3.2299 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{2615 \sqrt{1-2 x}}{28 \sqrt{5 x+3}}+\frac{173 \sqrt{1-2 x}}{28 (3 x+2) \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 \sqrt{5 x+3}}+\frac{17951 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

[Out]

(-2615*Sqrt[1 - 2*x])/(28*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*Sqrt[3 +
5*x]) + (173*Sqrt[1 - 2*x])/(28*(2 + 3*x)*Sqrt[3 + 5*x]) + (17951*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

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Rubi [A]  time = 0.234062, antiderivative size = 115, 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 \[ -\frac{2615 \sqrt{1-2 x}}{28 \sqrt{5 x+3}}+\frac{173 \sqrt{1-2 x}}{28 (3 x+2) \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 \sqrt{5 x+3}}+\frac{17951 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-2615*Sqrt[1 - 2*x])/(28*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*Sqrt[3 +
5*x]) + (173*Sqrt[1 - 2*x])/(28*(2 + 3*x)*Sqrt[3 + 5*x]) + (17951*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

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Rubi in Sympy [A]  time = 21.5785, size = 104, normalized size = 0.9 \[ - \frac{2615 \sqrt{- 2 x + 1}}{28 \sqrt{5 x + 3}} + \frac{173 \sqrt{- 2 x + 1}}{28 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{\sqrt{- 2 x + 1}}{2 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{17951 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{196} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2615*sqrt(-2*x + 1)/(28*sqrt(5*x + 3)) + 173*sqrt(-2*x + 1)/(28*(3*x + 2)*sqrt(
5*x + 3)) + sqrt(-2*x + 1)/(2*(3*x + 2)**2*sqrt(5*x + 3)) + 17951*sqrt(7)*atan(s
qrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/196

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Mathematica [A]  time = 0.0866831, size = 77, normalized size = 0.67 \[ \frac{17951 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{56 \sqrt{7}}-\frac{\sqrt{1-2 x} \left (23535 x^2+30861 x+10100\right )}{28 (3 x+2)^2 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(10100 + 30861*x + 23535*x^2))/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) +
(17951*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Maple [B]  time = 0.02, size = 202, normalized size = 1.8 \[ -{\frac{1}{392\, \left ( 2+3\,x \right ) ^{2}} \left ( 807795\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+1561737\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1005256\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+329490\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+215412\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +432054\,x\sqrt{-10\,{x}^{2}-x+3}+141400\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/392*(807795*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+15
61737*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1005256*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+329490*x^2*(-10*x^2-x+
3)^(1/2)+215412*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+43205
4*x*(-10*x^2-x+3)^(1/2)+141400*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^2/(-10
*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.51499, size = 193, normalized size = 1.68 \[ -\frac{17951}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2615 \, x}{14 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{8191}{84 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{7}{6 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{169}{12 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17951/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2615/14*x
/sqrt(-10*x^2 - x + 3) - 8191/84/sqrt(-10*x^2 - x + 3) + 7/6/(9*sqrt(-10*x^2 - x
+ 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 169/12/(3*sq
rt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.222791, size = 127, normalized size = 1.1 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (23535 \, x^{2} + 30861 \, x + 10100\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 17951 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{392 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/392*sqrt(7)*(2*sqrt(7)*(23535*x^2 + 30861*x + 10100)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) + 17951*(45*x^3 + 87*x^2 + 56*x + 12)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt
(5*x + 3)*sqrt(-2*x + 1))))/(45*x^3 + 87*x^2 + 56*x + 12)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293188, size = 427, normalized size = 3.71 \[ -\frac{1}{3920} \, \sqrt{5}{\left (17951 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + 9800 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{9240 \, \sqrt{2}{\left (313 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{69160 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{276640 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3920*sqrt(5)*(17951*sqrt(70)*sqrt(2)*(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)))) + 9800*sqrt(2)*((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))) + 9240*sqrt(2)*(31
3*((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)))^3 + 69160*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 276640*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)^2)