3.2324 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{2+3 x} \, dx\)

Optimal. Leaf size=128 \[ \frac{1}{9} (1-2 x)^{3/2} (5 x+3)^{3/2}+\frac{37}{180} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{1781 \sqrt{1-2 x} \sqrt{5 x+3}}{2160}+\frac{19573 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6480 \sqrt{10}}-\frac{14}{81} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-1781*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2160 + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/18
0 + ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/9 + (19573*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(6480*Sqrt[10]) - (14*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
81

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Rubi [A]  time = 0.29737, antiderivative size = 128, 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 \[ \frac{1}{9} (1-2 x)^{3/2} (5 x+3)^{3/2}+\frac{37}{180} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{1781 \sqrt{1-2 x} \sqrt{5 x+3}}{2160}+\frac{19573 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6480 \sqrt{10}}-\frac{14}{81} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1781*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2160 + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/18
0 + ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/9 + (19573*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(6480*Sqrt[10]) - (14*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
81

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Rubi in Sympy [A]  time = 30.0641, size = 117, normalized size = 0.91 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{9} - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{72} + \frac{661 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2160} + \frac{19573 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{64800} - \frac{14 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{81} \]

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),x)

[Out]

(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/9 - 37*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/72 + 6
61*sqrt(-2*x + 1)*sqrt(5*x + 3)/2160 + 19573*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3
)/11)/64800 - 14*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/81

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Mathematica [A]  time = 0.183909, size = 105, normalized size = 0.82 \[ \frac{-60 \sqrt{1-2 x} \sqrt{5 x+3} \left (2400 x^2-1980 x-271\right )-11200 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+19573 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{129600} \]

Antiderivative was successfully verified.

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

[Out]

(-60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-271 - 1980*x + 2400*x^2) - 11200*Sqrt[7]*ArcT
an[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 19573*Sqrt[10]*ArcTan[(1 + 2
0*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/129600

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Maple [A]  time = 0.017, size = 115, normalized size = 0.9 \[{\frac{1}{129600}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -144000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+11200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +19573\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +118800\,x\sqrt{-10\,{x}^{2}-x+3}+16260\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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),x)

[Out]

1/129600*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-144000*x^2*(-10*x^2-x+3)^(1/2)+11200*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+19573*10^(1/2)*arcsin(20/
11*x+1/11)+118800*x*(-10*x^2-x+3)^(1/2)+16260*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)
^(1/2)

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Maxima [A]  time = 1.51288, size = 112, normalized size = 0.88 \[ \frac{1}{9} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{37}{36} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{19573}{129600} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{7}{81} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{449}{2160} \, \sqrt{-10 \, x^{2} - x + 3} \]

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),x, algorithm="maxima")

[Out]

1/9*(-10*x^2 - x + 3)^(3/2) + 37/36*sqrt(-10*x^2 - x + 3)*x + 19573/129600*sqrt(
10)*arcsin(20/11*x + 1/11) + 7/81*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/ab
s(3*x + 2)) - 449/2160*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.231854, size = 128, normalized size = 1. \[ -\frac{1}{129600} \, \sqrt{10}{\left (6 \, \sqrt{10}{\left (2400 \, x^{2} - 1980 \, x - 271\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1120 \, \sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 19573 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/129600*sqrt(10)*(6*sqrt(10)*(2400*x^2 - 1980*x - 271)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) - 1120*sqrt(10)*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqr
t(-2*x + 1))) - 19573*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x +
1))))

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

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.286202, size = 251, normalized size = 1.96 \[ \frac{7}{810} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{10800} \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 81 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1781 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{19573}{129600} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/810*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
)) - 1/10800*(12*(8*sqrt(5)*(5*x + 3) - 81*sqrt(5))*(5*x + 3) + 1781*sqrt(5))*sq
rt(5*x + 3)*sqrt(-10*x + 5) + 19573/129600*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))))