3.2338 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=166 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac{35}{4} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{185}{27} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1945}{324} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{6829 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{324 \sqrt{7}} \]

[Out]

(185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/27 - (35*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4 - ((
1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(6*(2 + 3*x)^2) + (181*Sqrt[1 - 2*x]*(3 + 5*x)^(
5/2))/(36*(2 + 3*x)) + (1945*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/324 + (
6829*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(324*Sqrt[7])

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Rubi [A]  time = 0.379811, antiderivative size = 166, 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 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac{35}{4} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{185}{27} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1945}{324} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{6829 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{324 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/27 - (35*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4 - ((
1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(6*(2 + 3*x)^2) + (181*Sqrt[1 - 2*x]*(3 + 5*x)^(
5/2))/(36*(2 + 3*x)) + (1945*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/324 + (
6829*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(324*Sqrt[7])

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Rubi in Sympy [A]  time = 36.9443, size = 148, normalized size = 0.89 \[ - \frac{181 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{252 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{6 \left (3 x + 2\right )^{2}} - \frac{107 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{126} + \frac{185 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{27} + \frac{1945 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{648} + \frac{6829 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2268} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-181*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/(252*(3*x + 2)) - (-2*x + 1)**(3/2)*(5*x
+ 3)**(5/2)/(6*(3*x + 2)**2) - 107*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/126 + 185*sq
rt(-2*x + 1)*sqrt(5*x + 3)/27 + 1945*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/64
8 + 6829*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2268

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Mathematica [A]  time = 0.225719, size = 117, normalized size = 0.7 \[ \frac{\frac{84 \sqrt{1-2 x} \sqrt{5 x+3} \left (-900 x^3+1095 x^2+2985 x+1232\right )}{(3 x+2)^2}+13658 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+13615 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{9072} \]

Antiderivative was successfully verified.

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

[Out]

((84*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1232 + 2985*x + 1095*x^2 - 900*x^3))/(2 + 3*x)
^2 + 13658*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 13615
*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/9072

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Maple [A]  time = 0.019, size = 225, normalized size = 1.4 \[ -{\frac{1}{9072\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 122922\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-122535\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+75600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+163896\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-163380\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-91980\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+54632\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -54460\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -250740\,x\sqrt{-10\,{x}^{2}-x+3}-103488\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9072*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(122922*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^2-122535*10^(1/2)*arcsin(20/11*x+1/11)*x^2+75600*x^3*(-
10*x^2-x+3)^(1/2)+163896*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x-163380*10^(1/2)*arcsin(20/11*x+1/11)*x-91980*x^2*(-10*x^2-x+3)^(1/2)+54632
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-54460*10^(1/2)*arcsi
n(20/11*x+1/11)-250740*x*(-10*x^2-x+3)^(1/2)-103488*(-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 = 1.50483, size = 176, normalized size = 1.06 \[ -\frac{5}{63} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{535}{126} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{1945}{1296} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{6829}{4536} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{1627}{378} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{59 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{84 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/63*(-10*x^2 - x + 3)^(3/2) - 1/14*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4)
- 535/126*sqrt(-10*x^2 - x + 3)*x + 1945/1296*sqrt(10)*arcsin(20/11*x + 1/11) -
6829/4536*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1627/378*s
qrt(-10*x^2 - x + 3) - 59/84*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.232979, size = 194, normalized size = 1.17 \[ -\frac{\sqrt{7} \sqrt{2}{\left (6 \, \sqrt{7} \sqrt{2}{\left (900 \, x^{3} - 1095 \, x^{2} - 2985 \, x - 1232\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1945 \, \sqrt{7} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6829 \, \sqrt{2}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{9072 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9072*sqrt(7)*sqrt(2)*(6*sqrt(7)*sqrt(2)*(900*x^3 - 1095*x^2 - 2985*x - 1232)*
sqrt(5*x + 3)*sqrt(-2*x + 1) - 1945*sqrt(7)*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))) + 6829*sqrt(2)*(9*
x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)/(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)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.456067, size = 481, normalized size = 2.9 \[ -\frac{6829}{45360} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{108} \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 63 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{1945}{1296} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{55 \,{\left (17 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 5992 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{54 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6829/45360*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)))) - 1/108*(4*sqrt(5)*(5*x + 3) - 63*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5
) + 1945/1296*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)))) + 55/54*(
17*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 + 5992*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)/(s
qrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2