3.2294 \(\int \frac{\sqrt{1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{9}{100} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{2 (1-2 x)^{3/2}}{275 \sqrt{5 x+3}}+\frac{317 \sqrt{5 x+3} \sqrt{1-2 x}}{2200}+\frac{317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(3/2))/(275*Sqrt[3 + 5*x]) + (317*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/220
0 - (9*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/100 + (317*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(200*Sqrt[10])

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Rubi [A]  time = 0.114583, 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 \[ -\frac{9}{100} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{2 (1-2 x)^{3/2}}{275 \sqrt{5 x+3}}+\frac{317 \sqrt{5 x+3} \sqrt{1-2 x}}{2200}+\frac{317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2))/(275*Sqrt[3 + 5*x]) + (317*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/220
0 - (9*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/100 + (317*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(200*Sqrt[10])

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Rubi in Sympy [A]  time = 9.71293, size = 85, normalized size = 0.9 \[ - \frac{9 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{100} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{275 \sqrt{5 x + 3}} + \frac{317 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2200} + \frac{317 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/100 - 2*(-2*x + 1)**(3/2)/(275*sqrt(5*x + 3))
+ 317*sqrt(-2*x + 1)*sqrt(5*x + 3)/2200 + 317*sqrt(10)*asin(sqrt(22)*sqrt(5*x +
3)/11)/2000

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Mathematica [A]  time = 0.138073, size = 60, normalized size = 0.64 \[ \frac{\frac{10 \sqrt{1-2 x} \left (180 x^2+165 x+31\right )}{\sqrt{5 x+3}}-317 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(31 + 165*x + 180*x^2))/Sqrt[3 + 5*x] - 317*Sqrt[10]*ArcSin[S
qrt[5/11]*Sqrt[1 - 2*x]])/2000

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Maple [A]  time = 0.019, size = 99, normalized size = 1.1 \[{\frac{1}{4000} \left ( 1585\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+3600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+951\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +3300\,x\sqrt{-10\,{x}^{2}-x+3}+620\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4000*(1585*10^(1/2)*arcsin(20/11*x+1/11)*x+3600*x^2*(-10*x^2-x+3)^(1/2)+951*10
^(1/2)*arcsin(20/11*x+1/11)+3300*x*(-10*x^2-x+3)^(1/2)+620*(-10*x^2-x+3)^(1/2))*
(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.51334, size = 88, normalized size = 0.94 \[ \frac{317}{4000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{50} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{57}{1000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

317/4000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 9/50*sqrt(-10*x^2 - x + 3)*x +
57/1000*sqrt(-10*x^2 - x + 3) - 2/125*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.217571, size = 100, normalized size = 1.06 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (180 \, x^{2} + 165 \, x + 31\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 317 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{4000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4000*sqrt(10)*(2*sqrt(10)*(180*x^2 + 165*x + 31)*sqrt(5*x + 3)*sqrt(-2*x + 1)
+ 317*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))
/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270085, size = 150, normalized size = 1.6 \[ \frac{3}{5000} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 17 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{317}{2000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5000*(12*sqrt(5)*(5*x + 3) - 17*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 317/2
000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/1250*sqrt(10)*(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))