3.2325 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=135 \[ -\frac{1}{3} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{3 (3 x+2)}+\frac{107}{36} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1649 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{108 \sqrt{10}}+\frac{37}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/36 - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/3 - ((1 -
2*x)^(3/2)*(3 + 5*x)^(3/2))/(3*(2 + 3*x)) + (1649*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(108*Sqrt[10]) + (37*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
/27

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Rubi [A]  time = 0.30064, 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 \[ -\frac{1}{3} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{3 (3 x+2)}+\frac{107}{36} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1649 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{108 \sqrt{10}}+\frac{37}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/36 - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/3 - ((1 -
2*x)^(3/2)*(3 + 5*x)^(3/2))/(3*(2 + 3*x)) + (1649*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(108*Sqrt[10]) + (37*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
/27

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Rubi in Sympy [A]  time = 30.1971, size = 119, normalized size = 0.88 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right )} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{3} + \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36} + \frac{1649 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1080} + \frac{37 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/(3*(3*x + 2)) - sqrt(-2*x + 1)*(5*x + 3)**(3
/2)/3 + 107*sqrt(-2*x + 1)*sqrt(5*x + 3)/36 + 1649*sqrt(10)*asin(sqrt(22)*sqrt(5
*x + 3)/11)/1080 + 37*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/27

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Mathematica [A]  time = 0.170447, size = 112, normalized size = 0.83 \[ \frac{\frac{60 \sqrt{1-2 x} \sqrt{5 x+3} \left (-60 x^2+105 x+106\right )}{3 x+2}+1480 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+1649 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{2160} \]

Antiderivative was successfully verified.

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

[Out]

((60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(106 + 105*x - 60*x^2))/(2 + 3*x) + 1480*Sqrt[7
]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 1649*Sqrt[10]*ArcTan[(
1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/2160

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Maple [A]  time = 0.017, size = 163, normalized size = 1.2 \[ -{\frac{1}{4320+6480\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-4947\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+3600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -3298\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -6300\,x\sqrt{-10\,{x}^{2}-x+3}-6360\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2160*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(4440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x-4947*10^(1/2)*arcsin(20/11*x+1/11)*x+3600*x^2*(-10*x^2-x+
3)^(1/2)+2960*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-3298*10
^(1/2)*arcsin(20/11*x+1/11)-6300*x*(-10*x^2-x+3)^(1/2)-6360*(-10*x^2-x+3)^(1/2))
/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.51663, size = 122, normalized size = 0.9 \[ -\frac{5}{3} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{1649}{2160} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{37}{54} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{71}{36} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/3*sqrt(-10*x^2 - x + 3)*x + 1649/2160*sqrt(10)*arcsin(20/11*x + 1/11) - 37/54
*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 71/36*sqrt(-10*x^2
- x + 3) - 1/3*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.234974, size = 151, normalized size = 1.12 \[ -\frac{\sqrt{10}{\left (148 \, \sqrt{10} \sqrt{7}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{10}{\left (60 \, x^{2} - 105 \, x - 106\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1649 \,{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2160 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2160*sqrt(10)*(148*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)*(60*x^2 - 105*x - 106)*sqrt(5*x +
3)*sqrt(-2*x + 1) - 1649*(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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{\left (3 x + 2\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.35477, size = 394, normalized size = 2.92 \[ -\frac{37}{540} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{540} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 181 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{1649}{2160} \, \sqrt{10}{\left (\pi - 2 \, \arctan \left (\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{154 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{27 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37/540*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
)))) - 1/540*(12*sqrt(5)*(5*x + 3) - 181*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)
+ 1649/2160*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)))) + 154/27*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)