3.2619 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=181 \[ \frac{3443814775 \sqrt{1-2 x}}{60262356 \sqrt{5 x+3}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (5 x+3)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac{3715}{3234 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}-\frac{538245 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

-3715/(3234*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - 40765/(83006*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2)) - (34551425*Sqrt[1 - 2*x])/(5478396*(3 + 5*x)^(3/2)) + 3/(14*(1 - 2*x
)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + 111/(28*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*
x)^(3/2)) + (3443814775*Sqrt[1 - 2*x])/(60262356*Sqrt[3 + 5*x]) - (538245*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.492758, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3443814775 \sqrt{1-2 x}}{60262356 \sqrt{5 x+3}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (5 x+3)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac{3715}{3234 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}-\frac{538245 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-3715/(3234*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - 40765/(83006*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2)) - (34551425*Sqrt[1 - 2*x])/(5478396*(3 + 5*x)^(3/2)) + 3/(14*(1 - 2*x
)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + 111/(28*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*
x)^(3/2)) + (3443814775*Sqrt[1 - 2*x])/(60262356*Sqrt[3 + 5*x]) - (538245*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi in Sympy [A]  time = 42.7831, size = 167, normalized size = 0.92 \[ \frac{3443814775 \sqrt{- 2 x + 1}}{60262356 \sqrt{5 x + 3}} - \frac{34551425 \sqrt{- 2 x + 1}}{5478396 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{538245 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} - \frac{40765}{83006 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{3715}{3234 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{111}{28 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{3}{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3443814775*sqrt(-2*x + 1)/(60262356*sqrt(5*x + 3)) - 34551425*sqrt(-2*x + 1)/(54
78396*(5*x + 3)**(3/2)) - 538245*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x
+ 3)))/9604 - 40765/(83006*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) - 3715/(3234*(-2*x
+ 1)**(3/2)*(5*x + 3)**(3/2)) + 111/(28*(-2*x + 1)**(3/2)*(3*x + 2)*(5*x + 3)**(
3/2)) + 3/(14*(-2*x + 1)**(3/2)*(3*x + 2)**2*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.116463, size = 95, normalized size = 0.52 \[ \frac{\sqrt{1-2 x} \left (619886659500 x^5+564878517900 x^4-276089438305 x^3-297937101390 x^2+28838387211 x+39900939556\right )}{60262356 (5 x+3)^{3/2} \left (6 x^2+x-2\right )^2}-\frac{538245 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(39900939556 + 28838387211*x - 297937101390*x^2 - 276089438305*x^
3 + 564878517900*x^4 + 619886659500*x^5))/(60262356*(3 + 5*x)^(3/2)*(-2 + x + 6*
x^2)^2) - (538245*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(2744*S
qrt[7])

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Maple [B]  time = 0.026, size = 353, normalized size = 2. \[{\frac{1}{843672984\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 21277201621500\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-16123390562070\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-5366583075645\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-3865252136270\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-4171119419460\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/843672984*(1-2*x)^(1/2)*(21277201621500*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x^6+32625042486300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x^5+2576905529715*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^4+8678413233000*x^5*(-10*x^2-x+3)^(1/2)-16123390562070*7^(1/2)
*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+7908299250600*x^4*(-10*x
^2-x+3)^(1/2)-5366583075645*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^
(1/2))*x^2-3865252136270*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-4171119419460*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))+403737
420954*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(2+3*x)^2/(-1+2*x
)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.50737, size = 232, normalized size = 1.28 \[ \frac{538245}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{3443814775 \, x}{30131178 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3595841045}{60262356 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1022125 \, x}{35574 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{3}{14 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{111}{28 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{1103855}{71148 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

538245/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 3443814
775/30131178*x/sqrt(-10*x^2 - x + 3) + 3595841045/60262356/sqrt(-10*x^2 - x + 3)
+ 1022125/35574*x/(-10*x^2 - x + 3)^(3/2) + 3/14/(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)) + 111/28/(3*(-10*x^
2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 1103855/71148/(-10*x^2 - x + 3
)^(3/2)

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Fricas [A]  time = 0.234973, size = 188, normalized size = 1.04 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (619886659500 \, x^{5} + 564878517900 \, x^{4} - 276089438305 \, x^{3} - 297937101390 \, x^{2} + 28838387211 \, x + 39900939556\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 23641335135 \,{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{843672984 \,{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/843672984*sqrt(7)*(2*sqrt(7)*(619886659500*x^5 + 564878517900*x^4 - 2760894383
05*x^3 - 297937101390*x^2 + 28838387211*x + 39900939556)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) + 23641335135*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 3
6)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(900*x^6 + 1
380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)

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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/(1-2*x)**(5/2)/(2+3*x)**3/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.532899, size = 562, normalized size = 3.1 \[ -\frac{625}{702768} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{107649}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{58125}{29282} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{128 \,{\left (577 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3366 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{2636478075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{8019 \,{\left (159 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 38360 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4802 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-625/702768*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 107649/38416*sqrt(70)*sqrt(
10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 58125/29282*sqr
t(10)*((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))) - 128/2636478075*(577*sqrt(5)*(5*x + 3) - 3366
*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 8019/4802*(159*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^3 + 38360*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))))/(((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)))^2 + 280)^2