3.2346 \(\int \frac{(1-2 x)^{3/2} (2+3 x)}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{1}{10} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{3}{40} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{99}{400} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1089 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

[Out]

(99*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 + (3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/40 - ((
1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/10 + (1089*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*
Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.0959091, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{10} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{3}{40} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{99}{400} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1089 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(99*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 + (3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/40 - ((
1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/10 + (1089*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*
Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.20896, size = 83, normalized size = 0.88 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{10} + \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{40} + \frac{99 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{400} + \frac{1089 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*sqrt(5*x + 3)/10 + 3*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/40 + 99*
sqrt(-2*x + 1)*sqrt(5*x + 3)/400 + 1089*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)
/4000

_______________________________________________________________________________________

Mathematica [A]  time = 0.0938027, size = 60, normalized size = 0.64 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (-160 x^2+100 x+89\right )-1089 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(89 + 100*x - 160*x^2) - 1089*Sqrt[10]*ArcSin[Sq
rt[5/11]*Sqrt[1 - 2*x]])/4000

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 87, normalized size = 0.9 \[{\frac{1}{8000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -3200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1089\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2000\,x\sqrt{-10\,{x}^{2}-x+3}+1780\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-3200*x^2*(-10*x^2-x+3)^(1/2)+1089*10^(1/2)*
arcsin(20/11*x+1/11)+2000*x*(-10*x^2-x+3)^(1/2)+1780*(-10*x^2-x+3)^(1/2))/(-10*x
^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49544, size = 78, normalized size = 0.83 \[ -\frac{2}{5} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{1}{4} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1089}{8000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{89}{400} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*sqrt(-10*x^2 - x + 3)*x^2 + 1/4*sqrt(-10*x^2 - x + 3)*x - 1089/8000*sqrt(10
)*arcsin(-20/11*x - 1/11) + 89/400*sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.217047, size = 84, normalized size = 0.89 \[ -\frac{1}{8000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (160 \, x^{2} - 100 \, x - 89\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1089 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8000*sqrt(10)*(2*sqrt(10)*(160*x^2 - 100*x - 89)*sqrt(5*x + 3)*sqrt(-2*x + 1)
- 1089*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

_______________________________________________________________________________________

Sympy [A]  time = 33.6418, size = 223, normalized size = 2.37 \[ - \frac{7 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (\frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{968} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{8}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} + \frac{3 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} + \frac{3 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{16}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*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)))/2 + 3*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)*s
qrt(10*x + 6)*(20*x + 1)/1936 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + 5*asi
n(sqrt(55)*sqrt(-2*x + 1)/11)/16)/625, (x <= 1/2) & (x > -3/5)))/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232541, size = 189, normalized size = 2.01 \[ -\frac{1}{20000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{2000} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20000*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))) - 1/2000*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))) + 1/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(
5*x + 3)*sqrt(-10*x + 5))