3.2427 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ \frac{(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{81 (1-2 x)^{5/2}}{28 (3 x+2)^2 \sqrt{5 x+3}}+\frac{4455 (1-2 x)^{3/2}}{56 (3 x+2) \sqrt{5 x+3}}-\frac{147015 \sqrt{1-2 x}}{56 \sqrt{5 x+3}}+\frac{147015 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

[Out]

(-147015*Sqrt[1 - 2*x])/(56*Sqrt[3 + 5*x]) + (1 - 2*x)^(7/2)/(7*(2 + 3*x)^3*Sqrt
[3 + 5*x]) + (81*(1 - 2*x)^(5/2))/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (4455*(1 - 2*
x)^(3/2))/(56*(2 + 3*x)*Sqrt[3 + 5*x]) + (147015*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(8*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.214932, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{81 (1-2 x)^{5/2}}{28 (3 x+2)^2 \sqrt{5 x+3}}+\frac{4455 (1-2 x)^{3/2}}{56 (3 x+2) \sqrt{5 x+3}}-\frac{147015 \sqrt{1-2 x}}{56 \sqrt{5 x+3}}+\frac{147015 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

(-147015*Sqrt[1 - 2*x])/(56*Sqrt[3 + 5*x]) + (1 - 2*x)^(7/2)/(7*(2 + 3*x)^3*Sqrt
[3 + 5*x]) + (81*(1 - 2*x)^(5/2))/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (4455*(1 - 2*
x)^(3/2))/(56*(2 + 3*x)*Sqrt[3 + 5*x]) + (147015*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(8*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.8316, size = 138, normalized size = 0.96 \[ - \frac{10 \left (- 2 x + 1\right )^{\frac{7}{2}}}{11 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} - \frac{81 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{11 \left (3 x + 2\right )^{3}} - \frac{405 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4 \left (3 x + 2\right )^{2}} - \frac{13365 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8 \left (3 x + 2\right )} + \frac{147015 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{56} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**4/(3+5*x)**(3/2),x)

[Out]

-10*(-2*x + 1)**(7/2)/(11*(3*x + 2)**3*sqrt(5*x + 3)) - 81*(-2*x + 1)**(5/2)*sqr
t(5*x + 3)/(11*(3*x + 2)**3) - 405*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(4*(3*x + 2)*
*2) - 13365*sqrt(-2*x + 1)*sqrt(5*x + 3)/(8*(3*x + 2)) + 147015*sqrt(7)*atan(sqr
t(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/56

_______________________________________________________________________________________

Mathematica [A]  time = 0.0961095, size = 82, normalized size = 0.57 \[ \frac{147015 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{16 \sqrt{7}}-\frac{\sqrt{1-2 x} \left (578245 x^3+1143741 x^2+753654 x+165424\right )}{8 (3 x+2)^3 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

-(Sqrt[1 - 2*x]*(165424 + 753654*x + 1143741*x^2 + 578245*x^3))/(8*(2 + 3*x)^3*S
qrt[3 + 5*x]) + (147015*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(
16*Sqrt[7])

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 250, normalized size = 1.7 \[ -{\frac{1}{112\, \left ( 2+3\,x \right ) ^{3}} \left ( 19847025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+8095430\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+16012374\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3528360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+2315936\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^(3/2),x)

[Out]

-1/112*(19847025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+
51602265*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+50279130
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+8095430*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+16012374*x^2*(-10*x^2-x+3)^(1/2)+3528360*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)+2315936*(-10*x^2-x+3)^(
1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.50619, size = 285, normalized size = 1.98 \[ -\frac{147015}{112} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{578245 \, x}{108 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{603743}{216 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{81 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{10339}{324 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{87199}{216 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^4),x, algorithm="maxima")

[Out]

-147015/112*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 578245/1
08*x/sqrt(-10*x^2 - x + 3) - 603743/216/sqrt(-10*x^2 - x + 3) + 343/81/(27*sqrt(
-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x
+ 8*sqrt(-10*x^2 - x + 3)) + 10339/324/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-
10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 87199/216/(3*sqrt(-10*x^2 - x + 3
)*x + 2*sqrt(-10*x^2 - x + 3))

_______________________________________________________________________________________

Fricas [A]  time = 0.220414, size = 147, normalized size = 1.02 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (578245 \, x^{3} + 1143741 \, x^{2} + 753654 \, x + 165424\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 147015 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{112 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^4),x, algorithm="fricas")

[Out]

-1/112*sqrt(7)*(2*sqrt(7)*(578245*x^3 + 1143741*x^2 + 753654*x + 165424)*sqrt(5*
x + 3)*sqrt(-2*x + 1) + 147015*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*arctan
(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(135*x^4 + 351*x^3 +
342*x^2 + 148*x + 24)

_______________________________________________________________________________________

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)**4/(3+5*x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.376855, size = 509, normalized size = 3.53 \[ -\frac{29403}{224} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{121}{2} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{121 \,{\left (993 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 436800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 51352000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^4),x, algorithm="giac")

[Out]

-29403/224*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)))) - 121/2*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))) - 121/4*(993*sqrt(10)*((sqr
t(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 + 436800*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 + 51352
000*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)^
3