3.2292 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^5 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=151 \[ \frac{1479375 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}+\frac{14145 \sqrt{1-2 x} \sqrt{5 x+3}}{1568 (3 x+2)^2}+\frac{81 \sqrt{1-2 x} \sqrt{5 x+3}}{56 (3 x+2)^3}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{4 (3 x+2)^4}-\frac{16925425 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4*(2 + 3*x)^4) + (81*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
/(56*(2 + 3*x)^3) + (14145*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1568*(2 + 3*x)^2) + (14
79375*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (16925425*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.296856, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{1479375 \sqrt{1-2 x} \sqrt{5 x+3}}{21952 (3 x+2)}+\frac{14145 \sqrt{1-2 x} \sqrt{5 x+3}}{1568 (3 x+2)^2}+\frac{81 \sqrt{1-2 x} \sqrt{5 x+3}}{56 (3 x+2)^3}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{4 (3 x+2)^4}-\frac{16925425 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4*(2 + 3*x)^4) + (81*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
/(56*(2 + 3*x)^3) + (14145*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1568*(2 + 3*x)^2) + (14
79375*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (16925425*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi in Sympy [A]  time = 28.4997, size = 136, normalized size = 0.9 \[ \frac{1479375 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21952 \left (3 x + 2\right )} + \frac{14145 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1568 \left (3 x + 2\right )^{2}} + \frac{81 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{56 \left (3 x + 2\right )^{3}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4 \left (3 x + 2\right )^{4}} - \frac{16925425 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{153664} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1479375*sqrt(-2*x + 1)*sqrt(5*x + 3)/(21952*(3*x + 2)) + 14145*sqrt(-2*x + 1)*sq
rt(5*x + 3)/(1568*(3*x + 2)**2) + 81*sqrt(-2*x + 1)*sqrt(5*x + 3)/(56*(3*x + 2)*
*3) + sqrt(-2*x + 1)*sqrt(5*x + 3)/(4*(3*x + 2)**4) - 16925425*sqrt(7)*atan(sqrt
(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/153664

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Mathematica [A]  time = 0.109283, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (39943125 x^3+81668520 x^2+55729116 x+12696112\right )}{(3 x+2)^4}-16925425 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{307328} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12696112 + 55729116*x + 81668520*x^2 + 3994312
5*x^3))/(2 + 3*x)^4 - 16925425*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqr
t[3 + 5*x])])/307328

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Maple [B]  time = 0.02, size = 250, normalized size = 1.7 \[{\frac{1}{307328\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1370959425\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1143359280\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/307328*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1370959425*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+3655891800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^3+3655891800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^2+559203750*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+1143359280*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))/(-10*x^2-x+3)^(1/2)/(2+3*x)
^4

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Maxima [A]  time = 1.51629, size = 193, normalized size = 1.28 \[ \frac{16925425}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{4 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{81 \, \sqrt{-10 \, x^{2} - x + 3}}{56 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{14145 \, \sqrt{-10 \, x^{2} - x + 3}}{1568 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{1479375 \, \sqrt{-10 \, x^{2} - x + 3}}{21952 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16925425/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/4*
sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 81/56*sqrt(-10*
x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 14145/1568*sqrt(-10*x^2 - x + 3)/(9*
x^2 + 12*x + 4) + 1479375/21952*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.22616, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (39943125 \, x^{3} + 81668520 \, x^{2} + 55729116 \, x + 12696112\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 16925425 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{307328 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/307328*sqrt(7)*(2*sqrt(7)*(39943125*x^3 + 81668520*x^2 + 55729116*x + 12696112
)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 16925425*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 1
6)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 21
6*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.407318, size = 504, normalized size = 3.34 \[ \frac{55}{614656} \, \sqrt{5}{\left (61547 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{280 \, \sqrt{2}{\left (157973 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 83743800 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 17691640512 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{1351079744000 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{5404318976000 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

55/614656*sqrt(5)*(61547*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)))) + 280*sqrt(2)*(157973*((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 + 83743
800*((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 + 17691640512*((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 +
1351079744000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 5404318976000
*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)