3.2483 \(\int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{9}{50} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{2 \sqrt{1-2 x}}{275 \sqrt{5 x+3}}+\frac{123 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(275*Sqrt[3 + 5*x]) - (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/50 + (1
23*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sqrt[10])

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Rubi [A]  time = 0.098453, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{9}{50} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{2 \sqrt{1-2 x}}{275 \sqrt{5 x+3}}+\frac{123 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(275*Sqrt[3 + 5*x]) - (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/50 + (1
23*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sqrt[10])

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Rubi in Sympy [A]  time = 7.86429, size = 65, normalized size = 0.9 \[ - \frac{9 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{50} - \frac{2 \sqrt{- 2 x + 1}}{275 \sqrt{5 x + 3}} + \frac{123 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{500} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*sqrt(-2*x + 1)*sqrt(5*x + 3)/50 - 2*sqrt(-2*x + 1)/(275*sqrt(5*x + 3)) + 123*
sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/500

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Mathematica [A]  time = 0.122176, size = 55, normalized size = 0.76 \[ -\frac{\sqrt{1-2 x} (495 x+301)}{550 \sqrt{5 x+3}}-\frac{123 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(301 + 495*x))/(550*Sqrt[3 + 5*x]) - (123*ArcSin[Sqrt[5/11]*Sqrt
[1 - 2*x]])/(50*Sqrt[10])

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Maple [A]  time = 0.018, size = 82, normalized size = 1.1 \[{\frac{1}{11000} \left ( 6765\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+4059\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -9900\,x\sqrt{-10\,{x}^{2}-x+3}-6020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11000*(6765*10^(1/2)*arcsin(20/11*x+1/11)*x+4059*10^(1/2)*arcsin(20/11*x+1/11)
-9900*x*(-10*x^2-x+3)^(1/2)-6020*(-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 = 1.50235, size = 68, normalized size = 0.94 \[ \frac{123}{1000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{9}{50} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{275 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

123/1000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/50*sqrt(-10*x^2 - x + 3) - 2
/275*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.228156, size = 93, normalized size = 1.29 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (495 \, x + 301\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1353 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{11000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/11000*sqrt(10)*(2*sqrt(10)*(495*x + 301)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1353*
(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. \[ \int \frac{\left (3 x + 2\right )^{2}}{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.242901, size = 132, normalized size = 1.83 \[ -\frac{9}{250} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{123}{500} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2750 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{1375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/250*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 123/500*sqrt(10)*arcsin(1/11*sqrt
(22)*sqrt(5*x + 3)) - 1/2750*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(
5*x + 3) + 2/1375*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))