3.2547 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{1229 \sqrt{1-2 x}}{1210 \sqrt{5 x+3}}+\frac{49}{22 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{9 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]

[Out]

49/(22*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (1229*Sqrt[1 - 2*x])/(1210*Sqrt[3 + 5*x])
- (9*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])

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Rubi [A]  time = 0.0976486, 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 \[ -\frac{1229 \sqrt{1-2 x}}{1210 \sqrt{5 x+3}}+\frac{49}{22 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{9 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

49/(22*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (1229*Sqrt[1 - 2*x])/(1210*Sqrt[3 + 5*x])
- (9*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])

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Rubi in Sympy [A]  time = 8.65728, size = 65, normalized size = 0.9 \[ - \frac{9 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{50} + \frac{1229 \sqrt{5 x + 3}}{3025 \sqrt{- 2 x + 1}} - \frac{2}{275 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/50 + 1229*sqrt(5*x + 3)/(3025*sqrt(-
2*x + 1)) - 2/(275*sqrt(-2*x + 1)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.13626, size = 55, normalized size = 0.76 \[ \frac{1229 x+733}{605 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{9 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(733 + 1229*x)/(605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (9*ArcSin[Sqrt[5/11]*Sqrt[1 -
2*x]])/(5*Sqrt[10])

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Maple [A]  time = 0.02, size = 103, normalized size = 1.4 \[ -{\frac{1}{-12100+24200\,x}\sqrt{1-2\,x} \left ( 10890\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+1089\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3267\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +24580\,x\sqrt{-10\,{x}^{2}-x+3}+14660\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12100*(1-2*x)^(1/2)*(10890*10^(1/2)*arcsin(20/11*x+1/11)*x^2+1089*10^(1/2)*ar
csin(20/11*x+1/11)*x-3267*10^(1/2)*arcsin(20/11*x+1/11)+24580*x*(-10*x^2-x+3)^(1
/2)+14660*(-10*x^2-x+3)^(1/2))/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.4971, size = 55, normalized size = 0.76 \[ \frac{9}{100} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1229 \, x}{605 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{733}{605 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/100*sqrt(10)*arcsin(-20/11*x - 1/11) + 1229/605*x/sqrt(-10*x^2 - x + 3) + 733/
605/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.229446, size = 101, normalized size = 1.4 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (1229 \, x + 733\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1089 \,{\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{12100 \,{\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12100*sqrt(10)*(2*sqrt(10)*(1229*x + 733)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 1089
*(10*x^2 + x - 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))
))/(10*x^2 + x - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.246186, size = 142, normalized size = 1.97 \[ -\frac{9}{50} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{6050 \, \sqrt{5 \, x + 3}} - \frac{49 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{605 \,{\left (2 \, x - 1\right )}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{3025 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/50*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/6050*sqrt(10)*(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 49/605*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*
x + 5)/(2*x - 1) + 2/3025*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))