3.2455 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)} \, dx\)

Optimal. Leaf size=86 \[ -\frac{5}{6} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{29}{18} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9 \sqrt{7}} \]

[Out]

(-5*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6 + (29*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/18 - (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(9*Sqrt[7])

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Rubi [A]  time = 0.174828, 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 \[ -\frac{5}{6} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{29}{18} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6 + (29*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/18 - (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(9*Sqrt[7])

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Rubi in Sympy [A]  time = 16.2237, size = 78, normalized size = 0.91 \[ - \frac{5 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6} + \frac{29 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{36} - \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-2*x + 1)*sqrt(5*x + 3)/6 + 29*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/
36 - 2*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/63

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Mathematica [A]  time = 0.110822, size = 95, normalized size = 1.1 \[ \frac{1}{504} \left (-420 \sqrt{1-2 x} \sqrt{5 x+3}-8 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+203 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-420*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] - 8*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14
*x]*Sqrt[3 + 5*x])] + 203*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 +
50*x])])/504

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Maple [A]  time = 0.017, size = 83, normalized size = 1. \[{\frac{1}{504}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 8\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +203\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -420\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/504*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(8*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))+203*10^(1/2)*arcsin(20/11*x+1/11)-420*(-10*x^2-x+3)^(1/2))/(-10*
x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49811, size = 73, normalized size = 0.85 \[ \frac{29}{72} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1}{63} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{5}{6} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

29/72*sqrt(10)*arcsin(20/11*x + 1/11) + 1/63*sqrt(7)*arcsin(37/11*x/abs(3*x + 2)
+ 20/11/abs(3*x + 2)) - 5/6*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.228147, size = 131, normalized size = 1.52 \[ -\frac{1}{504} \, \sqrt{7} \sqrt{2}{\left (30 \, \sqrt{7} \sqrt{2} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 29 \, \sqrt{7} \sqrt{5} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 4 \, \sqrt{2} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/504*sqrt(7)*sqrt(2)*(30*sqrt(7)*sqrt(2)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 29*sqr
t(7)*sqrt(5)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1
))) - 4*sqrt(2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.276378, size = 216, normalized size = 2.51 \[ \frac{1}{630} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{29}{72} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{6} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/630*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
)) + 29/72*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)))) - 1/6*sqrt(5
)*sqrt(5*x + 3)*sqrt(-10*x + 5)