3.2359 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{(1-2 x)^{3/2}}{(3 x+2) \sqrt{5 x+3}}-\frac{33 \sqrt{1-2 x}}{\sqrt{5 x+3}}+33 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-33*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + (1 - 2*x)^(3/2)/((2 + 3*x)*Sqrt[3 + 5*x]) +
33*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi [A]  time = 0.12877, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(1-2 x)^{3/2}}{(3 x+2) \sqrt{5 x+3}}-\frac{33 \sqrt{1-2 x}}{\sqrt{5 x+3}}+33 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-33*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + (1 - 2*x)^(3/2)/((2 + 3*x)*Sqrt[3 + 5*x]) +
33*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi in Sympy [A]  time = 10.3629, size = 78, normalized size = 0.99 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{\left (3 x + 2\right ) \sqrt{5 x + 3}} - \frac{21 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 x + 2} + 33 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )} \]

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)**(3/2),x)

[Out]

-2*(-2*x + 1)**(3/2)/((3*x + 2)*sqrt(5*x + 3)) - 21*sqrt(-2*x + 1)*sqrt(5*x + 3)
/(3*x + 2) + 33*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))

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Mathematica [A]  time = 0.0926626, size = 70, normalized size = 0.89 \[ \frac{33}{2} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{\sqrt{1-2 x} (101 x+65)}{(3 x+2) \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[1 - 2*x]*(65 + 101*x))/((2 + 3*x)*Sqrt[3 + 5*x])) + (33*Sqrt[7]*ArcTan[(
-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/2

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Maple [B]  time = 0.02, size = 154, normalized size = 2. \[ -{\frac{1}{4+6\,x} \left ( 495\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+627\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+198\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +202\,x\sqrt{-10\,{x}^{2}-x+3}+130\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(495*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+627*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+198*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+202*x*(-10*x^2-x+3)^(1/2)+130*(-10*x^2-
x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.51331, size = 124, normalized size = 1.57 \[ -\frac{33}{2} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{202 \, x}{3 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{317}{9 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{49}{9 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33/2*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 202/3*x/sqrt(-
10*x^2 - x + 3) - 317/9/sqrt(-10*x^2 - x + 3) + 49/9/(3*sqrt(-10*x^2 - x + 3)*x
+ 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.221078, size = 103, normalized size = 1.3 \[ -\frac{33 \, \sqrt{7}{\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \,{\left (101 \, x + 65\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(33*sqrt(7)*(15*x^2 + 19*x + 6)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x +
3)*sqrt(-2*x + 1))) + 2*(101*x + 65)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(15*x^2 + 19
*x + 6)

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280477, size = 340, normalized size = 4.3 \[ -\frac{33}{20} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11}{10} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{154 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-33/20*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)
))) - 11/10*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 154*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)