3.2503 \(\int \frac{(2+3 x)^3 \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=110 \[ \frac{\sqrt{5 x+3} (3 x+2)^3}{\sqrt{1-2 x}}+\frac{7}{4} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{\sqrt{1-2 x} \sqrt{5 x+3} (73380 x+176833)}{3200}-\frac{1463447 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200 \sqrt{10}} \]

[Out]

(7*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/4 + ((2 + 3*x)^3*Sqrt[3 + 5*x])/Sqrt
[1 - 2*x] + (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(176833 + 73380*x))/3200 - (1463447*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*Sqrt[10])

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Rubi [A]  time = 0.184084, antiderivative size = 110, 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 \[ \frac{\sqrt{5 x+3} (3 x+2)^3}{\sqrt{1-2 x}}+\frac{7}{4} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{\sqrt{1-2 x} \sqrt{5 x+3} (73380 x+176833)}{3200}-\frac{1463447 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/4 + ((2 + 3*x)^3*Sqrt[3 + 5*x])/Sqrt
[1 - 2*x] + (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(176833 + 73380*x))/3200 - (1463447*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*Sqrt[10])

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Rubi in Sympy [A]  time = 18.7885, size = 102, normalized size = 0.93 \[ \frac{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{4} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{275175 x}{2} + \frac{2652495}{8}\right )}{6000} - \frac{1463447 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{32000} + \frac{\left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*sqrt(-2*x + 1)*(3*x + 2)**2*sqrt(5*x + 3)/4 + sqrt(-2*x + 1)*sqrt(5*x + 3)*(27
5175*x/2 + 2652495/8)/6000 - 1463447*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/32
000 + (3*x + 2)**3*sqrt(5*x + 3)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.0928808, size = 69, normalized size = 0.63 \[ \frac{1463447 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (14400 x^3+57960 x^2+142686 x-224833\right )}{32000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-224833 + 142686*x + 57960*x^2 + 14400*x^3) + 1463447*Sqrt[1
0 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(32000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.017, size = 123, normalized size = 1.1 \[ -{\frac{1}{-64000+128000\,x} \left ( -288000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+2926894\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1159200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-1463447\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -2853720\,x\sqrt{-10\,{x}^{2}-x+3}+4496660\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64000*(-288000*x^3*(-10*x^2-x+3)^(1/2)+2926894*10^(1/2)*arcsin(20/11*x+1/11)*
x-1159200*x^2*(-10*x^2-x+3)^(1/2)-1463447*10^(1/2)*arcsin(20/11*x+1/11)-2853720*
x*(-10*x^2-x+3)^(1/2)+4496660*(-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 = 1.51689, size = 107, normalized size = 0.97 \[ -\frac{1463447}{64000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{9}{40} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1593}{160} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{89793}{3200} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{343 \, \sqrt{-10 \, x^{2} - x + 3}}{8 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1463447/64000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/40*(-10*x^2 - x + 3)^(
3/2) + 1593/160*sqrt(-10*x^2 - x + 3)*x + 89793/3200*sqrt(-10*x^2 - x + 3) - 343
/8*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.223108, size = 107, normalized size = 0.97 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (14400 \, x^{3} + 57960 \, x^{2} + 142686 \, x - 224833\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1463447 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{64000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64000*sqrt(10)*(2*sqrt(10)*(14400*x^3 + 57960*x^2 + 142686*x - 224833)*sqrt(5*
x + 3)*sqrt(-2*x + 1) - 1463447*(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(-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*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230721, size = 113, normalized size = 1.03 \[ -\frac{1463447}{32000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (18 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 89 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4927 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 1463447 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{80000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1463447/32000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/80000*(18*(4*(8*
sqrt(5)*(5*x + 3) + 89*sqrt(5))*(5*x + 3) + 4927*sqrt(5))*(5*x + 3) - 1463447*sq
rt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)