3.2249 \(\int \sqrt{1-2 x} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=72 \[ -\frac{1}{4} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{11}{40} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{121 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40 \sqrt{10}} \]

[Out]

(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/40 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4 + (121*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40*Sqrt[10])

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Rubi [A]  time = 0.0628127, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{4} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{11}{40} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{121 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/40 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4 + (121*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40*Sqrt[10])

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Rubi in Sympy [A]  time = 6.47012, size = 63, normalized size = 0.88 \[ \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{10} - \frac{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{40} + \frac{121 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{400} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(5*x + 3)**(3/2)/10 - 11*sqrt(-2*x + 1)*sqrt(5*x + 3)/40 + 121*sq
rt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/400

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Mathematica [A]  time = 0.0419034, size = 55, normalized size = 0.76 \[ \frac{1}{400} \left (10 \sqrt{1-2 x} \sqrt{5 x+3} (20 x+1)-121 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1 + 20*x) - 121*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt
[1 - 2*x]])/400

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Maple [A]  time = 0.008, size = 72, normalized size = 1. \[{\frac{1}{10} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{11}{40}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{121\,\sqrt{10}}{800}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(3+5*x)^(3/2)*(1-2*x)^(1/2)-11/40*(1-2*x)^(1/2)*(3+5*x)^(1/2)+121/800*((1-2
*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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Maxima [A]  time = 1.50114, size = 55, normalized size = 0.76 \[ \frac{1}{2} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{121}{800} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1}{40} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-10*x^2 - x + 3)*x - 121/800*sqrt(10)*arcsin(-20/11*x - 1/11) + 1/40*sq
rt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.214407, size = 77, normalized size = 1.07 \[ \frac{1}{800} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 121 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/800*sqrt(10)*(2*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 121*arctan(
1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 4.33179, size = 184, normalized size = 2.56 \[ \begin{cases} \frac{5 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{10 x - 5}} - \frac{33 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4 \sqrt{10 x - 5}} + \frac{121 i \sqrt{x + \frac{3}{5}}}{40 \sqrt{10 x - 5}} - \frac{121 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{400} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{121 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{400} - \frac{5 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{- 10 x + 5}} + \frac{33 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4 \sqrt{- 10 x + 5}} - \frac{121 \sqrt{x + \frac{3}{5}}}{40 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((5*I*(x + 3/5)**(5/2)/sqrt(10*x - 5) - 33*I*(x + 3/5)**(3/2)/(4*sqrt(1
0*x - 5)) + 121*I*sqrt(x + 3/5)/(40*sqrt(10*x - 5)) - 121*sqrt(10)*I*acosh(sqrt(
110)*sqrt(x + 3/5)/11)/400, 10*Abs(x + 3/5)/11 > 1), (121*sqrt(10)*asin(sqrt(110
)*sqrt(x + 3/5)/11)/400 - 5*(x + 3/5)**(5/2)/sqrt(-10*x + 5) + 33*(x + 3/5)**(3/
2)/(4*sqrt(-10*x + 5)) - 121*sqrt(x + 3/5)/(40*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.244136, size = 61, normalized size = 0.85 \[ \frac{1}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/400*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1
/11*sqrt(22)*sqrt(5*x + 3)))