3.2362 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{63678595 \sqrt{1-2 x}}{9408 \sqrt{5 x+3}}+\frac{1403963 \sqrt{1-2 x}}{3136 (3 x+2) \sqrt{5 x+3}}+\frac{8063 \sqrt{1-2 x}}{224 (3 x+2)^2 \sqrt{5 x+3}}+\frac{33 \sqrt{1-2 x}}{8 (3 x+2)^3 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4 \sqrt{5 x+3}}+\frac{145708761 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

[Out]

(-63678595*Sqrt[1 - 2*x])/(9408*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)
^4*Sqrt[3 + 5*x]) + (33*Sqrt[1 - 2*x])/(8*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (8063*Sqr
t[1 - 2*x])/(224*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (1403963*Sqrt[1 - 2*x])/(3136*(2 +
3*x)*Sqrt[3 + 5*x]) + (145708761*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
/(3136*Sqrt[7])

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Rubi [A]  time = 0.404005, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{63678595 \sqrt{1-2 x}}{9408 \sqrt{5 x+3}}+\frac{1403963 \sqrt{1-2 x}}{3136 (3 x+2) \sqrt{5 x+3}}+\frac{8063 \sqrt{1-2 x}}{224 (3 x+2)^2 \sqrt{5 x+3}}+\frac{33 \sqrt{1-2 x}}{8 (3 x+2)^3 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4 \sqrt{5 x+3}}+\frac{145708761 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-63678595*Sqrt[1 - 2*x])/(9408*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)
^4*Sqrt[3 + 5*x]) + (33*Sqrt[1 - 2*x])/(8*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (8063*Sqr
t[1 - 2*x])/(224*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (1403963*Sqrt[1 - 2*x])/(3136*(2 +
3*x)*Sqrt[3 + 5*x]) + (145708761*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
/(3136*Sqrt[7])

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Rubi in Sympy [A]  time = 35.5762, size = 160, normalized size = 0.92 \[ - \frac{63678595 \sqrt{- 2 x + 1}}{9408 \sqrt{5 x + 3}} + \frac{1403963 \sqrt{- 2 x + 1}}{3136 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{8063 \sqrt{- 2 x + 1}}{224 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{33 \sqrt{- 2 x + 1}}{8 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} + \frac{7 \sqrt{- 2 x + 1}}{12 \left (3 x + 2\right )^{4} \sqrt{5 x + 3}} + \frac{145708761 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{21952} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-63678595*sqrt(-2*x + 1)/(9408*sqrt(5*x + 3)) + 1403963*sqrt(-2*x + 1)/(3136*(3*
x + 2)*sqrt(5*x + 3)) + 8063*sqrt(-2*x + 1)/(224*(3*x + 2)**2*sqrt(5*x + 3)) + 3
3*sqrt(-2*x + 1)/(8*(3*x + 2)**3*sqrt(5*x + 3)) + 7*sqrt(-2*x + 1)/(12*(3*x + 2)
**4*sqrt(5*x + 3)) + 145708761*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x +
3)))/21952

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Mathematica [A]  time = 0.134953, size = 87, normalized size = 0.5 \[ \frac{145708761 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{14 \sqrt{1-2 x} \left (1719322065 x^4+4546951839 x^3+4508028900 x^2+1985778980 x+327908240\right )}{(3 x+2)^4 \sqrt{5 x+3}}}{43904} \]

Antiderivative was successfully verified.

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

[Out]

((-14*Sqrt[1 - 2*x]*(327908240 + 1985778980*x + 4508028900*x^2 + 4546951839*x^3
+ 1719322065*x^4))/((2 + 3*x)^4*Sqrt[3 + 5*x]) + 145708761*Sqrt[7]*ArcTan[(-20 -
37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/43904

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Maple [B]  time = 0.023, size = 298, normalized size = 1.7 \[ -{\frac{1}{43904\, \left ( 2+3\,x \right ) ^{4}} \left ( 59012048205\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+63112404600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/43904*(59012048205*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x^5+192772690803*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4
[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+240
70508910*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^2+63657325746*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+63112404600*x^2*(-10*x
^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1
/2)/(2+3*x)^4/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.52125, size = 400, normalized size = 2.31 \[ -\frac{145708761}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{63678595 \, x}{4704 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{66486521}{9408 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{49}{36 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{665}{72 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{7799}{96 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{457237}{448 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

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)^5),x, algorithm="maxima")

[Out]

-145708761/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 636
78595/4704*x/sqrt(-10*x^2 - x + 3) - 66486521/9408/sqrt(-10*x^2 - x + 3) + 49/36
/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(-10*x^
2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 665/72
/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2
- x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 7799/96/(9*sqrt(-10*x^2 - x + 3)*x^2 + 1
2*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 457237/448/(3*sqrt(-10*x^
2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.22261, size = 167, normalized size = 0.97 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1719322065 \, x^{4} + 4546951839 \, x^{3} + 4508028900 \, x^{2} + 1985778980 \, x + 327908240\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 145708761 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{43904 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

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)^5),x, algorithm="fricas")

[Out]

-1/43904*sqrt(7)*(2*sqrt(7)*(1719322065*x^4 + 4546951839*x^3 + 4508028900*x^2 +
1985778980*x + 327908240)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 145708761*(405*x^5 + 13
23*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt
(5*x + 3)*sqrt(-2*x + 1))))/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x +
48)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.422662, size = 591, normalized size = 3.42 \[ -\frac{145708761}{439040} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{275}{2} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{11 \,{\left (13252949 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 8830442040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 2086818820800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 170309125952000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1568 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

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)^5),x, algorithm="giac")

[Out]

-145708761/439040*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)))) - 275/2*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))) - 11/1568*(13252949*
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)))^7 + 8830442040*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) - sq
rt(22)))^5 + 2086818820800*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 + 170309125952
000*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)^
4