3.2363 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{128 \sqrt{1-2 x} (3 x+2)^3}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac{378}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (1140 x+853)}{10000}+\frac{13153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (128*Sqrt[1 - 2*x]*(2 +
3*x)^3)/(25*Sqrt[3 + 5*x]) + (378*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/125 +
(21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(853 + 1140*x))/10000 + (13153*ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]])/(10000*Sqrt[10])

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Rubi [A]  time = 0.263219, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{128 \sqrt{1-2 x} (3 x+2)^3}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac{378}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (1140 x+853)}{10000}+\frac{13153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (128*Sqrt[1 - 2*x]*(2 +
3*x)^3)/(25*Sqrt[3 + 5*x]) + (378*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/125 +
(21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(853 + 1140*x))/10000 + (13153*ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]])/(10000*Sqrt[10])

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Rubi in Sympy [A]  time = 21.0067, size = 124, normalized size = 0.87 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{128 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}}{275 \sqrt{5 x + 3}} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3} \left (130410 x + \frac{363825}{4}\right )}{123750} + \frac{13153 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{110000} + \frac{13153 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{100000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)*(3*x + 2)**3/(15*(5*x + 3)**(3/2)) - 128*(-2*x + 1)**(3/2)*
(3*x + 2)**2/(275*sqrt(5*x + 3)) + (-2*x + 1)**(3/2)*sqrt(5*x + 3)*(130410*x + 3
63825/4)/123750 + 13153*sqrt(-2*x + 1)*sqrt(5*x + 3)/110000 + 13153*sqrt(10)*asi
n(sqrt(22)*sqrt(5*x + 3)/11)/100000

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Mathematica [A]  time = 0.201434, size = 70, normalized size = 0.49 \[ \frac{\frac{10 \sqrt{1-2 x} \left (-108000 x^4-83700 x^3+118395 x^2+129910 x+31171\right )}{(5 x+3)^{3/2}}-39459 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{300000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(31171 + 129910*x + 118395*x^2 - 83700*x^3 - 108000*x^4))/(3
+ 5*x)^(3/2) - 39459*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/300000

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Maple [A]  time = 0.02, size = 147, normalized size = 1. \[{\frac{1}{600000} \left ( -2160000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+986475\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-1674000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1183770\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+2367900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+355131\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2598200\,x\sqrt{-10\,{x}^{2}-x+3}+623420\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/600000*(-2160000*x^4*(-10*x^2-x+3)^(1/2)+986475*10^(1/2)*arcsin(20/11*x+1/11)*
x^2-1674000*x^3*(-10*x^2-x+3)^(1/2)+1183770*10^(1/2)*arcsin(20/11*x+1/11)*x+2367
900*x^2*(-10*x^2-x+3)^(1/2)+355131*10^(1/2)*arcsin(20/11*x+1/11)+2598200*x*(-10*
x^2-x+3)^(1/2)+623420*(-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 = 1.53462, size = 285, normalized size = 2.01 \[ -\frac{35937}{1000000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{7457}{250000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{625} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{297}{2500} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{6831}{50000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{891}{12500} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1875 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1250 \,{\left (5 \, x + 3\right )}} - \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{9375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{877 \, \sqrt{-10 \, x^{2} - x + 3}}{9375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-35937/1000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 7457/250000*sqrt(5)*s
qrt(2)*arcsin(20/11*x + 1/11) + 9/625*(-10*x^2 - x + 3)^(3/2) + 297/2500*sqrt(10
*x^2 + 23*x + 51/5)*x + 6831/50000*sqrt(10*x^2 + 23*x + 51/5) + 891/12500*sqrt(-
10*x^2 - x + 3) - 1/1875*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27
) + 9/625*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 27/1250*(-10*x^2 - x + 3
)^(3/2)/(5*x + 3) - 11/9375*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 877/9375
*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.22332, size = 127, normalized size = 0.89 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (108000 \, x^{4} + 83700 \, x^{3} - 118395 \, x^{2} - 129910 \, x - 31171\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 39459 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{600000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/600000*sqrt(10)*(2*sqrt(10)*(108000*x^4 + 83700*x^3 - 118395*x^2 - 129910*x -
31171)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 39459*(25*x^2 + 30*x + 9)*arctan(1/20*sqr
t(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)**(3/2)*(2+3*x)**3/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.313425, size = 255, normalized size = 1.8 \[ -\frac{9}{250000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 65 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 265 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{13153}{100000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{193 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{62500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{579 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{46875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/250000*(4*(8*sqrt(5)*(5*x + 3) - 65*sqrt(5))*(5*x + 3) - 265*sqrt(5))*sqrt(5*
x + 3)*sqrt(-10*x + 5) - 1/750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^
3/(5*x + 3)^(3/2) + 13153/100000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) -
193/62500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/46875*
(579*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