3.2517 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)} \, dx\)

Optimal. Leaf size=86 \[ \frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}-\frac{5}{3} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21 \sqrt{7}} \]

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) - (5*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/3 - (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21*Sqrt[7])

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Rubi [A]  time = 0.177204, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x}}-\frac{5}{3} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]) - (5*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/3 - (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21*Sqrt[7])

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Rubi in Sympy [A]  time = 15.9025, size = 78, normalized size = 0.91 \[ - \frac{5 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{6} - \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{147} + \frac{11 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/6 - 2*sqrt(7)*atan(sqrt(7)*sqrt(-2*x
+ 1)/(7*sqrt(5*x + 3)))/147 + 11*sqrt(5*x + 3)/(7*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.238029, size = 104, normalized size = 1.21 \[ \frac{11 \sqrt{1-2 x} \sqrt{5 x+3}}{7-14 x}-\frac{\tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{21 \sqrt{7}}-\frac{5}{6} \sqrt{\frac{5}{2}} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7 - 14*x) - ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14
*x]*Sqrt[3 + 5*x])]/(21*Sqrt[7]) - (5*Sqrt[5/2]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*
x]*Sqrt[30 + 50*x])])/6

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Maple [B]  time = 0.017, size = 131, normalized size = 1.5 \[ -{\frac{1}{-588+1176\,x} \left ( 490\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-8\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-245\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +4\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +924\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/588*(490*10^(1/2)*arcsin(20/11*x+1/11)*x-8*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x-245*10^(1/2)*arcsin(20/11*x+1/11)+4*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+924*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)
*(3+5*x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49756, size = 93, normalized size = 1.08 \[ -\frac{5}{12} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1}{147} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{55 \, x}{7 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{33}{7 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/12*sqrt(10)*arcsin(20/11*x + 1/11) + 1/147*sqrt(7)*arcsin(37/11*x/abs(3*x + 2
) + 20/11/abs(3*x + 2)) + 55/7*x/sqrt(-10*x^2 - x + 3) + 33/7/sqrt(-10*x^2 - x +
3)

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Fricas [A]  time = 0.233892, size = 154, normalized size = 1.79 \[ -\frac{\sqrt{7} \sqrt{2}{\left (35 \, \sqrt{7} \sqrt{5}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 2 \, \sqrt{2}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 66 \, \sqrt{7} \sqrt{2} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{588 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/588*sqrt(7)*sqrt(2)*(35*sqrt(7)*sqrt(5)*(2*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)
*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 2*sqrt(2)*(2*x - 1)*arctan(1/14*sq
rt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 66*sqrt(7)*sqrt(2)*sqrt(5*x
+ 3)*sqrt(-2*x + 1))/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.266378, size = 225, normalized size = 2.62 \[ \frac{1}{1470} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{5}{12} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{35 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1470*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
))) - 5/12*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 11/35*sqrt
(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)