3.2286 \(\int \frac{\sqrt{1-2 x} (2+3 x)}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{3}{20} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{41}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 - (3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/20 + (4
51*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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Rubi [A]  time = 0.0729696, 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 \[ -\frac{3}{20} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{41}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 - (3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/20 + (4
51*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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Rubi in Sympy [A]  time = 6.66826, size = 65, normalized size = 0.9 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{20} + \frac{41 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{200} + \frac{451 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/20 + 41*sqrt(-2*x + 1)*sqrt(5*x + 3)/200 + 45
1*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/2000

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Mathematica [A]  time = 0.053352, size = 55, normalized size = 0.76 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} (60 x+11)-451 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(11 + 60*x) - 451*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqr
t[1 - 2*x]])/2000

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Maple [A]  time = 0.013, size = 70, normalized size = 1. \[{\frac{1}{4000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 451\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1200\,x\sqrt{-10\,{x}^{2}-x+3}+220\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(451*10^(1/2)*arcsin(20/11*x+1/11)+1200*x*(-1
0*x^2-x+3)^(1/2)+220*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49121, size = 59, normalized size = 0.82 \[ \frac{451}{4000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{10} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{11}{200} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

451/4000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 3/10*sqrt(-10*x^2 - x + 3)*x +
11/200*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.222086, size = 77, normalized size = 1.07 \[ \frac{1}{4000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (60 \, x + 11\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 451 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4000*sqrt(10)*(2*sqrt(10)*(60*x + 11)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 451*arcta
n(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 7.79481, size = 165, normalized size = 2.29 \[ - \frac{7 \sqrt{2} \left (\begin{cases} \frac{11 \sqrt{5} \left (- \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{2}\right )}{25} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} + \frac{3 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (\frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{968} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{8}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*sqrt(2)*Piecewise((11*sqrt(5)*(-sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + as
in(sqrt(55)*sqrt(-2*x + 1)/11)/2)/25, (x <= 1/2) & (x > -3/5)))/2 + 3*sqrt(2)*Pi
ecewise((121*sqrt(5)*(sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/968 - sqr
t(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 + 3*asin(sqrt(55)*sqrt(-2*x + 1)/11)/8)/12
5, (x <= 1/2) & (x > -3/5)))/2

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GIAC/XCAS [A]  time = 0.230619, size = 116, normalized size = 1.61 \[ \frac{3}{2000} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2000*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin
(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*s
qrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))