3.2590 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=202 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac{139745 \sqrt{5 x+3}}{1613472 \sqrt{1-2 x}}-\frac{14135 \sqrt{5 x+3}}{153664 \sqrt{1-2 x} (3 x+2)}-\frac{2013 \sqrt{5 x+3}}{10976 \sqrt{1-2 x} (3 x+2)^2}-\frac{2717 \sqrt{5 x+3}}{8232 \sqrt{1-2 x} (3 x+2)^3}+\frac{43 \sqrt{5 x+3}}{588 \sqrt{1-2 x} (3 x+2)^4}-\frac{547745 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1075648 \sqrt{7}} \]

[Out]

(139745*Sqrt[3 + 5*x])/(1613472*Sqrt[1 - 2*x]) + (43*Sqrt[3 + 5*x])/(588*Sqrt[1
- 2*x]*(2 + 3*x)^4) - (2717*Sqrt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2
013*Sqrt[3 + 5*x])/(10976*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (14135*Sqrt[3 + 5*x])/(15
3664*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*
x)^4) - (547745*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1075648*Sqrt[7])

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Rubi [A]  time = 0.493227, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac{139745 \sqrt{5 x+3}}{1613472 \sqrt{1-2 x}}-\frac{14135 \sqrt{5 x+3}}{153664 \sqrt{1-2 x} (3 x+2)}-\frac{2013 \sqrt{5 x+3}}{10976 \sqrt{1-2 x} (3 x+2)^2}-\frac{2717 \sqrt{5 x+3}}{8232 \sqrt{1-2 x} (3 x+2)^3}+\frac{43 \sqrt{5 x+3}}{588 \sqrt{1-2 x} (3 x+2)^4}-\frac{547745 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1075648 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(139745*Sqrt[3 + 5*x])/(1613472*Sqrt[1 - 2*x]) + (43*Sqrt[3 + 5*x])/(588*Sqrt[1
- 2*x]*(2 + 3*x)^4) - (2717*Sqrt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2
013*Sqrt[3 + 5*x])/(10976*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (14135*Sqrt[3 + 5*x])/(15
3664*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*
x)^4) - (547745*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1075648*Sqrt[7])

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Rubi in Sympy [A]  time = 43.2809, size = 187, normalized size = 0.93 \[ - \frac{547745 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{7529536} + \frac{139745 \sqrt{5 x + 3}}{1613472 \sqrt{- 2 x + 1}} - \frac{14135 \sqrt{5 x + 3}}{153664 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{2013 \sqrt{5 x + 3}}{10976 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} - \frac{2717 \sqrt{5 x + 3}}{8232 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{43 \sqrt{5 x + 3}}{588 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}} \]

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)**5,x)

[Out]

-547745*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/7529536 + 139745*
sqrt(5*x + 3)/(1613472*sqrt(-2*x + 1)) - 14135*sqrt(5*x + 3)/(153664*sqrt(-2*x +
1)*(3*x + 2)) - 2013*sqrt(5*x + 3)/(10976*sqrt(-2*x + 1)*(3*x + 2)**2) - 2717*s
qrt(5*x + 3)/(8232*sqrt(-2*x + 1)*(3*x + 2)**3) + 43*sqrt(5*x + 3)/(588*sqrt(-2*
x + 1)*(3*x + 2)**4) + 11*(5*x + 3)**(3/2)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**4)

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Mathematica [A]  time = 0.151454, size = 92, normalized size = 0.46 \[ \frac{\frac{14 \sqrt{5 x+3} \left (-45277380 x^5-82071900 x^4-25673409 x^3+27318504 x^2+18627988 x+2906640\right )}{(1-2 x)^{3/2} (3 x+2)^4}-1643235 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{45177216} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(2906640 + 18627988*x + 27318504*x^2 - 25673409*x^3 - 8207190
0*x^4 - 45277380*x^5))/((1 - 2*x)^(3/2)*(2 + 3*x)^4) - 1643235*Sqrt[7]*ArcTan[(-
20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/45177216

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Maple [B]  time = 0.023, size = 353, normalized size = 1.8 \[{\frac{1}{45177216\, \left ( 2+3\,x \right ) ^{4} \left ( -1+2\,x \right ) ^{2}} \left ( 532408140\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-633883320\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-433814040\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-1149006600\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-170896440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-359427726\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+382459056\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^5,x)

[Out]

1/45177216*(532408140*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x^6+887346900*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+13
3102035*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-633883320
*x^5*(-10*x^2-x+3)^(1/2)-433814040*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))*x^3-1149006600*x^4*(-10*x^2-x+3)^(1/2)-170896440*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-359427726*x^3*(-10*x^2-x+3)^(1/2)+
52583520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+382459056*
x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x
)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50922, size = 439, normalized size = 2.17 \[ \frac{547745}{15059072} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{698725 \, x}{1613472 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343745}{3226944 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{633875 \, x}{691488 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{2268 \,{\left (81 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 96 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 16 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{331}{31752 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{9313}{98784 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{659891}{1778112 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{296615}{12446784 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

547745/15059072*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 6987
25/1613472*x/sqrt(-10*x^2 - x + 3) + 343745/3226944/sqrt(-10*x^2 - x + 3) + 6338
75/691488*x/(-10*x^2 - x + 3)^(3/2) - 1/2268/(81*(-10*x^2 - x + 3)^(3/2)*x^4 + 2
16*(-10*x^2 - x + 3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^(3/2)*x^2 + 96*(-10*x^2 -
x + 3)^(3/2)*x + 16*(-10*x^2 - x + 3)^(3/2)) + 331/31752/(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)) - 9313/98784/(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)) + 659891/1778112/(3*(-10*x^2 - x
+ 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 296615/12446784/(-10*x^2 - x + 3)^(3
/2)

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Fricas [A]  time = 0.226863, size = 188, normalized size = 0.93 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (45277380 \, x^{5} + 82071900 \, x^{4} + 25673409 \, x^{3} - 27318504 \, x^{2} - 18627988 \, x - 2906640\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1643235 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{45177216 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45177216*sqrt(7)*(2*sqrt(7)*(45277380*x^5 + 82071900*x^4 + 25673409*x^3 - 273
18504*x^2 - 18627988*x - 2906640)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1643235*(324*x^
6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*arctan(1/14*sqrt(7)*(37*x
+ 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 1
04*x^2 + 32*x + 16)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.750022, size = 564, normalized size = 2.79 \[ \frac{109549}{30118144} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{88 \,{\left (100 \, \sqrt{5}{\left (5 \, x + 3\right )} - 627 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1764735 \,{\left (2 \, x - 1\right )}^{2}} - \frac{55 \,{\left (79441 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 82486488 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 31196222400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 1487445568000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{3764768 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

109549/30118144*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)))) - 88/1764735*(100*sqrt(5)*(5*x + 3) - 627*sqrt(5))*sqrt(5*x + 3)*sq
rt(-10*x + 5)/(2*x - 1)^2 - 55/3764768*(79441*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)
))^7 + 82486488*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 + 31196222400*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^3 + 1487445568000*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)^4