3.2304 \(\int \frac{\sqrt{1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 (1-2 x)^{3/2}}{165 (5 x+3)^{3/2}}-\frac{6 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{6}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*(1 - 2*x)^(3/2))/(165*(3 + 5*x)^(3/2)) - (6*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x]
) - (6*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

_______________________________________________________________________________________

Rubi [A]  time = 0.0738079, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (1-2 x)^{3/2}}{165 (5 x+3)^{3/2}}-\frac{6 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{6}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2))/(165*(3 + 5*x)^(3/2)) - (6*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x]
) - (6*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.43451, size = 66, normalized size = 0.89 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{6 \sqrt{- 2 x + 1}}{25 \sqrt{5 x + 3}} - \frac{6 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)/(165*(5*x + 3)**(3/2)) - 6*sqrt(-2*x + 1)/(25*sqrt(5*x + 3)
) - 6*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/125

_______________________________________________________________________________________

Mathematica [A]  time = 0.124624, size = 57, normalized size = 0.77 \[ \frac{6}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{2 \sqrt{1-2 x} (485 x+302)}{825 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(302 + 485*x))/(825*(3 + 5*x)^(3/2)) + (6*Sqrt[2/5]*ArcSin[Sqr
t[5/11]*Sqrt[1 - 2*x]])/25

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 96, normalized size = 1.3 \[ -{\frac{1}{4125} \left ( 2475\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+2970\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+891\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +4850\,x\sqrt{-10\,{x}^{2}-x+3}+3020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4125*(2475*10^(1/2)*arcsin(20/11*x+1/11)*x^2+2970*10^(1/2)*arcsin(20/11*x+1/1
1)*x+891*10^(1/2)*arcsin(20/11*x+1/11)+4850*x*(-10*x^2-x+3)^(1/2)+3020*(-10*x^2-
x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.60761, size = 65, normalized size = 0.88 \[ -\frac{4 \, \sqrt{-10 \, x^{2} - x + 3}}{15 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{8 \, \sqrt{-10 \, x^{2} - x + 3}}{165 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/15*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) + 8/165*sqrt(-10*x^2 - x + 3)/(5
*x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.216955, size = 115, normalized size = 1.55 \[ -\frac{\sqrt{5}{\left (2 \, \sqrt{5}{\left (485 \, x + 302\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 99 \, \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{4125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4125*sqrt(5)*(2*sqrt(5)*(485*x + 302)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 99*sqrt(
2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqr
t(-2*x + 1))))/(25*x^2 + 30*x + 9)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.277727, size = 194, normalized size = 2.62 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{66000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{6}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{13 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1100 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{195 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{4125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/66000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 6/125
*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 13/1100*sqrt(10)*(sqrt(2)*sqrt(-
10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/4125*(195*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