3.2454 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{1}{4} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{33}{16} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16 \sqrt{10}} \]

[Out]

(-33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/16 - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4 + (363*
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(16*Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.0643038, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{4} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{33}{16} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/16 - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4 + (363*
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(16*Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.48045, size = 63, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{4} - \frac{33 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{16} + \frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{160} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(5*x + 3)**(3/2)/4 - 33*sqrt(-2*x + 1)*sqrt(5*x + 3)/16 + 363*sq
rt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/160

_______________________________________________________________________________________

Mathematica [A]  time = 0.0481386, size = 55, normalized size = 0.76 \[ \frac{1}{160} \left (-50 \sqrt{1-2 x} \sqrt{5 x+3} (4 x+9)-363 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-50*Sqrt[1 - 2*x]*(9 + 4*x)*Sqrt[3 + 5*x] - 363*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt
[1 - 2*x]])/160

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 72, normalized size = 1. \[ -{\frac{1}{4} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{33}{16}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{363\,\sqrt{10}}{320}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(3+5*x)^(3/2)*(1-2*x)^(1/2)-33/16*(1-2*x)^(1/2)*(3+5*x)^(1/2)+363/320*((1-2
*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

_______________________________________________________________________________________

Maxima [A]  time = 1.50983, size = 55, normalized size = 0.76 \[ -\frac{5}{4} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{363}{320} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{45}{16} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/4*sqrt(-10*x^2 - x + 3)*x - 363/320*sqrt(10)*arcsin(-20/11*x - 1/11) - 45/16*
sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.222764, size = 77, normalized size = 1.07 \[ -\frac{1}{320} \, \sqrt{10}{\left (10 \, \sqrt{10} \sqrt{5 \, x + 3}{\left (4 \, x + 9\right )} \sqrt{-2 \, x + 1} - 363 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/320*sqrt(10)*(10*sqrt(10)*sqrt(5*x + 3)*(4*x + 9)*sqrt(-2*x + 1) - 363*arctan
(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

_______________________________________________________________________________________

Sympy [A]  time = 6.1716, size = 187, normalized size = 2.6 \[ \begin{cases} - \frac{25 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{2 \sqrt{10 x - 5}} - \frac{55 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{8 \sqrt{10 x - 5}} + \frac{363 i \sqrt{x + \frac{3}{5}}}{16 \sqrt{10 x - 5}} - \frac{363 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{160} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{160} + \frac{25 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{2 \sqrt{- 10 x + 5}} + \frac{55 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{8 \sqrt{- 10 x + 5}} - \frac{363 \sqrt{x + \frac{3}{5}}}{16 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-25*I*(x + 3/5)**(5/2)/(2*sqrt(10*x - 5)) - 55*I*(x + 3/5)**(3/2)/(8*
sqrt(10*x - 5)) + 363*I*sqrt(x + 3/5)/(16*sqrt(10*x - 5)) - 363*sqrt(10)*I*acosh
(sqrt(110)*sqrt(x + 3/5)/11)/160, 10*Abs(x + 3/5)/11 > 1), (363*sqrt(10)*asin(sq
rt(110)*sqrt(x + 3/5)/11)/160 + 25*(x + 3/5)**(5/2)/(2*sqrt(-10*x + 5)) + 55*(x
+ 3/5)**(3/2)/(8*sqrt(-10*x + 5)) - 363*sqrt(x + 3/5)/(16*sqrt(-10*x + 5)), True
))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228622, size = 61, normalized size = 0.85 \[ -\frac{1}{160} \, \sqrt{5}{\left (10 \, \sqrt{5 \, x + 3}{\left (4 \, x + 9\right )} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/160*sqrt(5)*(10*sqrt(5*x + 3)*(4*x + 9)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(
1/11*sqrt(22)*sqrt(5*x + 3)))