3.2345 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^2}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{3}{40} (3 x+2) \sqrt{5 x+3} (1-2 x)^{5/2}-\frac{119}{800} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{301 \sqrt{5 x+3} (1-2 x)^{3/2}}{3200}+\frac{9933 \sqrt{5 x+3} \sqrt{1-2 x}}{32000}+\frac{109263 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{32000 \sqrt{10}} \]

[Out]

(9933*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32000 + (301*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3
200 - (119*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/800 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*Sqr
t[3 + 5*x])/40 + (109263*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(32000*Sqrt[10])

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Rubi [A]  time = 0.141259, antiderivative size = 121, 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 \[ -\frac{3}{40} (3 x+2) \sqrt{5 x+3} (1-2 x)^{5/2}-\frac{119}{800} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{301 \sqrt{5 x+3} (1-2 x)^{3/2}}{3200}+\frac{9933 \sqrt{5 x+3} \sqrt{1-2 x}}{32000}+\frac{109263 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{32000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(9933*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32000 + (301*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3
200 - (119*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/800 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*Sqr
t[3 + 5*x])/40 + (109263*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(32000*Sqrt[10])

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Rubi in Sympy [A]  time = 11.1113, size = 109, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3} \left (9 x + 6\right )}{40} - \frac{119 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{800} + \frac{301 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3200} + \frac{9933 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{32000} + \frac{109263 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{320000} \]

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)**(1/2),x)

[Out]

-(-2*x + 1)**(5/2)*sqrt(5*x + 3)*(9*x + 6)/40 - 119*(-2*x + 1)**(5/2)*sqrt(5*x +
3)/800 + 301*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/3200 + 9933*sqrt(-2*x + 1)*sqrt(5*
x + 3)/32000 + 109263*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/320000

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Mathematica [A]  time = 0.104766, size = 65, normalized size = 0.54 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (-28800 x^3-9440 x^2+25020 x+3383\right )-109263 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{320000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3383 + 25020*x - 9440*x^2 - 28800*x^3) - 109263
*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/320000

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Maple [A]  time = 0.014, size = 104, normalized size = 0.9 \[{\frac{1}{640000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-188800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+109263\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +500400\,x\sqrt{-10\,{x}^{2}-x+3}+67660\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/640000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-576000*x^3*(-10*x^2-x+3)^(1/2)-188800*x^2
*(-10*x^2-x+3)^(1/2)+109263*10^(1/2)*arcsin(20/11*x+1/11)+500400*x*(-10*x^2-x+3)
^(1/2)+67660*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50778, size = 101, normalized size = 0.83 \[ -\frac{9}{10} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{59}{200} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{1251}{1600} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{109263}{640000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3383}{32000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/10*sqrt(-10*x^2 - x + 3)*x^3 - 59/200*sqrt(-10*x^2 - x + 3)*x^2 + 1251/1600*s
qrt(-10*x^2 - x + 3)*x - 109263/640000*sqrt(10)*arcsin(-20/11*x - 1/11) + 3383/3
2000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.230167, size = 90, normalized size = 0.74 \[ -\frac{1}{640000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (28800 \, x^{3} + 9440 \, x^{2} - 25020 \, x - 3383\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 109263 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/640000*sqrt(10)*(2*sqrt(10)*(28800*x^3 + 9440*x^2 - 25020*x - 3383)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) - 109263*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt
(-2*x + 1))))

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Sympy [A]  time = 56.8946, size = 394, normalized size = 3.26 \[ - \frac{49 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (\frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{968} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{8}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} + \frac{21 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} + \frac{3 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{16}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} - \frac{9 \sqrt{2} \left (\begin{cases} \frac{14641 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{3993} + \frac{7 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{3872} + \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (12100 x - 2000 \left (- 2 x + 1\right )^{3} + 6600 \left (- 2 x + 1\right )^{2} - 4719\right )}{1874048} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{128}\right )}{3125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-49*sqrt(2)*Piecewise((121*sqrt(5)*(sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x
+ 1)/968 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + 3*asin(sqrt(55)*sqrt(-2*x
+ 1)/11)/8)/125, (x <= 1/2) & (x > -3/5)))/4 + 21*sqrt(2)*Piecewise((1331*sqrt(5
)*(5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/7986 + 3*sqrt(5)*sqrt(-2*x + 1)
*sqrt(10*x + 6)*(20*x + 1)/1936 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + 5*a
sin(sqrt(55)*sqrt(-2*x + 1)/11)/16)/625, (x <= 1/2) & (x > -3/5)))/2 - 9*sqrt(2)
*Piecewise((14641*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2)/3993 +
7*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/3872 + sqrt(5)*sqrt(-2*x + 1)
*sqrt(10*x + 6)*(12100*x - 2000*(-2*x + 1)**3 + 6600*(-2*x + 1)**2 - 4719)/18740
48 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + 35*asin(sqrt(55)*sqrt(-2*x + 1)/
11)/128)/3125, (x <= 1/2) & (x > -3/5)))/4

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GIAC/XCAS [A]  time = 0.238343, size = 274, normalized size = 2.26 \[ -\frac{3}{1600000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 119\right )}{\left (5 \, x + 3\right )} + 6163\right )}{\left (5 \, x + 3\right )} - 66189\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 184305 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{8000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{500} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{2}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/1600000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sq
rt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))
) - 1/8000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x
+ 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/500*sqrt(5)*(2*(20*
x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*
x + 3))) + 2/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt
(5*x + 3)*sqrt(-10*x + 5))