3.2440 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=224 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}+\frac{20529722435 \sqrt{1-2 x}}{18816 \sqrt{5 x+3}}+\frac{270667969 \sqrt{1-2 x}}{18816 (3 x+2) (5 x+3)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (3 x+2)^3 (5 x+3)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (3 x+2)^4 (5 x+3)^{3/2}}-\frac{754386765 \sqrt{1-2 x}}{6272 (5 x+3)^{3/2}}-\frac{46975917593 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

[Out]

(-754386765*Sqrt[1 - 2*x])/(6272*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(15*(2 +
3*x)^5*(3 + 5*x)^(3/2)) + (1001*Sqrt[1 - 2*x])/(120*(2 + 3*x)^4*(3 + 5*x)^(3/2)
) + (53009*Sqrt[1 - 2*x])/(720*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (3329689*Sqrt[1 -
2*x])/(4032*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (270667969*Sqrt[1 - 2*x])/(18816*(2 +
3*x)*(3 + 5*x)^(3/2)) + (20529722435*Sqrt[1 - 2*x])/(18816*Sqrt[3 + 5*x]) - (46
975917593*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.556578, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}+\frac{20529722435 \sqrt{1-2 x}}{18816 \sqrt{5 x+3}}+\frac{270667969 \sqrt{1-2 x}}{18816 (3 x+2) (5 x+3)^{3/2}}+\frac{3329689 \sqrt{1-2 x}}{4032 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{53009 \sqrt{1-2 x}}{720 (3 x+2)^3 (5 x+3)^{3/2}}+\frac{1001 \sqrt{1-2 x}}{120 (3 x+2)^4 (5 x+3)^{3/2}}-\frac{754386765 \sqrt{1-2 x}}{6272 (5 x+3)^{3/2}}-\frac{46975917593 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^(5/2)),x]

[Out]

(-754386765*Sqrt[1 - 2*x])/(6272*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(15*(2 +
3*x)^5*(3 + 5*x)^(3/2)) + (1001*Sqrt[1 - 2*x])/(120*(2 + 3*x)^4*(3 + 5*x)^(3/2)
) + (53009*Sqrt[1 - 2*x])/(720*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (3329689*Sqrt[1 -
2*x])/(4032*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (270667969*Sqrt[1 - 2*x])/(18816*(2 +
3*x)*(3 + 5*x)^(3/2)) + (20529722435*Sqrt[1 - 2*x])/(18816*Sqrt[3 + 5*x]) - (46
975917593*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 51.5714, size = 207, normalized size = 0.92 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{5} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{20529722435 \sqrt{- 2 x + 1}}{18816 \sqrt{5 x + 3}} - \frac{754386765 \sqrt{- 2 x + 1}}{6272 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{270667969 \sqrt{- 2 x + 1}}{18816 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{3329689 \sqrt{- 2 x + 1}}{4032 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{53009 \sqrt{- 2 x + 1}}{720 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{1001 \sqrt{- 2 x + 1}}{120 \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{46975917593 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{43904} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**6/(3+5*x)**(5/2),x)

[Out]

7*(-2*x + 1)**(3/2)/(15*(3*x + 2)**5*(5*x + 3)**(3/2)) + 20529722435*sqrt(-2*x +
1)/(18816*sqrt(5*x + 3)) - 754386765*sqrt(-2*x + 1)/(6272*(5*x + 3)**(3/2)) + 2
70667969*sqrt(-2*x + 1)/(18816*(3*x + 2)*(5*x + 3)**(3/2)) + 3329689*sqrt(-2*x +
1)/(4032*(3*x + 2)**2*(5*x + 3)**(3/2)) + 53009*sqrt(-2*x + 1)/(720*(3*x + 2)**
3*(5*x + 3)**(3/2)) + 1001*sqrt(-2*x + 1)/(120*(3*x + 2)**4*(5*x + 3)**(3/2)) -
46975917593*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/43904

_______________________________________________________________________________________

Mathematica [A]  time = 0.163061, size = 133, normalized size = 0.59 \[ \sqrt{1-2 x} \sqrt{5 x+3} \left (\frac{1672000}{15 x+9}-\frac{30250}{3 (5 x+3)^2}+\frac{2008587687}{6272 (3 x+2)}+\frac{10921833}{448 (3 x+2)^2}+\frac{169629}{80 (3 x+2)^3}+\frac{6811}{40 (3 x+2)^4}+\frac{49}{5 (3 x+2)^5}\right )-\frac{46975917593 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{12544 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^(5/2)),x]

[Out]

Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(49/(5*(2 + 3*x)^5) + 6811/(40*(2 + 3*x)^4) + 169629
/(80*(2 + 3*x)^3) + 10921833/(448*(2 + 3*x)^2) + 2008587687/(6272*(2 + 3*x)) - 3
0250/(3*(3 + 5*x)^2) + 1672000/(9 + 15*x)) - (46975917593*ArcTan[(-20 - 37*x)/(2
*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(12544*Sqrt[7])

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 394, normalized size = 1.8 \[{\frac{1}{1317120\, \left ( 2+3\,x \right ) ^{5}} \left ( 4280680490662125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{7}+19405751557668300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+37689013564451865\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+1746052893096750\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}+40650610289102550\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+6829311689562600\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+26297118668561400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+11125554365281230\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+10203169301199600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+9662658051124260\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+2198472943352400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+4718679545989416\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1228469050319504\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(5/2),x)

[Out]

1/1317120*(4280680490662125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^
(1/2))*x^7+19405751557668300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))*x^6+37689013564451865*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))*x^5+1746052893096750*x^6*(-10*x^2-x+3)^(1/2)+40650610289102550*7^(1/2)*
arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+6829311689562600*x^5*(-10
*x^2-x+3)^(1/2)+26297118668561400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^3+11125554365281230*x^4*(-10*x^2-x+3)^(1/2)+10203169301199600*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+9662658051124260*x^3
*(-10*x^2-x+3)^(1/2)+2198472943352400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))*x+4718679545989416*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))+1228469050319504*x*(-10*x
^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^5/(-10*
x^2-x+3)^(1/2)/(3+5*x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.53514, size = 576, normalized size = 2.57 \[ \frac{46975917593}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{20529722435 \, x}{9408 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{21434986553}{18816 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2211170555 \, x}{4032 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{405 \,{\left (243 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} + 810 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + 1080 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 720 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 240 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 32 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{43561}{1080 \,{\left (81 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 96 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 16 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{2438681}{6480 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{110694619}{25920 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{1309509421}{17280 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{21497905297}{72576 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(5/2)*(3*x + 2)^6),x, algorithm="maxima")

[Out]

46975917593/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 20
529722435/9408*x/sqrt(-10*x^2 - x + 3) + 21434986553/18816/sqrt(-10*x^2 - x + 3)
+ 2211170555/4032*x/(-10*x^2 - x + 3)^(3/2) + 2401/405/(243*(-10*x^2 - x + 3)^(
3/2)*x^5 + 810*(-10*x^2 - x + 3)^(3/2)*x^4 + 1080*(-10*x^2 - x + 3)^(3/2)*x^3 +
720*(-10*x^2 - x + 3)^(3/2)*x^2 + 240*(-10*x^2 - x + 3)^(3/2)*x + 32*(-10*x^2 -
x + 3)^(3/2)) + 43561/1080/(81*(-10*x^2 - x + 3)^(3/2)*x^4 + 216*(-10*x^2 - x +
3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^(3/2)*x^2 + 96*(-10*x^2 - x + 3)^(3/2)*x +
16*(-10*x^2 - x + 3)^(3/2)) + 2438681/6480/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*
(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)
^(3/2)) + 110694619/25920/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^
(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 1309509421/17280/(3*(-10*x^2 - x + 3)^(3/
2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 21497905297/72576/(-10*x^2 - x + 3)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.227324, size = 208, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (124718063792625 \, x^{6} + 487807977825900 \, x^{5} + 794682454662945 \, x^{4} + 690189860794590 \, x^{3} + 337048538999244 \, x^{2} + 87747789308536 \, x + 9514465420576\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 704638763895 \,{\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1317120 \,{\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(5/2)*(3*x + 2)^6),x, algorithm="fricas")

[Out]

1/1317120*sqrt(7)*(2*sqrt(7)*(124718063792625*x^6 + 487807977825900*x^5 + 794682
454662945*x^4 + 690189860794590*x^3 + 337048538999244*x^2 + 87747789308536*x + 9
514465420576)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 704638763895*(6075*x^7 + 27540*x^6
+ 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*arctan(1/14*sqrt
(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(6075*x^7 + 27540*x^6 + 53487*x
^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)

_______________________________________________________________________________________

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)/(2+3*x)**6/(3+5*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.632461, size = 756, normalized size = 3.38 \[ -\frac{275}{48} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{46975917593}{878080} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + 27775 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{11 \,{\left (3277500437 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 3147123544880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 1168996576419840 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 196941720284288000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 12621260024737280000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{3136 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(5/2)*(3*x + 2)^6),x, algorithm="giac")

[Out]

-275/48*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 + 46975917593/878080*sqrt(70)*sqr
t(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 27775*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))) + 11/3136*(3277500437*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) - sq
rt(22)))^9 + 3147123544880*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 + 116899657641
9840*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 + 196941720284288000*sqrt(10)*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^3 + 12621260024737280000*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)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5