} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{43}{30} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{2119 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{90 \sqrt{10}}-\frac{35}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-43*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/30 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3 - ((1 -
2*x)^(5/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)) - (2119*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(90*Sqrt[10]) - (35*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi [A]  time = 0.307301, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac{1}{3} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{43}{30} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{2119 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{90 \sqrt{10}}-\frac{35}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-43*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/30 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3 - ((1 -
2*x)^(5/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)) - (2119*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(90*Sqrt[10]) - (35*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi in Sympy [A]  time = 31.1963, size = 121, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3} - \frac{43 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{30} - \frac{2119 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{900} - \frac{35 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(3*(3*x + 2)) - (-2*x + 1)**(3/2)*sqrt(5*x + 3)
/3 - 43*sqrt(-2*x + 1)*sqrt(5*x + 3)/30 - 2119*sqrt(10)*asin(sqrt(22)*sqrt(5*x +
3)/11)/900 - 35*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/9

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Mathematica [A]  time = 0.158876, size = 112, normalized size = 0.83 \[ \frac{\frac{60 \sqrt{1-2 x} \sqrt{5 x+3} \left (20 x^2-79 x-116\right )}{3 x+2}-3500 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-2119 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{1800} \]

Antiderivative was successfully verified.

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

[Out]

((60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-116 - 79*x + 20*x^2))/(2 + 3*x) - 3500*Sqrt[7
]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 2119*Sqrt[10]*ArcTan[(
1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/1800

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Maple [A]  time = 0.017, size = 163, normalized size = 1.2 \[{\frac{1}{3600+5400\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 10500\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-6357\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+1200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+7000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -4238\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -4740\,x\sqrt{-10\,{x}^{2}-x+3}-6960\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1800*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(10500*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x-6357*10^(1/2)*arcsin(20/11*x+1/11)*x+1200*x^2*(-10*x^2-x+
3)^(1/2)+7000*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-4238*10
^(1/2)*arcsin(20/11*x+1/11)-4740*x*(-10*x^2-x+3)^(1/2)-6960*(-10*x^2-x+3)^(1/2))
/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.51054, size = 122, normalized size = 0.9 \[ \frac{2}{9} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{2119}{1800} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{35}{18} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{277}{270} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{27 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*sqrt(-10*x^2 - x + 3)*x - 2119/1800*sqrt(10)*arcsin(20/11*x + 1/11) + 35/18*
sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 277/270*sqrt(-10*x^2
- x + 3) - 49/27*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.228161, size = 151, normalized size = 1.12 \[ \frac{\sqrt{10}{\left (350 \, \sqrt{10} \sqrt{7}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{10}{\left (20 \, x^{2} - 79 \, x - 116\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2119 \,{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1800 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1800*sqrt(10)*(350*sqrt(10)*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(10)*(20*x^2 - 79*x - 116)*sqrt(5*x + 3)
*sqrt(-2*x + 1) - 2119*(3*x + 2)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*
sqrt(-2*x + 1))))/(3*x + 2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.366672, size = 394, normalized size = 2.92 \[ \frac{7}{36} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{1350} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 313 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{2119}{1800} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1078 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{27 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/36*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))
) + 1/1350*(12*sqrt(5)*(5*x + 3) - 313*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) -
2119/1800*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)))) - 1078/27*sqr
t(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))/(((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)))^2 + 280)