3.2424 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{22 (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{128}{75} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{338}{225} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{98}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-22*(1 - 2*x)^(3/2))/(5*Sqrt[3 + 5*x]) - (128*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/75 +
(338*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/225 + (98*Sqrt[7]*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

_______________________________________________________________________________________

Rubi [A]  time = 0.24108, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{22 (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{128}{75} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{338}{225} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{98}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-22*(1 - 2*x)^(3/2))/(5*Sqrt[3 + 5*x]) - (128*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/75 +
(338*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/225 + (98*Sqrt[7]*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.3839, size = 99, normalized size = 0.92 \[ - \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{5 \sqrt{5 x + 3}} - \frac{128 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{75} + \frac{338 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1125} + \frac{98 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-22*(-2*x + 1)**(3/2)/(5*sqrt(5*x + 3)) - 128*sqrt(-2*x + 1)*sqrt(5*x + 3)/75 +
338*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/1125 + 98*sqrt(7)*atan(sqrt(7)*sqrt
(-2*x + 1)/(7*sqrt(5*x + 3)))/9

_______________________________________________________________________________________

Mathematica [A]  time = 0.265712, size = 104, normalized size = 0.96 \[ \frac{2 \sqrt{1-2 x} (10 x-357)}{75 \sqrt{5 x+3}}+\frac{49}{9} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+\frac{169}{225} \sqrt{\frac{2}{5}} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*(-357 + 10*x))/(75*Sqrt[3 + 5*x]) + (49*Sqrt[7]*ArcTan[(-20 - 3
7*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/9 + (169*Sqrt[2/5]*ArcTan[(1 + 20*x)/(2*
Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/225

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 139, normalized size = 1.3 \[ -{\frac{1}{1125} \left ( 30625\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-845\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+18375\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -507\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -300\,x\sqrt{-10\,{x}^{2}-x+3}+10710\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1125*(30625*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-845*
10^(1/2)*arcsin(20/11*x+1/11)*x+18375*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))-507*10^(1/2)*arcsin(20/11*x+1/11)-300*x*(-10*x^2-x+3)^(1/2)+107
10*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.53065, size = 116, normalized size = 1.07 \[ -\frac{8 \, x^{2}}{15 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{169}{1125} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{49}{9} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{1448 \, x}{75 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{238}{25 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/15*x^2/sqrt(-10*x^2 - x + 3) + 169/1125*sqrt(10)*arcsin(20/11*x + 1/11) - 49/
9*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1448/75*x/sqrt(-10
*x^2 - x + 3) - 238/25/sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.23261, size = 153, normalized size = 1.42 \[ -\frac{\sqrt{5}{\left (1225 \, \sqrt{7} \sqrt{5}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{5}{\left (10 \, x - 357\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 169 \, \sqrt{2}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1125*sqrt(5)*(1225*sqrt(7)*sqrt(5)*(5*x + 3)*arctan(1/14*sqrt(7)*(37*x + 20)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 6*sqrt(5)*(10*x - 357)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) - 169*sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x +
3)*sqrt(-2*x + 1))))/(5*x + 3)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.294258, size = 296, normalized size = 2.74 \[ -\frac{49}{90} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{169}{1125} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{4}{375} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{121}{250} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-49/90*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)
))) + 169/1125*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)))) + 4/375*
sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 121/250*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(2
2)))