3.2484 \(\int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac{3}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \sqrt{1-2 x}}{55 \sqrt{5 x+3}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(55*Sqrt[3 + 5*x]) + (3*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/5

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Rubi [A]  time = 0.0576421, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{3}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \sqrt{1-2 x}}{55 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(55*Sqrt[3 + 5*x]) + (3*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/5

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Rubi in Sympy [A]  time = 5.39526, size = 44, normalized size = 0.85 \[ - \frac{2 \sqrt{- 2 x + 1}}{55 \sqrt{5 x + 3}} + \frac{3 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)/(55*sqrt(5*x + 3)) + 3*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11
)/25

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Mathematica [A]  time = 0.0744146, size = 52, normalized size = 1. \[ -\frac{2 \sqrt{1-2 x}}{55 \sqrt{5 x+3}}-\frac{3}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(55*Sqrt[3 + 5*x]) - (3*Sqrt[2/5]*ArcSin[Sqrt[5/11]*Sqrt[1 -
2*x]])/5

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Maple [A]  time = 0.017, size = 67, normalized size = 1.3 \[{\frac{1}{550} \left ( 165\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+99\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -20\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/550*(165*10^(1/2)*arcsin(20/11*x+1/11)*x+99*10^(1/2)*arcsin(20/11*x+1/11)-20*(
-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 = 1.49917, size = 49, normalized size = 0.94 \[ \frac{3}{50} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{55 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/50*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 2/55*sqrt(-10*x^2 - x + 3)/(5*x +
3)

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Fricas [A]  time = 0.229431, size = 95, normalized size = 1.83 \[ \frac{\sqrt{5}{\left (33 \, \sqrt{2}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 4 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{550 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/550*sqrt(5)*(33*sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt
(5*x + 3)*sqrt(-2*x + 1))) - 4*sqrt(5)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.243514, size = 107, normalized size = 2.06 \[ \frac{3}{25} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{550 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{275 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/25*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/550*sqrt(10)*(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/275*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))