3.2587 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=122 \[ \frac{2 (5 x+3)^{5/2}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac{10 (5 x+3)^{3/2}}{147 \sqrt{1-2 x} (3 x+2)}-\frac{5 \sqrt{1-2 x} \sqrt{5 x+3}}{343 (3 x+2)}-\frac{55 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

[Out]

(-5*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*(2 + 3*x)) - (10*(3 + 5*x)^(3/2))/(147*Sqr
t[1 - 2*x]*(2 + 3*x)) + (2*(3 + 5*x)^(5/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (55
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.174053, antiderivative size = 122, 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 \[ \frac{2 (5 x+3)^{5/2}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac{10 (5 x+3)^{3/2}}{147 \sqrt{1-2 x} (3 x+2)}-\frac{5 \sqrt{1-2 x} \sqrt{5 x+3}}{343 (3 x+2)}-\frac{55 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*(2 + 3*x)) - (10*(3 + 5*x)^(3/2))/(147*Sqr
t[1 - 2*x]*(2 + 3*x)) + (2*(3 + 5*x)^(5/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (55
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.0778, size = 97, normalized size = 0.8 \[ - \frac{55 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2401} - \frac{55 \sqrt{5 x + 3}}{343 \sqrt{- 2 x + 1}} + \frac{55 \left (5 x + 3\right )^{\frac{3}{2}}}{147 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{\left (5 x + 3\right )^{\frac{5}{2}}}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-55*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2401 - 55*sqrt(5*x +
3)/(343*sqrt(-2*x + 1)) + 55*(5*x + 3)**(3/2)/(147*(-2*x + 1)**(3/2)) - (5*x + 3
)**(5/2)/(7*(-2*x + 1)**(3/2)*(3*x + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.105553, size = 77, normalized size = 0.63 \[ \frac{\sqrt{5 x+3} \left (3090 x^2+3070 x+657\right )}{1029 (1-2 x)^{3/2} (3 x+2)}-\frac{55 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{686 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 5*x]*(657 + 3070*x + 3090*x^2))/(1029*(1 - 2*x)^(3/2)*(2 + 3*x)) - (55
*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(686*Sqrt[7])

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 209, normalized size = 1.7 \[{\frac{1}{ \left ( 28812+43218\,x \right ) \left ( -1+2\,x \right ) ^{2}} \left ( 1980\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-660\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-825\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+43260\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+330\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +42980\,x\sqrt{-10\,{x}^{2}-x+3}+9198\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14406*(1980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-660
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-825*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+43260*x^2*(-10*x^2-x+3)^(1/2)+3
30*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+42980*x*(-10*x^2-x
+3)^(1/2)+9198*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)/(-1+2*x)
^2/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50056, size = 186, normalized size = 1.52 \[ \frac{55}{4802} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2575 \, x}{1029 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{625 \, x^{2}}{18 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{135}{1372 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{138125 \, x}{5292 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{567 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{50315}{15876 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

55/4802*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2575/1029*x/
sqrt(-10*x^2 - x + 3) + 625/18*x^2/(-10*x^2 - x + 3)^(3/2) - 135/1372/sqrt(-10*x
^2 - x + 3) + 138125/5292*x/(-10*x^2 - x + 3)^(3/2) - 1/567/(3*(-10*x^2 - x + 3)
^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 50315/15876/(-10*x^2 - x + 3)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.222309, size = 127, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (3090 \, x^{2} + 3070 \, x + 657\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 165 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{14406 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14406*sqrt(7)*(2*sqrt(7)*(3090*x^2 + 3070*x + 657)*sqrt(5*x + 3)*sqrt(-2*x + 1
) + 165*(12*x^3 - 4*x^2 - 5*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3
)*sqrt(-2*x + 1))))/(12*x^3 - 4*x^2 - 5*x + 2)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.347802, size = 313, normalized size = 2.57 \[ \frac{11}{9604} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{22 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{343 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} + \frac{22 \,{\left (47 \, \sqrt{5}{\left (5 \, x + 3\right )} - 66 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{25725 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/9604*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
)))) - 22/343*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s
qrt(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) + 22/25725*(47*sqrt(5)*(5*x + 3) - 66*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x
+ 5)/(2*x - 1)^2