3.2400 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=188 \[ \frac{575}{162} \sqrt{1-2 x} (5 x+3)^{5/2}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac{785}{36} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{34145 \sqrt{1-2 x} \sqrt{5 x+3}}{1944}+\frac{81733 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5832}+\frac{21935 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2916} \]

[Out]

(34145*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1944 - (785*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/3
6 + (575*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/162 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/
(6*(2 + 3*x)^2) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(36*(2 + 3*x)) + (81733*
Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/5832 + (21935*Sqrt[7]*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/2916

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Rubi [A]  time = 0.456656, antiderivative size = 188, 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 \[ \frac{575}{162} \sqrt{1-2 x} (5 x+3)^{5/2}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac{785}{36} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{34145 \sqrt{1-2 x} \sqrt{5 x+3}}{1944}+\frac{81733 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5832}+\frac{21935 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2916} \]

Antiderivative was successfully verified.

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

[Out]

(34145*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1944 - (785*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/3
6 + (575*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/162 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/
(6*(2 + 3*x)^2) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(36*(2 + 3*x)) + (81733*
Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/5832 + (21935*Sqrt[7]*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/2916

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Rubi in Sympy [A]  time = 45.8297, size = 168, normalized size = 0.89 \[ - \frac{185 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{252 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{6 \left (3 x + 2\right )^{2}} - \frac{905 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1134} - \frac{185 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{81} + \frac{34145 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1944} + \frac{81733 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{11664} + \frac{21935 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2916} \]

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)**3,x)

[Out]

-185*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/(252*(3*x + 2)) - (-2*x + 1)**(5/2)*(5*x
+ 3)**(5/2)/(6*(3*x + 2)**2) - 905*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/1134 - 18
5*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/81 + 34145*sqrt(-2*x + 1)*sqrt(5*x + 3)/1944 +
81733*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/11664 + 21935*sqrt(7)*atan(sqrt(
7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2916

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Mathematica [A]  time = 0.209126, size = 122, normalized size = 0.65 \[ \frac{\frac{12 \sqrt{1-2 x} \sqrt{5 x+3} \left (21600 x^4-28980 x^3+31731 x^2+120534 x+53204\right )}{(3 x+2)^2}+87740 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+81733 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{23328} \]

Antiderivative was successfully verified.

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

[Out]

((12*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(53204 + 120534*x + 31731*x^2 - 28980*x^3 + 216
00*x^4))/(2 + 3*x)^2 + 87740*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[
3 + 5*x])] + 81733*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])]
)/23328

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Maple [A]  time = 0.019, size = 242, normalized size = 1.3 \[ -{\frac{1}{23328\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -259200\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+789660\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-735597\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+347760\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1052880\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-980796\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-380772\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+350960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -326932\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1446408\,x\sqrt{-10\,{x}^{2}-x+3}-638448\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^3,x)

[Out]

-1/23328*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-259200*x^4*(-10*x^2-x+3)^(1/2)+789660*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-735597*10^(1/2)*arcs
in(20/11*x+1/11)*x^2+347760*x^3*(-10*x^2-x+3)^(1/2)+1052880*7^(1/2)*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-980796*10^(1/2)*arcsin(20/11*x+1/11)*x-
380772*x^2*(-10*x^2-x+3)^(1/2)+350960*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))-326932*10^(1/2)*arcsin(20/11*x+1/11)-1446408*x*(-10*x^2-x+3)^(1
/2)-638448*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^2

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Maxima [A]  time = 1.52542, size = 215, normalized size = 1.14 \[ \frac{5}{21} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{925}{126} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{10135}{2268} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{28 \,{\left (3 \, x + 2\right )}} - \frac{925}{81} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{81733}{23328} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{21935}{5832} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{20825}{1944} \, \sqrt{-10 \, x^{2} - x + 3} \]

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)^3,x, algorithm="maxima")

[Out]

5/21*(-10*x^2 - x + 3)^(5/2) + 3/14*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) +
925/126*(-10*x^2 - x + 3)^(3/2)*x - 10135/2268*(-10*x^2 - x + 3)^(3/2) + 37/28*
(-10*x^2 - x + 3)^(5/2)/(3*x + 2) - 925/81*sqrt(-10*x^2 - x + 3)*x + 81733/23328
*sqrt(10)*arcsin(20/11*x + 1/11) - 21935/5832*sqrt(7)*arcsin(37/11*x/abs(3*x + 2
) + 20/11/abs(3*x + 2)) + 20825/1944*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.237654, size = 193, normalized size = 1.03 \[ -\frac{\sqrt{2}{\left (43870 \, \sqrt{7} \sqrt{2}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{2}{\left (21600 \, x^{4} - 28980 \, x^{3} + 31731 \, x^{2} + 120534 \, x + 53204\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 81733 \, \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{23328 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

-1/23328*sqrt(2)*(43870*sqrt(7)*sqrt(2)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(
37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 6*sqrt(2)*(21600*x^4 - 28980*x^3 +
31731*x^2 + 120534*x + 53204)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 81733*sqrt(5)*(9*x^
2 + 12*x + 4)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x +
1))))/(9*x^2 + 12*x + 4)

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.533514, size = 498, normalized size = 2.65 \[ -\frac{4387}{11664} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{3240} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 155 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 5245 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{81733}{23328} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{77 \,{\left (263 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 92120 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{486 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

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)^3,x, algorithm="giac")

[Out]

-4387/11664*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)))) + 1/3240*(4*(8*sqrt(5)*(5*x + 3) - 155*sqrt(5))*(5*x + 3) + 5245*sqrt(5
))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 81733/23328*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)))) + 77/486*(263*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 + 9
2120*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)
^2