3.2295 \(\int \frac{\sqrt{1-2 x} (2+3 x)}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{2 (1-2 x)^{3/2}}{55 \sqrt{5 x+3}}+\frac{29}{275} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{29 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(3/2))/(55*Sqrt[3 + 5*x]) + (29*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/275 +
(29*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25*Sqrt[10])

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Rubi [A]  time = 0.0759131, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (1-2 x)^{3/2}}{55 \sqrt{5 x+3}}+\frac{29}{275} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{29 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2))/(55*Sqrt[3 + 5*x]) + (29*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/275 +
(29*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25*Sqrt[10])

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Rubi in Sympy [A]  time = 7.12049, size = 65, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{55 \sqrt{5 x + 3}} + \frac{29 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{275} + \frac{29 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{250} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(3/2)/(55*sqrt(5*x + 3)) + 29*sqrt(-2*x + 1)*sqrt(5*x + 3)/275 +
29*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/250

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Mathematica [A]  time = 0.101911, size = 55, normalized size = 0.76 \[ \frac{\sqrt{1-2 x} (15 x+7)}{25 \sqrt{5 x+3}}-\frac{29 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(7 + 15*x))/(25*Sqrt[3 + 5*x]) - (29*ArcSin[Sqrt[5/11]*Sqrt[1 - 2
*x]])/(25*Sqrt[10])

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Maple [A]  time = 0.015, size = 82, normalized size = 1.1 \[{\frac{1}{500} \left ( 145\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+87\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +300\,x\sqrt{-10\,{x}^{2}-x+3}+140\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/500*(145*10^(1/2)*arcsin(20/11*x+1/11)*x+87*10^(1/2)*arcsin(20/11*x+1/11)+300*
x*(-10*x^2-x+3)^(1/2)+140*(-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 = 1.50721, size = 68, normalized size = 0.94 \[ \frac{29}{500} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{25} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

29/500*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 3/25*sqrt(-10*x^2 - x + 3) - 2/2
5*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.218924, size = 93, normalized size = 1.29 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (15 \, x + 7\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 29 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{500 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/500*sqrt(10)*(2*sqrt(10)*(15*x + 7)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 29*(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. \[ \int \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.249004, size = 132, normalized size = 1.83 \[ \frac{3}{125} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{29}{250} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/125*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 29/250*sqrt(10)*arcsin(1/11*sqrt(2
2)*sqrt(5*x + 3)) - 1/250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x
+ 3) + 2/125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))