3.2436 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}-\frac{55 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac{385 \sqrt{1-2 x}}{\sqrt{5 x+3}}-385 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-55*(1 - 2*x)^(3/2))/(3*(3 + 5*x)^(3/2)) + (1 - 2*x)^(5/2)/((2 + 3*x)*(3 + 5*x)
^(3/2)) + (385*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] - 385*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi [A]  time = 0.171903, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}-\frac{55 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac{385 \sqrt{1-2 x}}{\sqrt{5 x+3}}-385 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-55*(1 - 2*x)^(3/2))/(3*(3 + 5*x)^(3/2)) + (1 - 2*x)^(5/2)/((2 + 3*x)*(3 + 5*x)
^(3/2)) + (385*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] - 385*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi in Sympy [A]  time = 13.9196, size = 105, normalized size = 1.04 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{3 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{70 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{245 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3 x + 2} - 385 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)/(3*(3*x + 2)*(5*x + 3)**(3/2)) + 70*(-2*x + 1)**(3/2)/(3*(3
*x + 2)*sqrt(5*x + 3)) + 245*sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*x + 2) - 385*sqrt(7
)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))

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Mathematica [A]  time = 0.0905577, size = 77, normalized size = 0.76 \[ \frac{\sqrt{1-2 x} \left (17667 x^2+21988 x+6823\right )}{3 (3 x+2) (5 x+3)^{3/2}}-\frac{385}{2} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(6823 + 21988*x + 17667*x^2))/(3*(2 + 3*x)*(3 + 5*x)^(3/2)) - (38
5*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 = 202, normalized size = 2. \[{\frac{1}{12+18\,x} \left ( 86625\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+161700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+100485\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+35334\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+20790\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +43976\,x\sqrt{-10\,{x}^{2}-x+3}+13646\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(86625*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+161700
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+100485*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+35334*x^2*(-10*x^2-x+3)^(1/2
)+20790*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+43976*x*(-10*
x^2-x+3)^(1/2)+13646*(-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)^(3/2)

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Maxima [A]  time = 1.51487, size = 186, normalized size = 1.84 \[ \frac{385}{2} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{3926 \, x}{5 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{16 \, x^{2}}{45 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{30743}{75 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{133642 \, x}{675 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{81 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{217433}{2025 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

385/2*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 3926/5*x/sqrt(
-10*x^2 - x + 3) - 16/45*x^2/(-10*x^2 - x + 3)^(3/2) + 30743/75/sqrt(-10*x^2 - x
+ 3) + 133642/675*x/(-10*x^2 - x + 3)^(3/2) + 2401/81/(3*(-10*x^2 - x + 3)^(3/2
)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 217433/2025/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.22135, size = 123, normalized size = 1.22 \[ \frac{1155 \, \sqrt{7}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \,{\left (17667 \, x^{2} + 21988 \, x + 6823\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{6 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(1155*sqrt(7)*(75*x^3 + 140*x^2 + 87*x + 18)*arctan(1/14*sqrt(7)*(37*x + 20)
/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 2*(17667*x^2 + 21988*x + 6823)*sqrt(5*x + 3)*
sqrt(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 18)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.327495, size = 423, normalized size = 4.19 \[ -\frac{11}{1200} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{77}{4} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{77}{5} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{1078 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11/1200*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)))^3 + 77/4*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)))) + 77/5*sqrt(10)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22))) + 1078*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) - sq
rt(22)))^2 + 280)