3.2510 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=151 \[ \frac{565 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{5 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^2}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^3}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{7435 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(7*(2 + 3*x)^3) - (5*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)^2) + (565*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (7435*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(2744*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.293805, 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 \[ \frac{565 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{5 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^2}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^3}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{7435 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 + 5*x]/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(7*(2 + 3*x)^3) - (5*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)^2) + (565*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (7435*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(2744*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.5261, size = 136, normalized size = 0.9 \[ \frac{565 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2744 \left (3 x + 2\right )} - \frac{5 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{196 \left (3 x + 2\right )^{2}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{7 \left (3 x + 2\right )^{3}} - \frac{7435 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} + \frac{2 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(1/2)/(1-2*x)**(3/2)/(2+3*x)**4,x)

[Out]

565*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2744*(3*x + 2)) - 5*sqrt(-2*x + 1)*sqrt(5*x +
3)/(196*(3*x + 2)**2) - sqrt(-2*x + 1)*sqrt(5*x + 3)/(7*(3*x + 2)**3) - 7435*sqr
t(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/19208 + 2*sqrt(5*x + 3)/(7*s
qrt(-2*x + 1)*(3*x + 2)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.110148, size = 82, normalized size = 0.54 \[ \frac{\frac{14 \sqrt{5 x+3} \left (-10170 x^3-8055 x^2+3114 x+2512\right )}{\sqrt{1-2 x} (3 x+2)^3}-7435 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{38416} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 + 5*x]/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

((14*Sqrt[3 + 5*x]*(2512 + 3114*x - 8055*x^2 - 10170*x^3))/(Sqrt[1 - 2*x]*(2 + 3
*x)^3) - 7435*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/384
16

_______________________________________________________________________________________

Maple [B]  time = 0.022, size = 257, normalized size = 1.7 \[{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) } \left ( 401490\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+602235\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+133830\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+142380\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-148700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+112770\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-59480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -43596\,x\sqrt{-10\,{x}^{2}-x+3}-35168\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^4,x)

[Out]

1/38416*(401490*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+6
02235*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+133830*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+142380*x^3*(-10*x^2-x
+3)^(1/2)-148700*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+11
2770*x^2*(-10*x^2-x+3)^(1/2)-59480*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))-43596*x*(-10*x^2-x+3)^(1/2)-35168*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/
2)*(3+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51286, size = 285, normalized size = 1.89 \[ \frac{7435}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2825 \, x}{4116 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1145}{2744 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{63 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{23}{252 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{125}{1176 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/((3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

7435/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2825/4116
*x/sqrt(-10*x^2 - x + 3) + 1145/2744/sqrt(-10*x^2 - x + 3) + 1/63/(27*sqrt(-10*x
^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*
sqrt(-10*x^2 - x + 3)) - 23/252/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 -
x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 125/1176/(3*sqrt(-10*x^2 - x + 3)*x + 2*s
qrt(-10*x^2 - x + 3))

_______________________________________________________________________________________

Fricas [A]  time = 0.235589, size = 147, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (10170 \, x^{3} + 8055 \, x^{2} - 3114 \, x - 2512\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 7435 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{38416 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/((3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

1/38416*sqrt(7)*(2*sqrt(7)*(10170*x^3 + 8055*x^2 - 3114*x - 2512)*sqrt(5*x + 3)*
sqrt(-2*x + 1) + 7435*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*arctan(1/14*sqrt(7)*
(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x -
8)

_______________________________________________________________________________________

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)**(3/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.468808, size = 464, normalized size = 3.07 \[ \frac{1487}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{16 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{12005 \,{\left (2 \, x - 1\right )}} - \frac{99 \,{\left (527 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 253120 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 36299200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9604 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)/((3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

1487/76832*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)))) - 16/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 99/9604*(52
7*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 - 253120*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 - 36299200*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)^3