3.2293 \(\int \frac{\sqrt{1-2 x} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{7}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{7 (73-60 x) \sqrt{1-2 x} \sqrt{5 x+3}}{4000}+\frac{10409 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) - (7*(73 - 60*x)*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/4000 + (7*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/25 + (10409*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4000*Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.17946, antiderivative size = 113, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{7}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{7 (73-60 x) \sqrt{1-2 x} \sqrt{5 x+3}}{4000}+\frac{10409 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) - (7*(73 - 60*x)*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/4000 + (7*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/25 + (10409*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4000*Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.2844, size = 104, normalized size = 0.92 \[ - \frac{\left (- 1575 x + \frac{7665}{4}\right ) \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15000} - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{5 \sqrt{5 x + 3}} + \frac{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{25} + \frac{10409 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{40000} \]

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)**(3/2),x)

[Out]

-(-1575*x + 7665/4)*sqrt(-2*x + 1)*sqrt(5*x + 3)/15000 - 2*sqrt(-2*x + 1)*(3*x +
2)**3/(5*sqrt(5*x + 3)) + 7*sqrt(-2*x + 1)*(3*x + 2)**2*sqrt(5*x + 3)/25 + 1040
9*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/40000

_______________________________________________________________________________________

Mathematica [A]  time = 0.17307, size = 65, normalized size = 0.58 \[ \frac{\frac{10 \sqrt{1-2 x} \left (7200 x^3+13140 x^2+3825 x-893\right )}{\sqrt{5 x+3}}-10409 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{40000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(-893 + 3825*x + 13140*x^2 + 7200*x^3))/Sqrt[3 + 5*x] - 10409
*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/40000

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 116, normalized size = 1. \[{\frac{1}{80000} \left ( 144000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+52045\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+262800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+31227\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +76500\,x\sqrt{-10\,{x}^{2}-x+3}-17860\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/80000*(144000*x^3*(-10*x^2-x+3)^(1/2)+52045*10^(1/2)*arcsin(20/11*x+1/11)*x+26
2800*x^2*(-10*x^2-x+3)^(1/2)+31227*10^(1/2)*arcsin(20/11*x+1/11)+76500*x*(-10*x^
2-x+3)^(1/2)-17860*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x
)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51098, size = 107, normalized size = 0.95 \[ \frac{10409}{80000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{9}{250} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{81}{200} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{693}{20000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{625 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10409/80000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/250*(-10*x^2 - x + 3)^(3/
2) + 81/200*sqrt(-10*x^2 - x + 3)*x + 693/20000*sqrt(-10*x^2 - x + 3) - 2/625*sq
rt(-10*x^2 - x + 3)/(5*x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.224102, size = 107, normalized size = 0.95 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (7200 \, x^{3} + 13140 \, x^{2} + 3825 \, x - 893\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 10409 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{80000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/80000*sqrt(10)*(2*sqrt(10)*(7200*x^3 + 13140*x^2 + 3825*x - 893)*sqrt(5*x + 3)
*sqrt(-2*x + 1) + 10409*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)
*sqrt(-2*x + 1))))/(5*x + 3)

_______________________________________________________________________________________

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((2+3*x)**3*(1-2*x)**(1/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.298086, size = 165, normalized size = 1.46 \[ \frac{9}{100000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 463 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{10409}{40000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{6250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{3125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/100000*(4*(8*sqrt(5)*(5*x + 3) + sqrt(5))*(5*x + 3) - 463*sqrt(5))*sqrt(5*x +
3)*sqrt(-10*x + 5) + 10409/40000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) -
1/6250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/3125*sqrt
(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))