3.2284 \(\int \frac{\sqrt{1-2 x} (2+3 x)^3}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{3}{40} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (216 x+335)}{6400}+\frac{47761 \sqrt{1-2 x} \sqrt{5 x+3}}{64000}+\frac{525371 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000 \sqrt{10}} \]

[Out]

(47761*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[
3 + 5*x])/40 - (21*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]*(335 + 216*x))/6400 + (525371*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000*Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.134832, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{40} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (216 x+335)}{6400}+\frac{47761 \sqrt{1-2 x} \sqrt{5 x+3}}{64000}+\frac{525371 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(47761*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[
3 + 5*x])/40 - (21*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]*(335 + 216*x))/6400 + (525371*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000*Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.3783, size = 97, normalized size = 0.92 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{40} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3} \left (17010 x + \frac{105525}{4}\right )}{24000} + \frac{47761 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{64000} + \frac{525371 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{640000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(3/2)*(3*x + 2)**2*sqrt(5*x + 3)/40 - (-2*x + 1)**(3/2)*sqrt(5*x
+ 3)*(17010*x + 105525/4)/24000 + 47761*sqrt(-2*x + 1)*sqrt(5*x + 3)/64000 + 525
371*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/640000

_______________________________________________________________________________________

Mathematica [A]  time = 0.121001, size = 65, normalized size = 0.61 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (86400 x^3+162720 x^2+76140 x-41789\right )-525371 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{640000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-41789 + 76140*x + 162720*x^2 + 86400*x^3) - 52
5371*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/640000

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 104, normalized size = 1. \[{\frac{1}{1280000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1728000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+3254400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+525371\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1522800\,x\sqrt{-10\,{x}^{2}-x+3}-835780\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1280000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1728000*x^3*(-10*x^2-x+3)^(1/2)+3254400*x
^2*(-10*x^2-x+3)^(1/2)+525371*10^(1/2)*arcsin(20/11*x+1/11)+1522800*x*(-10*x^2-x
+3)^(1/2)-835780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50913, size = 99, normalized size = 0.93 \[ -\frac{27}{200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{525371}{1280000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{963}{4000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{21663}{16000} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{887}{12800} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/200*(-10*x^2 - x + 3)^(3/2)*x + 525371/1280000*sqrt(5)*sqrt(2)*arcsin(20/11*
x + 1/11) - 963/4000*(-10*x^2 - x + 3)^(3/2) + 21663/16000*sqrt(-10*x^2 - x + 3)
*x + 887/12800*sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.22177, size = 90, normalized size = 0.85 \[ \frac{1}{1280000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (86400 \, x^{3} + 162720 \, x^{2} + 76140 \, x - 41789\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 525371 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1280000*sqrt(10)*(2*sqrt(10)*(86400*x^3 + 162720*x^2 + 76140*x - 41789)*sqrt(5
*x + 3)*sqrt(-2*x + 1) + 525371*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*s
qrt(-2*x + 1))))

_______________________________________________________________________________________

Sympy [A]  time = 15.4911, size = 462, normalized size = 4.36 \[ - \frac{343 \sqrt{2} \left (\begin{cases} \frac{11 \sqrt{5} \left (- \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{2}\right )}{25} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} + \frac{441 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (\frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{968} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{8}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} - \frac{189 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} + \frac{3 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{16}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} + \frac{27 \sqrt{2} \left (\begin{cases} \frac{14641 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{3993} + \frac{7 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{3872} + \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (12100 x - 2000 \left (- 2 x + 1\right )^{3} + 6600 \left (- 2 x + 1\right )^{2} - 4719\right )}{1874048} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{128}\right )}{3125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-343*sqrt(2)*Piecewise((11*sqrt(5)*(-sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 +
asin(sqrt(55)*sqrt(-2*x + 1)/11)/2)/25, (x <= 1/2) & (x > -3/5)))/8 + 441*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)))/8 - 189*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*asin(sqrt(
55)*sqrt(-2*x + 1)/11)/16)/625, (x <= 1/2) & (x > -3/5)))/8 + 27*sqrt(2)*Piecewi
se((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)/1874048 - sqr
t(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)))/8

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.248749, size = 274, normalized size = 2.58 \[ \frac{9}{3200000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 119\right )}{\left (5 \, x + 3\right )} + 6163\right )}{\left (5 \, x + 3\right )} - 66189\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 184305 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{20000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{500} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{4}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/3200000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqr
t(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
+ 9/20000*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))) + 9/500*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))) + 4/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt
(5*x + 3)*sqrt(-10*x + 5))