3.2401 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=195 \[ -\frac{10385 \sqrt{1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac{2075}{72} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{48625 \sqrt{1-2 x} \sqrt{5 x+3}}{1944}-\frac{21935 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1458}-\frac{408665 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5832 \sqrt{7}} \]

[Out]

(-48625*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1944 + (2075*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))
/72 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) + (185*(1 - 2*x)^(3/2)*(
3 + 5*x)^(5/2))/(108*(2 + 3*x)^2) - (10385*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(648*(
2 + 3*x)) - (21935*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/1458 - (408665*Ar
cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5832*Sqrt[7])

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Rubi [A]  time = 0.4584, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{10385 \sqrt{1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac{2075}{72} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{48625 \sqrt{1-2 x} \sqrt{5 x+3}}{1944}-\frac{21935 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1458}-\frac{408665 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5832 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-48625*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1944 + (2075*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))
/72 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^3) + (185*(1 - 2*x)^(3/2)*(
3 + 5*x)^(5/2))/(108*(2 + 3*x)^2) - (10385*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(648*(
2 + 3*x)) - (21935*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/1458 - (408665*Ar
cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5832*Sqrt[7])

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Rubi in Sympy [A]  time = 44.7042, size = 177, normalized size = 0.91 \[ - \frac{22595 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{31752 \left (3 x + 2\right )} - \frac{185 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{756 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{9 \left (3 x + 2\right )^{3}} - \frac{20015 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{15876} - \frac{34145 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6804} - \frac{21935 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2916} - \frac{408665 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{40824} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-22595*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(31752*(3*x + 2)) - 185*(-2*x + 1)**(5/2)
*(5*x + 3)**(3/2)/(756*(3*x + 2)**2) - (-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/(9*(3*
x + 2)**3) - 20015*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/15876 - 34145*sqrt(-2*x + 1)*
sqrt(5*x + 3)/6804 - 21935*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/2916 - 40866
5*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/40824

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Mathematica [A]  time = 0.249445, size = 122, normalized size = 0.63 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (32400 x^4-93420 x^3-420531 x^2-391014 x-107984\right )}{(3 x+2)^3}-408665 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-307090 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{81648} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-107984 - 391014*x - 420531*x^2 - 93420*x^3 +
32400*x^4))/(2 + 3*x)^3 - 408665*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*S
qrt[3 + 5*x])] - 307090*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50
*x])])/81648

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Maple [A]  time = 0.019, size = 287, normalized size = 1.5 \[{\frac{1}{81648\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 11033955\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-8291430\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+1360800\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-16582860\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-3923640\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-11055240\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-17662302\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3269320\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2456720\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -16422588\,x\sqrt{-10\,{x}^{2}-x+3}-4535328\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/81648*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(11033955*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^3-8291430*10^(1/2)*arcsin(20/11*x+1/11)*x^3+1360800*x
^4*(-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^2-16582860*10^(1/2)*arcsin(20/11*x+1/11)*x^2-3923640*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-
11055240*10^(1/2)*arcsin(20/11*x+1/11)*x-17662302*x^2*(-10*x^2-x+3)^(1/2)+326932
0*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-2456720*10^(1/2)*ar
csin(20/11*x+1/11)-16422588*x*(-10*x^2-x+3)^(1/2)-4535328*(-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 = 1.52836, size = 257, normalized size = 1.32 \[ -\frac{185}{882} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{7 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{16075}{1764} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{189865}{31752} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{6347 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3528 \,{\left (3 \, x + 2\right )}} + \frac{41225}{2268} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{21935}{5832} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{408665}{81648} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{191965}{13608} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-185/882*(-10*x^2 - x + 3)^(5/2) + 1/7*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2
+ 36*x + 8) - 37/196*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 16075/1764*(-1
0*x^2 - x + 3)^(3/2)*x + 189865/31752*(-10*x^2 - x + 3)^(3/2) - 6347/3528*(-10*x
^2 - x + 3)^(5/2)/(3*x + 2) + 41225/2268*sqrt(-10*x^2 - x + 3)*x - 21935/5832*sq
rt(10)*arcsin(20/11*x + 1/11) + 408665/81648*sqrt(7)*arcsin(37/11*x/abs(3*x + 2)
+ 20/11/abs(3*x + 2)) - 191965/13608*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.233413, size = 221, normalized size = 1.13 \[ \frac{\sqrt{7} \sqrt{2}{\left (6 \, \sqrt{7} \sqrt{2}{\left (32400 \, x^{4} - 93420 \, x^{3} - 420531 \, x^{2} - 391014 \, x - 107984\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 87740 \, \sqrt{7} \sqrt{5}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 408665 \, \sqrt{2}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{163296 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/163296*sqrt(7)*sqrt(2)*(6*sqrt(7)*sqrt(2)*(32400*x^4 - 93420*x^3 - 420531*x^2
- 391014*x - 107984)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 87740*sqrt(7)*sqrt(5)*(27*x^
3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqr
t(-2*x + 1))) + 408665*sqrt(2)*(27*x^3 + 54*x^2 + 36*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)**(5/2)/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.603531, size = 563, normalized size = 2.89 \[ \frac{81733}{163296} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{486} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 329 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{21935}{5832} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (2803 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1982400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 411208000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{324 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81733/163296*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) + 1/486*(12*sqrt(5)*(5*x + 3) - 329*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x
+ 5) - 21935/5832*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)))) - 11/
324*(2803*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 + 1982400*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)))^3 + 411208000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*s
qrt(-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