3.2390 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=178 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{216 (3 x+2)^3}+\frac{2675 (5 x+3)^{3/2} \sqrt{1-2 x}}{864 (3 x+2)^2}-\frac{97235 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)}-\frac{40}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{3244595 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108864 \sqrt{7}} \]

[Out]

(-97235*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36288*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5
*x)^(3/2))/(12*(2 + 3*x)^4) + (115*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(216*(2 + 3*
x)^3) + (2675*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(864*(2 + 3*x)^2) - (40*Sqrt[10]*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (3244595*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/(108864*Sqrt[7])

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Rubi [A]  time = 0.379528, antiderivative size = 178, 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 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{216 (3 x+2)^3}+\frac{2675 (5 x+3)^{3/2} \sqrt{1-2 x}}{864 (3 x+2)^2}-\frac{97235 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)}-\frac{40}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{3244595 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108864 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-97235*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36288*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5
*x)^(3/2))/(12*(2 + 3*x)^4) + (115*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(216*(2 + 3*
x)^3) + (2675*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(864*(2 + 3*x)^2) - (40*Sqrt[10]*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (3244595*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/(108864*Sqrt[7])

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Rubi in Sympy [A]  time = 36.4446, size = 162, normalized size = 0.91 \[ - \frac{115 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1512 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{12 \left (3 x + 2\right )^{4}} + \frac{265 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{672 \left (3 x + 2\right )^{2}} + \frac{79315 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36288 \left (3 x + 2\right )} - \frac{40 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{243} - \frac{3244595 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{762048} \]

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)**5,x)

[Out]

-115*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(1512*(3*x + 2)**3) - (-2*x + 1)**(5/2)*(5*
x + 3)**(3/2)/(12*(3*x + 2)**4) + 265*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(672*(3*x
+ 2)**2) + 79315*sqrt(-2*x + 1)*sqrt(5*x + 3)/(36288*(3*x + 2)) - 40*sqrt(10)*as
in(sqrt(22)*sqrt(5*x + 3)/11)/243 - 3244595*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/
(7*sqrt(5*x + 3)))/762048

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Mathematica [A]  time = 0.232911, size = 117, normalized size = 0.66 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (1790325 x^3+4103592 x^2+2947548 x+677168\right )}{(3 x+2)^4}-3244595 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-125440 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{1524096} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(677168 + 2947548*x + 4103592*x^2 + 1790325*x^3
))/(2 + 3*x)^4 - 3244595*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 +
5*x])] - 125440*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/1
524096

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Maple [B]  time = 0.02, size = 315, normalized size = 1.8 \[{\frac{1}{1524096\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 262812195\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-10160640\,\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) \sqrt{10}{x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-27095040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-27095040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-12042240\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+172350864\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2007040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^5,x)

[Out]

1/1524096*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(262812195*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-10160640*arcsin(20/11*x+1/11)*10^(1/2)*x^4+70083
2520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-27095040*10^
(1/2)*arcsin(20/11*x+1/11)*x^3+700832520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^2-27095040*10^(1/2)*arcsin(20/11*x+1/11)*x^2+75193650*x^3*
(-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))*x-12042240*10^(1/2)*arcsin(20/11*x+1/11)*x+172350864*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))-200704
0*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^
2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]  time = 1.53204, size = 266, normalized size = 1.49 \[ \frac{21775}{21168} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{95 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{168 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{4355 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4704 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{539675}{42336} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{20}{243} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3244595}{1524096} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1460395}{254016} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{18245 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{28224 \,{\left (3 \, x + 2\right )}} \]

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)^5,x, algorithm="maxima")

[Out]

21775/21168*(-10*x^2 - x + 3)^(3/2) + 1/4*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*
x^3 + 216*x^2 + 96*x + 16) + 95/168*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 3
6*x + 8) + 4355/4704*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 539675/42336*s
qrt(-10*x^2 - x + 3)*x - 20/243*sqrt(10)*arcsin(20/11*x + 1/11) + 3244595/152409
6*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1460395/254016*sqr
t(-10*x^2 - x + 3) + 18245/28224*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.235273, size = 219, normalized size = 1.23 \[ -\frac{\sqrt{7}{\left (17920 \, \sqrt{10} \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{7}{\left (1790325 \, x^{3} + 4103592 \, x^{2} + 2947548 \, x + 677168\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3244595 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1524096 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

-1/1524096*sqrt(7)*(17920*sqrt(10)*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x +
16)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 6*sqrt(7)*
(1790325*x^3 + 4103592*x^2 + 2947548*x + 677168)*sqrt(5*x + 3)*sqrt(-2*x + 1) -
3244595*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)
/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 216*x^3 + 216*x^2 + 96*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((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.526482, size = 602, normalized size = 3.38 \[ \frac{648919}{3048192} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{20}{243} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{55 \,{\left (19447 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 19946472 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 6199166400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 348224576000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{18144 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

648919/3048192*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)))) - 20/243*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)))) -
55/18144*(19447*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 + 19946472*sqrt(10)*((sqr
t(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 - 6199166400*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 - 3
48224576000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2
+ 280)^4