3.2504 \(\int \frac{(2+3 x)^2 \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{9}{40} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{49 (5 x+3)^{3/2}}{22 \sqrt{1-2 x}}+\frac{17951 \sqrt{1-2 x} \sqrt{5 x+3}}{1760}-\frac{17951 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{160 \sqrt{10}} \]

[Out]

(17951*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1760 + (49*(3 + 5*x)^(3/2))/(22*Sqrt[1 - 2*x
]) + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/40 - (17951*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(160*Sqrt[10])

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Rubi [A]  time = 0.117366, antiderivative size = 94, 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 \[ \frac{9}{40} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{49 (5 x+3)^{3/2}}{22 \sqrt{1-2 x}}+\frac{17951 \sqrt{1-2 x} \sqrt{5 x+3}}{1760}-\frac{17951 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{160 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(17951*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1760 + (49*(3 + 5*x)^(3/2))/(22*Sqrt[1 - 2*x
]) + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/40 - (17951*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(160*Sqrt[10])

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Rubi in Sympy [A]  time = 10.0096, size = 85, normalized size = 0.9 \[ \frac{9 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{40} + \frac{17951 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1760} - \frac{17951 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1600} + \frac{49 \left (5 x + 3\right )^{\frac{3}{2}}}{22 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/40 + 17951*sqrt(-2*x + 1)*sqrt(5*x + 3)/1760 -
17951*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/1600 + 49*(5*x + 3)**(3/2)/(22*s
qrt(-2*x + 1))

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Mathematica [A]  time = 0.0781072, size = 64, normalized size = 0.68 \[ \frac{17951 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (360 x^2+1518 x-2809\right )}{1600 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-2809 + 1518*x + 360*x^2) + 17951*Sqrt[10 - 20*x]*ArcSin[Sqr
t[5/11]*Sqrt[1 - 2*x]])/(1600*Sqrt[1 - 2*x])

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Maple [A]  time = 0.018, size = 106, normalized size = 1.1 \[ -{\frac{1}{-3200+6400\,x} \left ( 35902\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-7200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-17951\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -30360\,x\sqrt{-10\,{x}^{2}-x+3}+56180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3200*(35902*10^(1/2)*arcsin(20/11*x+1/11)*x-7200*x^2*(-10*x^2-x+3)^(1/2)-1795
1*10^(1/2)*arcsin(20/11*x+1/11)-30360*x*(-10*x^2-x+3)^(1/2)+56180*(-10*x^2-x+3)^
(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50409, size = 88, normalized size = 0.94 \[ -\frac{17951}{3200} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{8} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{849}{160} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{4 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17951/3200*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 9/8*sqrt(-10*x^2 - x + 3)*x
+ 849/160*sqrt(-10*x^2 - x + 3) - 49/4*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.227821, size = 100, normalized size = 1.06 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (360 \, x^{2} + 1518 \, x - 2809\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 17951 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3200 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3200*sqrt(10)*(2*sqrt(10)*(360*x^2 + 1518*x - 2809)*sqrt(5*x + 3)*sqrt(-2*x +
1) - 17951*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x +
1))))/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229743, size = 96, normalized size = 1.02 \[ -\frac{17951}{1600} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 181 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 17951 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17951/1600*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/4000*(6*(12*sqrt(5)
*(5*x + 3) + 181*sqrt(5))*(5*x + 3) - 17951*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1)