3.2402 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=200 \[ \frac{1165 \sqrt{1-2 x} (5 x+3)^{5/2}}{2592 (3 x+2)^2}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{3485 \sqrt{1-2 x} (5 x+3)^{3/2}}{4032 (3 x+2)}+\frac{249575 \sqrt{1-2 x} \sqrt{5 x+3}}{108864}+\frac{1850}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{3304795 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{326592 \sqrt{7}} \]

[Out]

(249575*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/108864 - (3485*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
))/(4032*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(12*(2 + 3*x)^4) + (185*
(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(216*(2 + 3*x)^3) + (1165*Sqrt[1 - 2*x]*(3 + 5*
x)^(5/2))/(2592*(2 + 3*x)^2) + (1850*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/
729 + (3304795*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(326592*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.457579, antiderivative size = 200, 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 \[ \frac{1165 \sqrt{1-2 x} (5 x+3)^{5/2}}{2592 (3 x+2)^2}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{3485 \sqrt{1-2 x} (5 x+3)^{3/2}}{4032 (3 x+2)}+\frac{249575 \sqrt{1-2 x} \sqrt{5 x+3}}{108864}+\frac{1850}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{3304795 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{326592 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(249575*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/108864 - (3485*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
))/(4032*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(12*(2 + 3*x)^4) + (185*
(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(216*(2 + 3*x)^3) + (1165*Sqrt[1 - 2*x]*(3 + 5*
x)^(5/2))/(2592*(2 + 3*x)^2) + (1850*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/
729 + (3304795*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(326592*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 44.4145, size = 182, normalized size = 0.91 \[ - \frac{23255 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{127008 \left (3 x + 2\right )^{2}} - \frac{185 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1512 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{12 \left (3 x + 2\right )^{4}} + \frac{517345 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{254016 \left (3 x + 2\right )} + \frac{778885 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{381024} + \frac{1850 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{729} + \frac{3304795 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2286144} \]

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)**5,x)

[Out]

-23255*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(127008*(3*x + 2)**2) - 185*(-2*x + 1)**(
5/2)*(5*x + 3)**(3/2)/(1512*(3*x + 2)**3) - (-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/(
12*(3*x + 2)**4) + 517345*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(254016*(3*x + 2)) + 7
78885*sqrt(-2*x + 1)*sqrt(5*x + 3)/381024 + 1850*sqrt(10)*asin(sqrt(22)*sqrt(5*x
+ 3)/11)/729 + 3304795*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2
286144

_______________________________________________________________________________________

Mathematica [A]  time = 0.274615, size = 122, normalized size = 0.61 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (3628800 x^4+29475315 x^3+45563928 x^2+25998852 x+5093072\right )}{(3 x+2)^4}+3304795 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+5801600 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{4572288} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5093072 + 25998852*x + 45563928*x^2 + 29475315
*x^3 + 3628800*x^4))/(2 + 3*x)^4 + 3304795*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7
- 14*x]*Sqrt[3 + 5*x])] + 5801600*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*S
qrt[30 + 50*x])])/4572288

_______________________________________________________________________________________

Maple [B]  time = 0.018, size = 332, normalized size = 1.7 \[ -{\frac{1}{4572288\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 267688395\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-469929600\,\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) \sqrt{10}{x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-1253145600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-152409600\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-1253145600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-1237963230\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-556953600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1913684976\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -92825600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1091951784\,x\sqrt{-10\,{x}^{2}-x+3}-213909024\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^5,x)

[Out]

-1/4572288*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(267688395*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-469929600*arcsin(20/11*x+1/11)*10^(1/2)*x^4+713
835720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-1253145600
*10^(1/2)*arcsin(20/11*x+1/11)*x^3-152409600*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^2-1253145600*10^(1/2
)*arcsin(20/11*x+1/11)*x^2-1237963230*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-556953600*10^(1/2)*arcsin(2
0/11*x+1/11)*x-1913684976*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))-92825600*10^(1/2)*arcsin(20/11*x+1/11)-109
1951784*x*(-10*x^2-x+3)^(1/2)-213909024*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)
/(2+3*x)^4

_______________________________________________________________________________________

Maxima [A]  time = 1.52945, size = 305, normalized size = 1.52 \[ \frac{5755}{49392} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{392 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1151 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{182225}{98784} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{1488395}{1778112} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{44881 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{197568 \,{\left (3 \, x + 2\right )}} - \frac{28675}{127008} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{925}{729} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{3304795}{4572288} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{1643795}{762048} \, \sqrt{-10 \, x^{2} - x + 3} \]

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)^5,x, algorithm="maxima")

[Out]

5755/49392*(-10*x^2 - x + 3)^(5/2) + 3/28*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*
x^3 + 216*x^2 + 96*x + 16) + 37/392*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 3
6*x + 8) + 1151/10976*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) + 182225/98784*
(-10*x^2 - x + 3)^(3/2)*x - 1488395/1778112*(-10*x^2 - x + 3)^(3/2) + 44881/1975
68*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) - 28675/127008*sqrt(-10*x^2 - x + 3)*x + 92
5/729*sqrt(10)*arcsin(20/11*x + 1/11) - 3304795/4572288*sqrt(7)*arcsin(37/11*x/a
bs(3*x + 2) + 20/11/abs(3*x + 2)) + 1643795/762048*sqrt(-10*x^2 - x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.234629, size = 225, normalized size = 1.12 \[ \frac{\sqrt{7}{\left (828800 \, \sqrt{10} \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (3628800 \, x^{4} + 29475315 \, x^{3} + 45563928 \, x^{2} + 25998852 \, x + 5093072\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3304795 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{4572288 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4572288*sqrt(7)*(828800*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)*
(3628800*x^4 + 29475315*x^3 + 45563928*x^2 + 25998852*x + 5093072)*sqrt(5*x + 3)
*sqrt(-2*x + 1) - 3304795*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*s
qrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 216*x^3 + 216*x^2
+ 96*x + 16)

_______________________________________________________________________________________

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)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.641027, size = 628, normalized size = 3.14 \[ -\frac{660959}{9144576} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{925}{729} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{20}{243} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{55 \,{\left (8191 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 7386792 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 2164545600 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 2731201984000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{54432 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-660959/9144576*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)))) + 925/729*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))))
+ 20/243*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 55/54432*(8191*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 + 7386792*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 + 216454
5600*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 + 2731201984000*sqrt(10)*((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))))/(((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)^4