3.2356 \(\int \frac{(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2 (1-2 x)^{5/2}}{55 \sqrt{5 x+3}}+\frac{1}{22} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{3}{20} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{33 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2))/(55*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 + (
(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/22 + (33*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20*Sq
rt[10])

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Rubi [A]  time = 0.0957117, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (1-2 x)^{5/2}}{55 \sqrt{5 x+3}}+\frac{1}{22} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{3}{20} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{33 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2))/(55*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 + (
(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/22 + (33*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20*Sq
rt[10])

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Rubi in Sympy [A]  time = 8.81795, size = 83, normalized size = 0.88 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{55 \sqrt{5 x + 3}} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{22} + \frac{3 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{20} + \frac{33 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{200} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)/(55*sqrt(5*x + 3)) + (-2*x + 1)**(3/2)*sqrt(5*x + 3)/22 + 3
*sqrt(-2*x + 1)*sqrt(5*x + 3)/20 + 33*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/2
00

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Mathematica [A]  time = 0.119962, size = 60, normalized size = 0.64 \[ \frac{\sqrt{1-2 x} \left (-12 x^2+17 x+11\right )}{20 \sqrt{5 x+3}}-\frac{33 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{20 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(11 + 17*x - 12*x^2))/(20*Sqrt[3 + 5*x]) - (33*ArcSin[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(20*Sqrt[10])

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Maple [A]  time = 0.016, size = 99, normalized size = 1.1 \[{\frac{1}{400} \left ( 165\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-240\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+99\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +340\,x\sqrt{-10\,{x}^{2}-x+3}+220\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/400*(165*10^(1/2)*arcsin(20/11*x+1/11)*x-240*x^2*(-10*x^2-x+3)^(1/2)+99*10^(1/
2)*arcsin(20/11*x+1/11)+340*x*(-10*x^2-x+3)^(1/2)+220*(-10*x^2-x+3)^(1/2))*(1-2*
x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.50666, size = 131, normalized size = 1.39 \[ \frac{33}{400} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{99}{500} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{25 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{50 \,{\left (5 \, x + 3\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

33/400*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 99/500*sqrt(-10*x^2 - x + 3) + 1
/25*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 3/50*(-10*x^2 - x + 3)^(3/2)/(
5*x + 3) - 33/125*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.222838, size = 100, normalized size = 1.06 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (12 \, x^{2} - 17 \, x - 11\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 33 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{400 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/400*sqrt(10)*(2*sqrt(10)*(12*x^2 - 17*x - 11)*sqrt(5*x + 3)*sqrt(-2*x + 1) -
33*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(5
*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.267344, size = 150, normalized size = 1.6 \[ -\frac{1}{2500} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 157 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{33}{200} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1250 \, \sqrt{5 \, x + 3}} + \frac{22 \, \sqrt{10} \sqrt{5 \, x + 3}}{625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2500*(12*sqrt(5)*(5*x + 3) - 157*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 33/
200*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 11/1250*sqrt(10)*(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))