3.2536 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=72 \[ \frac{9}{20} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{49 \sqrt{5 x+3}}{22 \sqrt{1-2 x}}-\frac{321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]

[Out]

(49*Sqrt[3 + 5*x])/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 - (32
1*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20*Sqrt[10])

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Rubi [A]  time = 0.0984511, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{9}{20} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{49 \sqrt{5 x+3}}{22 \sqrt{1-2 x}}-\frac{321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(49*Sqrt[3 + 5*x])/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 - (32
1*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20*Sqrt[10])

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Rubi in Sympy [A]  time = 8.26021, size = 65, normalized size = 0.9 \[ \frac{9 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{20} - \frac{321 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{200} + \frac{49 \sqrt{5 x + 3}}{22 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(-2*x + 1)*sqrt(5*x + 3)/20 - 321*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)
/200 + 49*sqrt(5*x + 3)/(22*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0719178, size = 59, normalized size = 0.82 \[ \frac{10 \sqrt{5 x+3} (589-198 x)+3531 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2200 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(10*(589 - 198*x)*Sqrt[3 + 5*x] + 3531*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1
- 2*x]])/(2200*Sqrt[1 - 2*x])

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Maple [A]  time = 0.019, size = 89, normalized size = 1.2 \[ -{\frac{1}{-4400+8800\,x} \left ( 7062\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3531\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3960\,x\sqrt{-10\,{x}^{2}-x+3}+11780\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4400*(7062*10^(1/2)*arcsin(20/11*x+1/11)*x-3531*10^(1/2)*arcsin(20/11*x+1/11)
-3960*x*(-10*x^2-x+3)^(1/2)+11780*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/
2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49016, size = 68, normalized size = 0.94 \[ -\frac{321}{400} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{20} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{22 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-321/400*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 9/20*sqrt(-10*x^2 - x + 3) - 4
9/22*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.226755, size = 93, normalized size = 1.29 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (198 \, x - 589\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3531 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{4400 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4400*sqrt(10)*(2*sqrt(10)*(198*x - 589)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3531*(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. \[ \int \frac{\left (3 x + 2\right )^{2}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.233553, size = 78, normalized size = 1.08 \[ -\frac{321}{200} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (198 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3539 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{5500 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-321/200*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/5500*(198*sqrt(5)*(5*x
+ 3) - 3539*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)