3.2574 \(\int \frac{(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{938 (5 x+3)^{5/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{40787 \sqrt{1-2 x} (5 x+3)^{3/2}}{5808}-\frac{40787}{704} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{40787 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64 \sqrt{10}} \]

[Out]

(-40787*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/704 - (40787*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))
/5808 + (49*(3 + 5*x)^(5/2))/(66*(1 - 2*x)^(3/2)) - (938*(3 + 5*x)^(5/2))/(363*S
qrt[1 - 2*x]) + (40787*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64*Sqrt[10])

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Rubi [A]  time = 0.141227, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{938 (5 x+3)^{5/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac{40787 \sqrt{1-2 x} (5 x+3)^{3/2}}{5808}-\frac{40787}{704} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{40787 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-40787*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/704 - (40787*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))
/5808 + (49*(3 + 5*x)^(5/2))/(66*(1 - 2*x)^(3/2)) - (938*(3 + 5*x)^(5/2))/(363*S
qrt[1 - 2*x]) + (40787*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64*Sqrt[10])

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Rubi in Sympy [A]  time = 12.3842, size = 105, normalized size = 0.91 \[ - \frac{40787 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{5808} - \frac{40787 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{704} + \frac{40787 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{640} - \frac{938 \left (5 x + 3\right )^{\frac{5}{2}}}{363 \sqrt{- 2 x + 1}} + \frac{49 \left (5 x + 3\right )^{\frac{5}{2}}}{66 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-40787*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/5808 - 40787*sqrt(-2*x + 1)*sqrt(5*x + 3)
/704 + 40787*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/640 - 938*(5*x + 3)**(5/2)
/(363*sqrt(-2*x + 1)) + 49*(5*x + 3)**(5/2)/(66*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.148764, size = 74, normalized size = 0.64 \[ \frac{122361 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (2160 x^3+12780 x^2-52256 x+18351\right )}{1920 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(18351 - 52256*x + 12780*x^2 + 2160*x^3) + 122361*Sqrt[10 - 2
0*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1920*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.019, size = 137, normalized size = 1.2 \[{\frac{1}{3840\, \left ( -1+2\,x \right ) ^{2}} \left ( 489444\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-43200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-489444\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-255600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+122361\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1045120\,x\sqrt{-10\,{x}^{2}-x+3}-367020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840*(489444*10^(1/2)*arcsin(20/11*x+1/11)*x^2-43200*x^3*(-10*x^2-x+3)^(1/2)-4
89444*10^(1/2)*arcsin(20/11*x+1/11)*x-255600*x^2*(-10*x^2-x+3)^(1/2)+122361*10^(
1/2)*arcsin(20/11*x+1/11)+1045120*x*(-10*x^2-x+3)^(1/2)-367020*(-10*x^2-x+3)^(1/
2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5333, size = 208, normalized size = 1.79 \[ \frac{40787}{1280} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{297}{64} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{24 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (2 \, x - 1\right )}} + \frac{539 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{5873 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

40787/1280*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 297/64*sqrt(-10*x^2 - x + 3)
- 49/24*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 21/4*(-10*x^2 - x
+ 3)^(3/2)/(4*x^2 - 4*x + 1) + 9/16*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 539/48*s
qrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 5873/48*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.226328, size = 120, normalized size = 1.03 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (2160 \, x^{3} + 12780 \, x^{2} - 52256 \, x + 18351\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 122361 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3840 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3840*sqrt(10)*(2*sqrt(10)*(2160*x^3 + 12780*x^2 - 52256*x + 18351)*sqrt(5*x +
3)*sqrt(-2*x + 1) - 122361*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(s
qrt(5*x + 3)*sqrt(-2*x + 1))))/(4*x^2 - 4*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)**2*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23517, size = 113, normalized size = 0.97 \[ \frac{40787}{640} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 247 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 81574 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1345971 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{24000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

40787/640*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/24000*(4*(9*(12*sqrt(
5)*(5*x + 3) + 247*sqrt(5))*(5*x + 3) - 81574*sqrt(5))*(5*x + 3) + 1345971*sqrt(
5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2