3.2570 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=144 \[ \frac{415 \sqrt{5 x+3}}{22638 \sqrt{1-2 x}}+\frac{5 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)}-\frac{\sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{765 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

(415*Sqrt[3 + 5*x])/(22638*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2
)*(2 + 3*x)^2) - Sqrt[3 + 5*x]/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*Sqrt[3 + 5*x]
)/(196*Sqrt[1 - 2*x]*(2 + 3*x)) - (765*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.323628, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{415 \sqrt{5 x+3}}{22638 \sqrt{1-2 x}}+\frac{5 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)}-\frac{\sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{765 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(415*Sqrt[3 + 5*x])/(22638*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2
)*(2 + 3*x)^2) - Sqrt[3 + 5*x]/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*Sqrt[3 + 5*x]
)/(196*Sqrt[1 - 2*x]*(2 + 3*x)) - (765*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(1372*Sqrt[7])

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Rubi in Sympy [A]  time = 29.4351, size = 131, normalized size = 0.91 \[ - \frac{765 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} + \frac{415 \sqrt{5 x + 3}}{22638 \sqrt{- 2 x + 1}} + \frac{5 \sqrt{5 x + 3}}{196 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{\sqrt{5 x + 3}}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{2 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-765*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/9604 + 415*sqrt(5*x
+ 3)/(22638*sqrt(-2*x + 1)) + 5*sqrt(5*x + 3)/(196*sqrt(-2*x + 1)*(3*x + 2)) - s
qrt(5*x + 3)/(14*sqrt(-2*x + 1)*(3*x + 2)**2) + 2*sqrt(5*x + 3)/(21*(-2*x + 1)**
(3/2)*(3*x + 2)**2)

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Mathematica [A]  time = 0.0946718, size = 85, normalized size = 0.59 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (-14940 x^3-19380 x^2+8633 x+6708\right )}{45276 \left (6 x^2+x-2\right )^2}-\frac{765 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(6708 + 8633*x - 19380*x^2 - 14940*x^3))/(45276*(-2
+ x + 6*x^2)^2) - (765*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(
2744*Sqrt[7])

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Maple [B]  time = 0.021, size = 257, normalized size = 1.8 \[{\frac{1}{633864\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) ^{2}} \left ( 908820\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+302940\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-580635\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-209160\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-100980\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-271320\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+100980\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +120862\,x\sqrt{-10\,{x}^{2}-x+3}+93912\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/633864*(908820*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+
302940*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-580635*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-209160*x^3*(-10*x^2-
x+3)^(1/2)-100980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-2
71320*x^2*(-10*x^2-x+3)^(1/2)+100980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))+120862*x*(-10*x^2-x+3)^(1/2)+93912*(-10*x^2-x+3)^(1/2))*(1-2*x)^
(1/2)*(3+5*x)^(1/2)/(2+3*x)^2/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.51825, size = 232, normalized size = 1.61 \[ \frac{765}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2075 \, x}{22638 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4415}{45276 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{125 \, x}{294 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{126 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{23}{252 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{5}{1764 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

765/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2075/22638
*x/sqrt(-10*x^2 - x + 3) + 4415/45276/sqrt(-10*x^2 - x + 3) + 125/294*x/(-10*x^2
- x + 3)^(3/2) - 1/126/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3
/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 23/252/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-1
0*x^2 - x + 3)^(3/2)) - 5/1764/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.229796, size = 147, normalized size = 1.02 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (14940 \, x^{3} + 19380 \, x^{2} - 8633 \, x - 6708\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 25245 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{633864 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/633864*sqrt(7)*(2*sqrt(7)*(14940*x^3 + 19380*x^2 - 8633*x - 6708)*sqrt(5*x +
3)*sqrt(-2*x + 1) - 25245*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*arctan(1/14*sqrt(
7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x
+ 4)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.419847, size = 400, normalized size = 2.78 \[ \frac{153}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{8 \,{\left (524 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3267 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1980825 \,{\left (2 \, x - 1\right )}^{2}} - \frac{297 \,{\left (19 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 840 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4802 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

153/38416*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)))) - 8/1980825*(524*sqrt(5)*(5*x + 3) - 3267*sqrt(5))*sqrt(5*x + 3)*sqrt(-10
*x + 5)/(2*x - 1)^2 - 297/4802*(19*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 - 840*
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))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2