3.2336 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx\)

Optimal. Leaf size=150 \[ \frac{1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac{23}{216} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{53}{192} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{15863 \sqrt{1-2 x} \sqrt{5 x+3}}{20736}+\frac{648919 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{62208 \sqrt{10}}+\frac{14}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-15863*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20736 - (53*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
192 + (23*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/216 + ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))
/12 + (648919*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(62208*Sqrt[10]) + (14*Sqrt[7]*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

_______________________________________________________________________________________

Rubi [A]  time = 0.369633, antiderivative size = 150, 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 \[ \frac{1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac{23}{216} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{53}{192} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{15863 \sqrt{1-2 x} \sqrt{5 x+3}}{20736}+\frac{648919 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{62208 \sqrt{10}}+\frac{14}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-15863*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20736 - (53*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
192 + (23*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/216 + ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))
/12 + (648919*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(62208*Sqrt[10]) + (14*Sqrt[7]*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.7146, size = 138, normalized size = 0.92 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{12} - \frac{115 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{432} + \frac{535 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{1728} - \frac{15863 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{20736} + \frac{648919 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{622080} + \frac{14 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{243} \]

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),x)

[Out]

(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/12 - 115*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/4
32 + 535*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/1728 - 15863*sqrt(-2*x + 1)*sqrt(5*x +
3)/20736 + 648919*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/622080 + 14*sqrt(7)*a
tan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/243

_______________________________________________________________________________________

Mathematica [A]  time = 0.209729, size = 110, normalized size = 0.73 \[ \frac{-60 \sqrt{1-2 x} \sqrt{5 x+3} \left (86400 x^3+5280 x^2-58356 x-2389\right )+35840 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+648919 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{1244160} \]

Antiderivative was successfully verified.

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

[Out]

(-60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-2389 - 58356*x + 5280*x^2 + 86400*x^3) + 3584
0*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 648919*Sqrt[10
]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/1244160

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 132, normalized size = 0.9 \[{\frac{1}{1244160}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -5184000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-316800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+648919\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -35840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3501360\,x\sqrt{-10\,{x}^{2}-x+3}+143340\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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),x)

[Out]

1/1244160*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-5184000*x^3*(-10*x^2-x+3)^(1/2)-316800*x
^2*(-10*x^2-x+3)^(1/2)+648919*10^(1/2)*arcsin(20/11*x+1/11)-35840*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+3501360*x*(-10*x^2-x+3)^(1/2)+14334
0*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50027, size = 132, normalized size = 0.88 \[ \frac{5}{12} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{7}{432} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{2675}{1728} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{648919}{1244160} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{7}{243} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3397}{20736} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/12*(-10*x^2 - x + 3)^(3/2)*x - 7/432*(-10*x^2 - x + 3)^(3/2) + 2675/1728*sqrt(
-10*x^2 - x + 3)*x + 648919/1244160*sqrt(10)*arcsin(20/11*x + 1/11) - 7/243*sqrt
(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3397/20736*sqrt(-10*x^2
- x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.238215, size = 135, normalized size = 0.9 \[ -\frac{1}{1244160} \, \sqrt{10}{\left (6 \, \sqrt{10}{\left (86400 \, x^{3} + 5280 \, x^{2} - 58356 \, x - 2389\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3584 \, \sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 648919 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

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),x, algorithm="fricas")

[Out]

-1/1244160*sqrt(10)*(6*sqrt(10)*(86400*x^3 + 5280*x^2 - 58356*x - 2389)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) + 3584*sqrt(10)*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)/(sq
rt(5*x + 3)*sqrt(-2*x + 1))) - 648919*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x
+ 3)*sqrt(-2*x + 1))))

_______________________________________________________________________________________

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),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.30653, size = 269, normalized size = 1.79 \[ -\frac{7}{2430} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{518400} \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} - 313 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 2385 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 79315 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{648919}{1244160} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

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),x, algorithm="giac")

[Out]

-7/2430*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
)))) - 1/518400*(12*(8*(36*sqrt(5)*(5*x + 3) - 313*sqrt(5))*(5*x + 3) + 2385*sqr
t(5))*(5*x + 3) + 79315*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 648919/1244160*
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))))