3.2537 \(\int \frac{2+3 x}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=48 \[ \frac{7 \sqrt{5 x+3}}{11 \sqrt{1-2 x}}-\frac{3 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{\sqrt{10}} \]

[Out]

(7*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) - (3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/Sqrt
[10]

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Rubi [A]  time = 0.0568325, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{7 \sqrt{5 x+3}}{11 \sqrt{1-2 x}}-\frac{3 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) - (3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/Sqrt
[10]

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Rubi in Sympy [A]  time = 5.52265, size = 44, normalized size = 0.92 \[ - \frac{3 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{10} + \frac{7 \sqrt{5 x + 3}}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/10 + 7*sqrt(5*x + 3)/(11*sqrt(-2*x +
1))

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Mathematica [A]  time = 0.0588202, size = 48, normalized size = 1. \[ \frac{7 \sqrt{5 x+3}}{11 \sqrt{1-2 x}}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt
[10]

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Maple [B]  time = 0.016, size = 74, normalized size = 1.5 \[ -{\frac{1}{-220+440\,x} \left ( 66\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-33\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +140\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/220*(66*10^(1/2)*arcsin(20/11*x+1/11)*x-33*10^(1/2)*arcsin(20/11*x+1/11)+140*
(-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 = 1.51403, size = 49, normalized size = 1.02 \[ -\frac{3}{20} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{11 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/20*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 7/11*sqrt(-10*x^2 - x + 3)/(2*x -
1)

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Fricas [A]  time = 0.227639, size = 86, normalized size = 1.79 \[ -\frac{\sqrt{10}{\left (33 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 14 \, \sqrt{10} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{220 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/220*sqrt(10)*(33*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqr
t(-2*x + 1))) + 14*sqrt(10)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226805, size = 61, normalized size = 1.27 \[ -\frac{3}{10} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{7 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{55 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/10*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 7/55*sqrt(5)*sqrt(5*x + 3)*
sqrt(-10*x + 5)/(2*x - 1)