3.2569 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=115 \[ \frac{850 \sqrt{5 x+3}}{11319 \sqrt{1-2 x}}-\frac{5 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac{75 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

[Out]

(850*Sqrt[3 + 5*x])/(11319*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2
)*(2 + 3*x)) - (5*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)) - (75*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.243769, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{850 \sqrt{5 x+3}}{11319 \sqrt{1-2 x}}-\frac{5 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac{75 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(850*Sqrt[3 + 5*x])/(11319*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2
)*(2 + 3*x)) - (5*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)) - (75*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.4389, size = 104, normalized size = 0.9 \[ - \frac{75 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2401} + \frac{850 \sqrt{5 x + 3}}{11319 \sqrt{- 2 x + 1}} - \frac{5 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{2 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-75*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2401 + 850*sqrt(5*x +
3)/(11319*sqrt(-2*x + 1)) - 5*sqrt(5*x + 3)/(49*sqrt(-2*x + 1)*(3*x + 2)) + 2*s
qrt(5*x + 3)/(21*(-2*x + 1)**(3/2)*(3*x + 2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0956743, size = 77, normalized size = 0.67 \[ \frac{\sqrt{5 x+3} \left (-5100 x^2+1460 x+1623\right )}{11319 (1-2 x)^{3/2} (3 x+2)}-\frac{75 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{686 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 5*x]*(1623 + 1460*x - 5100*x^2))/(11319*(1 - 2*x)^(3/2)*(2 + 3*x)) - (
75*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 = 1.8 \[{\frac{1}{ \left ( 316932+475398\,x \right ) \left ( -1+2\,x \right ) ^{2}} \left ( 29700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-9900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-12375\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-71400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4950\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +20440\,x\sqrt{-10\,{x}^{2}-x+3}+22722\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^2,x)

[Out]

1/158466*(29700*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-9
900*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-12375*7^(1/2)
*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-71400*x^2*(-10*x^2-x+3)^(1
/2)+4950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+20440*x*(-10
*x^2-x+3)^(1/2)+22722*(-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 = 1.51058, size = 163, normalized size = 1.42 \[ \frac{75}{4802} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{4250 \, x}{11319 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{625}{11319 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{100 \, x}{147 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{63 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{215}{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75/4802*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 4250/11319*x
/sqrt(-10*x^2 - x + 3) + 625/11319/sqrt(-10*x^2 - x + 3) + 100/147*x/(-10*x^2 -
x + 3)^(3/2) - 1/63/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) +
215/441/(-10*x^2 - x + 3)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.232698, size = 127, normalized size = 1.1 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (5100 \, x^{2} - 1460 \, x - 1623\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2475 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{158466 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/158466*sqrt(7)*(2*sqrt(7)*(5100*x^2 - 1460*x - 1623)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) - 2475*(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(-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)**2,x)

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.331367, size = 313, normalized size = 2.72 \[ \frac{15}{9604} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{198 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{343 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} - \frac{8 \,{\left (163 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1089 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{282975 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/9604*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
)))) - 198/343*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) - 8/282975*(163*sqrt(5)*(5*x + 3) - 1089*sqrt(5))*sqrt(5*x + 3)*sqrt(-
10*x + 5)/(2*x - 1)^2