3.2571 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=173 \[ -\frac{16985 \sqrt{5 x+3}}{316932 \sqrt{1-2 x}}+\frac{605 \sqrt{5 x+3}}{2744 \sqrt{1-2 x} (3 x+2)}-\frac{\sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)^2}-\frac{3 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^3}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac{25365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

[Out]

(-16985*Sqrt[3 + 5*x])/(316932*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^
(3/2)*(2 + 3*x)^3) - (3*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^3) - Sqrt[3 +
5*x]/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (605*Sqrt[3 + 5*x])/(2744*Sqrt[1 - 2*x]*
(2 + 3*x)) - (25365*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7
])

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Rubi [A]  time = 0.406856, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{16985 \sqrt{5 x+3}}{316932 \sqrt{1-2 x}}+\frac{605 \sqrt{5 x+3}}{2744 \sqrt{1-2 x} (3 x+2)}-\frac{\sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)^2}-\frac{3 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^3}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac{25365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-16985*Sqrt[3 + 5*x])/(316932*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^
(3/2)*(2 + 3*x)^3) - (3*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^3) - Sqrt[3 +
5*x]/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (605*Sqrt[3 + 5*x])/(2744*Sqrt[1 - 2*x]*
(2 + 3*x)) - (25365*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7
])

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Rubi in Sympy [A]  time = 36.6081, size = 158, normalized size = 0.91 \[ - \frac{25365 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{134456} - \frac{16985 \sqrt{5 x + 3}}{316932 \sqrt{- 2 x + 1}} + \frac{605 \sqrt{5 x + 3}}{2744 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{\sqrt{5 x + 3}}{196 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} - \frac{3 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{2 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25365*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/134456 - 16985*sqr
t(5*x + 3)/(316932*sqrt(-2*x + 1)) + 605*sqrt(5*x + 3)/(2744*sqrt(-2*x + 1)*(3*x
+ 2)) - sqrt(5*x + 3)/(196*sqrt(-2*x + 1)*(3*x + 2)**2) - 3*sqrt(5*x + 3)/(49*s
qrt(-2*x + 1)*(3*x + 2)**3) + 2*sqrt(5*x + 3)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**3
)

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Mathematica [A]  time = 0.148931, size = 87, normalized size = 0.5 \[ \frac{\frac{14 \sqrt{5 x+3} \left (1834380 x^4+235980 x^3-1465461 x^2-39530 x+302352\right )}{(1-2 x)^{3/2} (3 x+2)^3}-837045 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{8874096} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(302352 - 39530*x - 1465461*x^2 + 235980*x^3 + 1834380*x^4))/
((1 - 2*x)^(3/2)*(2 + 3*x)^3) - 837045*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 1
4*x]*Sqrt[3 + 5*x])])/8874096

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Maple [B]  time = 0.022, size = 305, normalized size = 1.8 \[{\frac{1}{8874096\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) ^{2}} \left ( 90400860\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-37667025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-48548610\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+3303720\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+3348180\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-20516454\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+6696360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -553420\,x\sqrt{-10\,{x}^{2}-x+3}+4232928\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^4,x)

[Out]

1/8874096*(90400860*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x
^5+90400860*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-37667
025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+25681320*x^4*
(-10*x^2-x+3)^(1/2)-48548610*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))*x^2+3303720*x^3*(-10*x^2-x+3)^(1/2)+3348180*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-20516454*x^2*(-10*x^2-x+3)^(1/2)+6696360*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-553420*x*(-10*x^2-x+3)^(1/2
)+4232928*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)^2/
(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5141, size = 324, normalized size = 1.87 \[ \frac{25365}{268912} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{84925 \, x}{316932 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{131015}{633864 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{375 \, x}{1372 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{189 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{11}{196 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{377}{3528 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{3215}{74088 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25365/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 84925/3
16932*x/sqrt(-10*x^2 - x + 3) + 131015/633864/sqrt(-10*x^2 - x + 3) + 375/1372*x
/(-10*x^2 - x + 3)^(3/2) - 1/189/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 -
x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) +
11/196/(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)) - 377/3528/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^
(3/2)) - 3215/74088/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.231431, size = 167, normalized size = 0.97 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1834380 \, x^{4} + 235980 \, x^{3} - 1465461 \, x^{2} - 39530 \, x + 302352\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 837045 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{8874096 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8874096*sqrt(7)*(2*sqrt(7)*(1834380*x^4 + 235980*x^3 - 1465461*x^2 - 39530*x +
302352)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 837045*(108*x^5 + 108*x^4 - 45*x^3 - 58*
x^2 + 4*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/
(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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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((3+5*x)**(1/2)/(1-2*x)**(5/2)/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.542411, size = 482, normalized size = 2.79 \[ \frac{5073}{537824} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{32 \,{\left (361 \, \sqrt{5}{\left (5 \, x + 3\right )} - 2178 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{13865775 \,{\left (2 \, x - 1\right )}^{2}} - \frac{297 \,{\left (603 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 235200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 37240000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{67228 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5073/537824*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))) - 32/13865775*(361*sqrt(5)*(5*x + 3) - 2178*sqrt(5))*sqrt(5*x + 3)*sqrt
(-10*x + 5)/(2*x - 1)^2 - 297/67228*(603*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5
- 235200*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 - 37240000*sqrt(10)*((sqrt(2)*sq
rt(-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