3.2431 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=164 \[ -\frac{376 (1-2 x)^{3/2} (3 x+2)^3}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac{741}{250} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2+\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (4392 x+3185)}{40000}+\frac{69713 \sqrt{1-2 x} \sqrt{5 x+3}}{400000}+\frac{766843 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (376*(1 - 2*x)^(3/2)*(2
+ 3*x)^3)/(75*Sqrt[3 + 5*x]) + (69713*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400000 + (741
*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/250 + (21*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x]*(3185 + 4392*x))/40000 + (766843*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400000*S
qrt[10])

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Rubi [A]  time = 0.281026, antiderivative size = 164, 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 \[ -\frac{376 (1-2 x)^{3/2} (3 x+2)^3}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac{741}{250} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2+\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (4392 x+3185)}{40000}+\frac{69713 \sqrt{1-2 x} \sqrt{5 x+3}}{400000}+\frac{766843 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (376*(1 - 2*x)^(3/2)*(2
+ 3*x)^3)/(75*Sqrt[3 + 5*x]) + (69713*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400000 + (741
*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/250 + (21*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x]*(3185 + 4392*x))/40000 + (766843*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400000*S
qrt[10])

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Rubi in Sympy [A]  time = 23.4168, size = 144, normalized size = 0.88 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{376 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2}}{825 \sqrt{5 x + 3}} + \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3} \left (252045 x + \frac{706959}{4}\right )}{247500} + \frac{69713 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1320000} + \frac{69713 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{400000} + \frac{766843 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**3/(15*(5*x + 3)**(3/2)) - 376*(-2*x + 1)**(5/2)*
(3*x + 2)**2/(825*sqrt(5*x + 3)) + (-2*x + 1)**(5/2)*sqrt(5*x + 3)*(252045*x + 7
06959/4)/247500 + 69713*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/1320000 + 69713*sqrt(-2*
x + 1)*sqrt(5*x + 3)/400000 + 766843*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/40
00000

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Mathematica [A]  time = 0.196162, size = 75, normalized size = 0.46 \[ \frac{\frac{10 \sqrt{1-2 x} \left (6480000 x^5+972000 x^4-7724700 x^3+3074745 x^2+7876210 x+2322001\right )}{(5 x+3)^{3/2}}-2300529 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12000000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(2322001 + 7876210*x + 3074745*x^2 - 7724700*x^3 + 972000*x^4
+ 6480000*x^5))/(3 + 5*x)^(3/2) - 2300529*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2
*x]])/12000000

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Maple [A]  time = 0.021, size = 164, normalized size = 1. \[{\frac{1}{24000000} \left ( 129600000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-154494000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+61494900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24000000*(129600000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)+5
7513225*10^(1/2)*arcsin(20/11*x+1/11)*x^2-154494000*x^3*(-10*x^2-x+3)^(1/2)+6901
5870*10^(1/2)*arcsin(20/11*x+1/11)*x+61494900*x^2*(-10*x^2-x+3)^(1/2)[phone]*1
0^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x
+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.53545, size = 439, normalized size = 2.68 \[ -\frac{395307}{8000000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{23221}{500000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{99}{5000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{625 \,{\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1250 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2500 \,{\left (5 \, x + 3\right )}} + \frac{3267}{20000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{75141}{400000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{3267}{25000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{11 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3750 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{99 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2500 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{99 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2500 \,{\left (5 \, x + 3\right )}} - \frac{121 \, \sqrt{-10 \, x^{2} - x + 3}}{18750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{9493 \, \sqrt{-10 \, x^{2} - x + 3}}{37500 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-395307/8000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 23221/500000*sqrt(5)
*sqrt(2)*arcsin(20/11*x + 1/11) + 99/5000*(-10*x^2 - x + 3)^(3/2) + 1/625*(-10*x
^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*x^2 + 540*x + 81) + 9/1250*(-10*x^2
- x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 9/625*(-10*x^2 - x + 3)^(5/2)
/(25*x^2 + 30*x + 9) + 27/2500*(-10*x^2 - x + 3)^(5/2)/(5*x + 3) + 3267/20000*sq
rt(10*x^2 + 23*x + 51/5)*x + 75141/400000*sqrt(10*x^2 + 23*x + 51/5) + 3267/2500
0*sqrt(-10*x^2 - x + 3) - 11/3750*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 1
35*x + 27) + 99/2500*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 99/2500*(-10*
x^2 - x + 3)^(3/2)/(5*x + 3) - 121/18750*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x +
9) - 9493/37500*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.224731, size = 134, normalized size = 0.82 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (6480000 \, x^{5} + 972000 \, x^{4} - 7724700 \, x^{3} + 3074745 \, x^{2} + 7876210 \, x + 2322001\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 2300529 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{24000000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24000000*sqrt(10)*(2*sqrt(10)*(6480000*x^5 + 972000*x^4 - 7724700*x^3 + 307474
5*x^2 + 7876210*x + 2322001)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 2300529*(25*x^2 + 30
*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(25*x^2
+ 30*x + 9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.343743, size = 273, normalized size = 1.66 \[ \frac{1}{10000000} \,{\left (36 \,{\left (24 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 57 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4915 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 338795 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{3750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{766843}{4000000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{2079 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{312500 \, \sqrt{5 \, x + 3}} + \frac{11 \,{\left (\frac{567 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{234375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10000000*(36*(24*(4*sqrt(5)*(5*x + 3) - 57*sqrt(5))*(5*x + 3) + 4915*sqrt(5))*
(5*x + 3) + 338795*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/3750000*sqrt(10)*
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 766843/4000000*sqrt(10)
*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 2079/312500*sqrt(10)*(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) + 11/234375*(567*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5
) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))^3