3.2544 \(\int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^3}{605 \sqrt{5 x+3}}+\frac{8463 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{12100}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (841380 x+2027201)}{1936000}-\frac{2911419 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16000 \sqrt{10}} \]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^4)/(11*Sqrt[1
- 2*x]*Sqrt[3 + 5*x]) + (8463*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/12100 +
(21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2027201 + 841380*x))/1936000 - (2911419*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/(16000*Sqrt[10])

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Rubi [A]  time = 0.264112, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^3}{605 \sqrt{5 x+3}}+\frac{8463 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{12100}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (841380 x+2027201)}{1936000}-\frac{2911419 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^4)/(11*Sqrt[1
- 2*x]*Sqrt[3 + 5*x]) + (8463*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/12100 +
(21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2027201 + 841380*x))/1936000 - (2911419*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/(16000*Sqrt[10])

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Rubi in Sympy [A]  time = 26.8862, size = 133, normalized size = 0.94 \[ - \frac{37 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{605 \sqrt{5 x + 3}} + \frac{8463 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{12100} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{66258675 x}{4} + \frac{638568315}{16}\right )}{1815000} - \frac{2911419 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{160000} + \frac{7 \left (3 x + 2\right )^{4}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37*sqrt(-2*x + 1)*(3*x + 2)**3/(605*sqrt(5*x + 3)) + 8463*sqrt(-2*x + 1)*(3*x +
2)**2*sqrt(5*x + 3)/12100 + sqrt(-2*x + 1)*sqrt(5*x + 3)*(66258675*x/4 + 638568
315/16)/1815000 - 2911419*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/160000 + 7*(3
*x + 2)**4/(11*sqrt(-2*x + 1)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.155565, size = 86, normalized size = 0.61 \[ \frac{352281699 \sqrt{10-20 x} (5 x+3) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (15681600 x^4+75663720 x^3+208989990 x^2-169670279 x-162727423\right )}{19360000 \sqrt{1-2 x} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-162727423 - 169670279*x + 208989990*x^2 + 75663720*x^3 + 15
681600*x^4) + 352281699*Sqrt[10 - 20*x]*(3 + 5*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x
]])/(19360000*Sqrt[1 - 2*x]*(3 + 5*x))

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Maple [A]  time = 0.023, size = 154, normalized size = 1.1 \[ -{\frac{1}{-38720000+77440000\,x}\sqrt{1-2\,x} \left ( -313632000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-1513274400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-4179799800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-1056845097\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/38720000*(1-2*x)^(1/2)*(-313632000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2
)*arcsin(20/11*x+1/11)*x^2-1513274400*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)
*arcsin(20/11*x+1/11)*x-4179799800*x^2*(-10*x^2-x+3)^(1/2)-1056845097*10^(1/2)*a
rcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1
/2))/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.52328, size = 124, normalized size = 0.87 \[ -\frac{81 \, x^{4}}{10 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{15633 \, x^{3}}{400 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{172719 \, x^{2}}{1600 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2911419}{320000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{169670279 \, x}{1936000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{162727423}{1936000 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/10*x^4/sqrt(-10*x^2 - x + 3) - 15633/400*x^3/sqrt(-10*x^2 - x + 3) - 172719/
1600*x^2/sqrt(-10*x^2 - x + 3) + 2911419/320000*sqrt(10)*arcsin(-20/11*x - 1/11)
+ 169670279/1936000*x/sqrt(-10*x^2 - x + 3) + 162727423/1936000/sqrt(-10*x^2 -
x + 3)

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Fricas [A]  time = 0.229246, size = 122, normalized size = 0.86 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (15681600 \, x^{4} + 75663720 \, x^{3} + 208989990 \, x^{2} - 169670279 \, x - 162727423\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 352281699 \,{\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{38720000 \,{\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/38720000*sqrt(10)*(2*sqrt(10)*(15681600*x^4 + 75663720*x^3 + 208989990*x^2 - 1
69670279*x - 162727423)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 352281699*(10*x^2 + x - 3
)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(10*x^2 + x -
3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.26187, size = 194, normalized size = 1.37 \[ -\frac{2911419}{160000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6534 \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 97 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 16325 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 1761451247 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{242000000 \,{\left (2 \, x - 1\right )}} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{756250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{378125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2911419/160000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/242000000*(6534
*(12*(8*sqrt(5)*(5*x + 3) + 97*sqrt(5))*(5*x + 3) + 16325*sqrt(5))*(5*x + 3) - 1
761451247*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 1/756250*sqrt(10)*(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/378125*sqrt(10)*sqrt(5*x +
3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))