3.2327 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=149 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{36 (3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}-\frac{661 \sqrt{1-2 x} \sqrt{5 x+3}}{1512 (3 x+2)}+\frac{20}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{19573 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4536 \sqrt{7}} \]

[Out]

(-661*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1512*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)
^(3/2))/(9*(2 + 3*x)^3) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(36*(2 + 3*x)^2) +
(20*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (19573*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(4536*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.305559, antiderivative size = 149, 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 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{36 (3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}-\frac{661 \sqrt{1-2 x} \sqrt{5 x+3}}{1512 (3 x+2)}+\frac{20}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{19573 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4536 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(-661*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1512*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)
^(3/2))/(9*(2 + 3*x)^3) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(36*(2 + 3*x)^2) +
(20*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (19573*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(4536*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.6123, size = 134, normalized size = 0.9 \[ - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{252 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{3}} + \frac{1781 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1512 \left (3 x + 2\right )} + \frac{20 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{81} - \frac{19573 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{31752} \]

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)**4,x)

[Out]

-37*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(252*(3*x + 2)**2) - (-2*x + 1)**(3/2)*(5*x
+ 3)**(3/2)/(9*(3*x + 2)**3) + 1781*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1512*(3*x + 2)
) + 20*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/81 - 19573*sqrt(7)*atan(sqrt(7)*
sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/31752

_______________________________________________________________________________________

Mathematica [A]  time = 0.184632, size = 112, normalized size = 0.75 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (19041 x^2+21762 x+6176\right )}{(3 x+2)^3}-19573 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+7840 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{63504} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(6176 + 21762*x + 19041*x^2))/(2 + 3*x)^3 - 195
73*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 7840*Sqrt[10]
*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/63504

_______________________________________________________________________________________

Maple [B]  time = 0.017, size = 253, normalized size = 1.7 \[{\frac{1}{63504\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 528471\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+211680\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+1056942\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+423360\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+704628\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+282240\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+799722\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+156584\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +62720\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +914004\,x\sqrt{-10\,{x}^{2}-x+3}+259392\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^4,x)

[Out]

1/63504*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(528471*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^3+211680*10^(1/2)*arcsin(20/11*x+1/11)*x^3+1056942*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+423360*10^(1/2)*arcsi
n(20/11*x+1/11)*x^2+704628*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))*x+282240*10^(1/2)*arcsin(20/11*x+1/11)*x+799722*x^2*(-10*x^2-x+3)^(1/2)+15
6584*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+62720*10^(1/2)*a
rcsin(20/11*x+1/11)+914004*x*(-10*x^2-x+3)^(1/2)+259392*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)/(2+3*x)^3

_______________________________________________________________________________________

Maxima [A]  time = 1.49036, size = 217, normalized size = 1.46 \[ \frac{185}{882} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{7 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{4045}{1764} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{10}{81} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{19573}{63504} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{8573}{10584} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{83 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1176 \,{\left (3 \, x + 2\right )}} \]

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)^4,x, algorithm="maxima")

[Out]

185/882*(-10*x^2 - x + 3)^(3/2) + 1/7*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 +
36*x + 8) + 37/196*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 4045/1764*sqrt(
-10*x^2 - x + 3)*x + 10/81*sqrt(10)*arcsin(20/11*x + 1/11) + 19573/63504*sqrt(7)
*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 8573/10584*sqrt(-10*x^2 - x
+ 3) + 83/1176*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.232917, size = 192, normalized size = 1.29 \[ \frac{\sqrt{7}{\left (1120 \, \sqrt{10} \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (19041 \, x^{2} + 21762 \, x + 6176\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 19573 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{63504 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

1/63504*sqrt(7)*(1120*sqrt(10)*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*
sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)*(19041*x^2 + 217
62*x + 6176)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 19573*(27*x^3 + 54*x^2 + 36*x + 8)*a
rctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(27*x^3 + 54*x^2
+ 36*x + 8)

_______________________________________________________________________________________

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)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**4,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.395137, size = 520, normalized size = 3.49 \[ \frac{19573}{635040} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{10}{81} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (661 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 499520 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 139630400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{756 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

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)^4,x, algorithm="giac")

[Out]

19573/635040*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) + 10/81*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)))) - 11/
756*(661*sqrt(10)*((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)))^5 + 499520*sqrt(10)*((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)))^3 - 139630400*sqrt(10)*((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)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5
) - sqrt(22)))^2 + 280)^3