3.2349 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^2 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=93 \[ \frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)}+\frac{4}{9} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{29}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)) + (4*Sqrt[2/5]*ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]])/9 - (29*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi [A]  time = 0.173728, antiderivative size = 93, 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 \[ \frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)}+\frac{4}{9} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{29}{9} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)) + (4*Sqrt[2/5]*ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]])/9 - (29*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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Rubi in Sympy [A]  time = 15.5386, size = 82, normalized size = 0.88 \[ \frac{7 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )} + \frac{4 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{45} - \frac{29 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*(3*x + 2)) + 4*sqrt(10)*asin(sqrt(22)*sqrt(5*x
+ 3)/11)/45 - 29*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/9

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Mathematica [A]  time = 0.14447, size = 104, normalized size = 1.12 \[ \frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{9 x+6}-\frac{29}{18} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+\frac{2}{9} \sqrt{\frac{2}{5}} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6 + 9*x) - (29*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*S
qrt[7 - 14*x]*Sqrt[3 + 5*x])])/18 + (2*Sqrt[2/5]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2
*x]*Sqrt[30 + 50*x])])/9

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Maple [A]  time = 0.02, size = 131, normalized size = 1.4 \[{\frac{1}{180+270\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 435\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+12\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+290\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +8\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +210\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(435*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))*x+12*10^(1/2)*arcsin(20/11*x+1/11)*x+290*7^(1/2)*arctan(1/14*(3
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+8*10^(1/2)*arcsin(20/11*x+1/11)+210*(-10*x^
2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.49578, size = 82, normalized size = 0.88 \[ \frac{2}{45} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{29}{18} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{3 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*sqrt(10)*arcsin(20/11*x + 1/11) + 29/18*sqrt(7)*arcsin(37/11*x/abs(3*x + 2)
+ 20/11/abs(3*x + 2)) + 7/3*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.234941, size = 146, normalized size = 1.57 \[ \frac{\sqrt{5}{\left (29 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 4 \, \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 42 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{90 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90*sqrt(5)*(29*sqrt(7)*sqrt(5)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt
(5*x + 3)*sqrt(-2*x + 1))) + 4*sqrt(2)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20
*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 42*sqrt(5)*sqrt(5*x + 3)*sqrt(-2*x + 1
))/(3*x + 2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.289642, size = 351, normalized size = 3.77 \[ \frac{29}{180} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{2}{45} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{154 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{3 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

29/180*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)
))) + 2/45*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)))) + 154/3*sqrt
(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt
(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x
+ 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)