3.2355 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3}{50} \sqrt{5 x+3} (1-2 x)^{5/2}-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{5 x+3}}+\frac{119 \sqrt{5 x+3} (1-2 x)^{3/2}}{2200}+\frac{357 \sqrt{5 x+3} \sqrt{1-2 x}}{2000}+\frac{3927 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2))/(275*Sqrt[3 + 5*x]) + (357*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200
0 + (119*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/2200 - (3*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])
/50 + (3927*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2000*Sqrt[10])

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Rubi [A]  time = 0.142834, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{50} \sqrt{5 x+3} (1-2 x)^{5/2}-\frac{2 (1-2 x)^{5/2}}{275 \sqrt{5 x+3}}+\frac{119 \sqrt{5 x+3} (1-2 x)^{3/2}}{2200}+\frac{357 \sqrt{5 x+3} \sqrt{1-2 x}}{2000}+\frac{3927 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2))/(275*Sqrt[3 + 5*x]) + (357*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200
0 + (119*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/2200 - (3*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])
/50 + (3927*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2000*Sqrt[10])

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Rubi in Sympy [A]  time = 11.5709, size = 105, normalized size = 0.91 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{50} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{275 \sqrt{5 x + 3}} + \frac{119 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2200} + \frac{357 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2000} + \frac{3927 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{20000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/50 - 2*(-2*x + 1)**(5/2)/(275*sqrt(5*x + 3))
+ 119*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/2200 + 357*sqrt(-2*x + 1)*sqrt(5*x + 3)/20
00 + 3927*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/20000

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Mathematica [A]  time = 0.164305, size = 65, normalized size = 0.56 \[ \frac{\frac{10 \sqrt{1-2 x} \left (-2400 x^3-180 x^2+2575 x+1021\right )}{\sqrt{5 x+3}}-3927 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{20000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(1021 + 2575*x - 180*x^2 - 2400*x^3))/Sqrt[3 + 5*x] - 3927*Sq
rt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/20000

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Maple [A]  time = 0.018, size = 116, normalized size = 1. \[{\frac{1}{40000} \left ( -48000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+19635\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+11781\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +51500\,x\sqrt{-10\,{x}^{2}-x+3}+20420\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40000*(-48000*x^3*(-10*x^2-x+3)^(1/2)+19635*10^(1/2)*arcsin(20/11*x+1/11)*x-36
00*x^2*(-10*x^2-x+3)^(1/2)+11781*10^(1/2)*arcsin(20/11*x+1/11)+51500*x*(-10*x^2-
x+3)^(1/2)+20420*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^
(1/2)

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Maxima [A]  time = 1.50396, size = 208, normalized size = 1.79 \[ -\frac{11979}{200000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{957}{25000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{125} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{99}{500} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{2277}{10000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{99}{1250} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{125 \,{\left (5 \, x + 3\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11979/200000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 957/25000*sqrt(5)*sqrt
(2)*arcsin(20/11*x + 1/11) + 3/125*(-10*x^2 - x + 3)^(3/2) + 99/500*sqrt(10*x^2
+ 23*x + 51/5)*x + 2277/10000*sqrt(10*x^2 + 23*x + 51/5) + 99/1250*sqrt(-10*x^2
- x + 3) + 1/125*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 3/125*(-10*x^2 -
x + 3)^(3/2)/(5*x + 3) - 33/625*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.227614, size = 107, normalized size = 0.92 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (2400 \, x^{3} + 180 \, x^{2} - 2575 \, x - 1021\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3927 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{40000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40000*sqrt(10)*(2*sqrt(10)*(2400*x^3 + 180*x^2 - 2575*x - 1021)*sqrt(5*x + 3)
*sqrt(-2*x + 1) - 3927*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*
sqrt(-2*x + 1))))/(5*x + 3)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279232, size = 167, normalized size = 1.44 \[ -\frac{1}{50000} \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 69 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 199 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{3927}{20000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{6250 \, \sqrt{5 \, x + 3}} + \frac{22 \, \sqrt{10} \sqrt{5 \, x + 3}}{3125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/50000*(12*(8*sqrt(5)*(5*x + 3) - 69*sqrt(5))*(5*x + 3) - 199*sqrt(5))*sqrt(5*
x + 3)*sqrt(-10*x + 5) + 3927/20000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))
- 11/6250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/3125
*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))