3.2425 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt{5 x+3}}-\frac{1111 \sqrt{1-2 x}}{15 \sqrt{5 x+3}}-\frac{8}{45} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{665}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-1111*Sqrt[1 - 2*x])/(15*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*Sqrt
[3 + 5*x]) - (8*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/45 + (665*Sqrt[7]*Ar
cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi [A]  time = 0.243313, antiderivative size = 115, 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 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt{5 x+3}}-\frac{1111 \sqrt{1-2 x}}{15 \sqrt{5 x+3}}-\frac{8}{45} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{665}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1111*Sqrt[1 - 2*x])/(15*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*Sqrt
[3 + 5*x]) - (8*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/45 + (665*Sqrt[7]*Ar
cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi in Sympy [A]  time = 23.872, size = 104, normalized size = 0.9 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right ) \sqrt{5 x + 3}} - \frac{1111 \sqrt{- 2 x + 1}}{15 \sqrt{5 x + 3}} - \frac{8 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{225} + \frac{665 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(3*(3*x + 2)*sqrt(5*x + 3)) - 1111*sqrt(-2*x + 1)/(15*sqrt(5
*x + 3)) - 8*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/225 + 665*sqrt(7)*atan(sqr
t(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/9

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Mathematica [A]  time = 0.227844, size = 107, normalized size = 0.93 \[ \frac{1}{450} \left (-\frac{30 \sqrt{1-2 x} (3403 x+2187)}{(3 x+2) \sqrt{5 x+3}}+16625 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-8 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-30*Sqrt[1 - 2*x]*(2187 + 3403*x))/((2 + 3*x)*Sqrt[3 + 5*x]) + 16625*Sqrt[7]*A
rcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 8*Sqrt[10]*ArcTan[(1 + 20
*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/450

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Maple [B]  time = 0.02, size = 191, normalized size = 1.7 \[ -{\frac{1}{900+1350\,x} \left ( 120\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+249375\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+152\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+315875\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+48\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +99750\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +102090\,x\sqrt{-10\,{x}^{2}-x+3}+65610\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/450*(120*10^(1/2)*arcsin(20/11*x+1/11)*x^2+249375*7^(1/2)*arctan(1/14*(37*x+2
0)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+152*10^(1/2)*arcsin(20/11*x+1/11)*x+315875*7
^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+48*10^(1/2)*arcsin(2
0/11*x+1/11)+99750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+10
2090*x*(-10*x^2-x+3)^(1/2)+65610*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)/(-10
*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.50716, size = 139, normalized size = 1.21 \[ -\frac{4}{225} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{665}{18} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{6806 \, x}{45 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{10699}{135 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{27 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

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)^2),x, algorithm="maxima")

[Out]

-4/225*sqrt(10)*arcsin(20/11*x + 1/11) - 665/18*sqrt(7)*arcsin(37/11*x/abs(3*x +
2) + 20/11/abs(3*x + 2)) + 6806/45*x/sqrt(-10*x^2 - x + 3) - 10699/135/sqrt(-10
*x^2 - x + 3) + 343/27/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.229609, size = 173, normalized size = 1.5 \[ -\frac{\sqrt{5}{\left (3325 \, \sqrt{7} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{5}{\left (3403 \, x + 2187\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 8 \, \sqrt{2}{\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{450 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \]

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)^2),x, algorithm="fricas")

[Out]

-1/450*sqrt(5)*(3325*sqrt(7)*sqrt(5)*(15*x^2 + 19*x + 6)*arctan(1/14*sqrt(7)*(37
*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(5)*(3403*x + 2187)*sqrt(5*x +
3)*sqrt(-2*x + 1) + 8*sqrt(2)*(15*x^2 + 19*x + 6)*arctan(1/20*sqrt(5)*sqrt(2)*(2
0*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(15*x^2 + 19*x + 6)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.325859, size = 431, normalized size = 3.75 \[ -\frac{133}{36} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{4}{225} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{121}{50} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{1078 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{3 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-133/36*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
)))) - 4/225*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)))) - 121/50*s
qrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 1078/3*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)*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)