3.2444 \(\int \frac{(2+3 x) \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{3}{20} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{107}{80} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1177 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80 \sqrt{10}} \]

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/80 - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/20 + (
1177*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10])

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Rubi [A]  time = 0.0717162, 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 \[ -\frac{3}{20} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{107}{80} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1177 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{80 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/80 - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/20 + (
1177*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10])

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Rubi in Sympy [A]  time = 6.88674, size = 65, normalized size = 0.9 \[ - \frac{3 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{20} - \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{80} + \frac{1177 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{800} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/20 - 107*sqrt(-2*x + 1)*sqrt(5*x + 3)/80 + 11
77*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/800

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Mathematica [A]  time = 0.0496415, size = 55, normalized size = 0.76 \[ \frac{1}{800} \left (-10 \sqrt{1-2 x} \sqrt{5 x+3} (60 x+143)-1177 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(143 + 60*x) - 1177*Sqrt[10]*ArcSin[Sqrt[5/11]*
Sqrt[1 - 2*x]])/800

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Maple [A]  time = 0.013, size = 70, normalized size = 1. \[{\frac{1}{1600}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1177\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1200\,x\sqrt{-10\,{x}^{2}-x+3}-2860\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1600*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(1177*10^(1/2)*arcsin(20/11*x+1/11)-1200*x*(-
10*x^2-x+3)^(1/2)-2860*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49729, size = 59, normalized size = 0.82 \[ \frac{1177}{1600} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{3}{4} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{143}{80} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1177/1600*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 3/4*sqrt(-10*x^2 - x + 3)*x -
143/80*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.215348, size = 77, normalized size = 1.07 \[ -\frac{1}{1600} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (60 \, x + 143\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1177 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1600*sqrt(10)*(2*sqrt(10)*(60*x + 143)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1177*ar
ctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 7.68106, size = 167, normalized size = 2.32 \[ \frac{2 \sqrt{5} \left (\begin{cases} \frac{11 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2}\right )}{4} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} + \frac{6 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*Piecewise((11*sqrt(2)*(-sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + asi
n(sqrt(22)*sqrt(5*x + 3)/11)/2)/4, (x >= -3/5) & (x < 1/2)))/25 + 6*sqrt(5)*Piec
ewise((121*sqrt(2)*(sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/968 - sqrt
(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (
x >= -3/5) & (x < 1/2)))/25

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GIAC/XCAS [A]  time = 0.227266, size = 61, normalized size = 0.85 \[ -\frac{1}{800} \, \sqrt{5}{\left (2 \,{\left (60 \, x + 143\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 1177 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/800*sqrt(5)*(2*(60*x + 143)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1177*sqrt(2)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3)))