3.2389 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=171 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{108 (3 x+2)^2}+\frac{365 (5 x+3)^{3/2} \sqrt{1-2 x}}{216 (3 x+2)}-\frac{845}{648} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{362}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{215 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1944 \sqrt{7}} \]

[Out]

(-845*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/648 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(9*(2
+ 3*x)^3) + (115*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(108*(2 + 3*x)^2) + (365*Sqrt
[1 - 2*x]*(3 + 5*x)^(3/2))/(216*(2 + 3*x)) + (362*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]])/243 + (215*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1944*Sqr
t[7])

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Rubi [A]  time = 0.375341, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{108 (3 x+2)^2}+\frac{365 (5 x+3)^{3/2} \sqrt{1-2 x}}{216 (3 x+2)}-\frac{845}{648} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{362}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{215 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1944 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-845*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/648 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(9*(2
+ 3*x)^3) + (115*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(108*(2 + 3*x)^2) + (365*Sqrt
[1 - 2*x]*(3 + 5*x)^(3/2))/(216*(2 + 3*x)) + (362*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]])/243 + (215*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1944*Sqr
t[7])

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Rubi in Sympy [A]  time = 36.9047, size = 155, normalized size = 0.91 \[ - \frac{115 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{756 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{3}} + \frac{2165 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1512 \left (3 x + 2\right )} + \frac{3065 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2268} + \frac{362 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{243} + \frac{215 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{13608} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-115*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(756*(3*x + 2)**2) - (-2*x + 1)**(5/2)*(5*x
+ 3)**(3/2)/(9*(3*x + 2)**3) + 2165*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(1512*(3*x
+ 2)) + 3065*sqrt(-2*x + 1)*sqrt(5*x + 3)/2268 + 362*sqrt(10)*asin(sqrt(22)*sqrt
(5*x + 3)/11)/243 + 215*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/1
3608

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Mathematica [A]  time = 0.221127, size = 117, normalized size = 0.68 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (4320 x^3+34341 x^2+36234 x+10304\right )}{(3 x+2)^3}+215 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+20272 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{27216} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(10304 + 36234*x + 34341*x^2 + 4320*x^3))/(2 +
3*x)^3 + 215*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 202
72*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/27216

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Maple [B]  time = 0.017, size = 270, normalized size = 1.6 \[ -{\frac{1}{27216\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 5805\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-547344\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+11610\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-1094688\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-181440\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-729792\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1442322\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -162176\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1521828\,x\sqrt{-10\,{x}^{2}-x+3}-432768\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27216*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5805*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^3-547344*10^(1/2)*arcsin(20/11*x+1/11)*x^3+11610*7^(1/2)
*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-1094688*10^(1/2)*arcsin(
20/11*x+1/11)*x^2-181440*x^3*(-10*x^2-x+3)^(1/2)+7740*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-729792*10^(1/2)*arcsin(20/11*x+1/11)*x-144232
2*x^2*(-10*x^2-x+3)^(1/2)+1720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))-162176*10^(1/2)*arcsin(20/11*x+1/11)-1521828*x*(-10*x^2-x+3)^(1/2)-432
768*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.51975, size = 217, normalized size = 1.27 \[ \frac{125}{378} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{25 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{84 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{1825}{756} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{181}{243} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{215}{27216} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{655}{4536} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{65 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{504 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/378*(-10*x^2 - x + 3)^(3/2) + 1/3*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 +
36*x + 8) + 25/84*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 1825/756*sqrt(-1
0*x^2 - x + 3)*x + 181/243*sqrt(10)*arcsin(20/11*x + 1/11) - 215/27216*sqrt(7)*a
rcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 655/4536*sqrt(-10*x^2 - x + 3
) - 65/504*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.235583, size = 198, normalized size = 1.16 \[ \frac{\sqrt{7}{\left (2896 \, \sqrt{10} \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (4320 \, x^{3} + 34341 \, x^{2} + 36234 \, x + 10304\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 215 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{27216 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27216*sqrt(7)*(2896*sqrt(10)*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*
sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)*(4320*x^3 + 3434
1*x^2 + 36234*x + 10304)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 215*(27*x^3 + 54*x^2 + 3
6*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(27*x^
3 + 54*x^2 + 36*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((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.497523, size = 545, normalized size = 3.19 \[ -\frac{43}{54432} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{181}{243} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{4}{81} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \,{\left (67 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 56000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 65464000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{108 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-43/54432*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)))) + 181/243*sqrt(10)*(pi + 2*arctan(-1/4*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)))) + 4/81
*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/108*(67*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) - sq
rt(22)))^5 + 56000*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 - 65464000*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))))/(((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