3.2353 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^6 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=180 \[ \frac{87374783 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{835409 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{23909 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{293 \sqrt{1-2 x} \sqrt{5 x+3}}{120 (3 x+2)^4}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{333216939 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (293*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(120*(2 + 3*x)^4) + (23909*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1680*(2 + 3*x)^3)
+ (835409*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9408*(2 + 3*x)^2) + (87374783*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(131712*(2 + 3*x)) - (333216939*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]
*Sqrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.371039, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{87374783 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{835409 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{23909 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{293 \sqrt{1-2 x} \sqrt{5 x+3}}{120 (3 x+2)^4}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{333216939 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(3/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (293*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(120*(2 + 3*x)^4) + (23909*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1680*(2 + 3*x)^3)
+ (835409*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9408*(2 + 3*x)^2) + (87374783*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(131712*(2 + 3*x)) - (333216939*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]
*Sqrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi in Sympy [A]  time = 35.4352, size = 165, normalized size = 0.92 \[ \frac{87374783 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{131712 \left (3 x + 2\right )} + \frac{835409 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{9408 \left (3 x + 2\right )^{2}} + \frac{23909 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1680 \left (3 x + 2\right )^{3}} + \frac{293 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{120 \left (3 x + 2\right )^{4}} + \frac{7 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{5}} - \frac{333216939 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{307328} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)/(2+3*x)**6/(3+5*x)**(1/2),x)

[Out]

87374783*sqrt(-2*x + 1)*sqrt(5*x + 3)/(131712*(3*x + 2)) + 835409*sqrt(-2*x + 1)
*sqrt(5*x + 3)/(9408*(3*x + 2)**2) + 23909*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1680*(3
*x + 2)**3) + 293*sqrt(-2*x + 1)*sqrt(5*x + 3)/(120*(3*x + 2)**4) + 7*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(15*(3*x + 2)**5) - 333216939*sqrt(7)*atan(sqrt(7)*sqrt(-2*x
+ 1)/(7*sqrt(5*x + 3)))/307328

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Mathematica [A]  time = 0.155126, size = 103, normalized size = 0.57 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (436873915 (3 x+2)^4+58478630 (3 x+2)^3+9372328 (3 x+2)^2+1607984 (3 x+2)+307328\right )}{(3 x+2)^5}-4998254085 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{9219840} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(307328 + 1607984*(2 + 3*x) + 9372328*(2 + 3*x)
^2 + 58478630*(2 + 3*x)^3 + 436873915*(2 + 3*x)^4))/(2 + 3*x)^5 - 4998254085*Sqr
t[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/9219840

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Maple [B]  time = 0.023, size = 298, normalized size = 1.7 \[{\frac{1}{3073280\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 404858580885\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+455499158856\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x)^(1/2),x)

[Out]

1/3073280*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(404858580885*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+1349528602950*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+1799371470600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^3+165138339870*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+447737213700*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+455499158856*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)+349
94513344*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]  time = 1.51473, size = 248, normalized size = 1.38 \[ \frac{333216939}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{15 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{293 \, \sqrt{-10 \, x^{2} - x + 3}}{120 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{23909 \, \sqrt{-10 \, x^{2} - x + 3}}{1680 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{835409 \, \sqrt{-10 \, x^{2} - x + 3}}{9408 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{87374783 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/(sqrt(5*x + 3)*(3*x + 2)^6),x, algorithm="maxima")

[Out]

333216939/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 7/1
5*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) +
293/120*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 23909/1
680*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 835409/9408*sqrt(-10*x^
2 - x + 3)/(9*x^2 + 12*x + 4) + 87374783/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.231882, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (11795595705 \, x^{4} + 31981229550 \, x^{3} + 32535654204 \, x^{2} + 14720806136 \, x + 2499608096\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1666084695 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3073280 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/(sqrt(5*x + 3)*(3*x + 2)^6),x, algorithm="fricas")

[Out]

1/3073280*sqrt(7)*(2*sqrt(7)*(11795595705*x^4 + 31981229550*x^3 + 32535654204*x^
2 + 14720806136*x + 2499608096)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 1666084695*(243*x
^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)

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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)**6/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.462996, size = 594, normalized size = 3.3 \[ \frac{333216939}{6146560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{121 \,{\left (8222141 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 5797080240 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 1842336276480 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 282112659584000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 16926759575040000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{21952 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/(sqrt(5*x + 3)*(3*x + 2)^6),x, algorithm="giac")

[Out]

333216939/6146560*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)))) + 121/21952*(8222141*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)))^9 + 579
7080240*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 + 1842336276480*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)))^5 + 282112659584000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 +
16926759575040000*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)^5