3.2609 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=159 \[ -\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{5 x+3}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{81}{28 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 \sqrt{5 x+3}}+\frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

-2725/(3234*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - 89945/(249018*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x]) - (46307675*Sqrt[1 - 2*x])/(5478396*Sqrt[3 + 5*x]) + 3/(14*(1 - 2*x)^(3/
2)*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 81/(28*(1 - 2*x)^(3/2)*(2 + 3*x)*Sqrt[3 + 5*x])
+ (79515*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.415338, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{5 x+3}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{81}{28 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 \sqrt{5 x+3}}+\frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-2725/(3234*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - 89945/(249018*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x]) - (46307675*Sqrt[1 - 2*x])/(5478396*Sqrt[3 + 5*x]) + 3/(14*(1 - 2*x)^(3/
2)*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 81/(28*(1 - 2*x)^(3/2)*(2 + 3*x)*Sqrt[3 + 5*x])
+ (79515*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi in Sympy [A]  time = 35.5864, size = 146, normalized size = 0.92 \[ - \frac{46307675 \sqrt{- 2 x + 1}}{5478396 \sqrt{5 x + 3}} + \frac{79515 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} - \frac{89945}{249018 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{2725}{3234 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{81}{28 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{3}{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} \]

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)**(3/2),x)

[Out]

-46307675*sqrt(-2*x + 1)/(5478396*sqrt(5*x + 3)) + 79515*sqrt(7)*atan(sqrt(7)*sq
rt(-2*x + 1)/(7*sqrt(5*x + 3)))/9604 - 89945/(249018*sqrt(-2*x + 1)*sqrt(5*x + 3
)) - 2725/(3234*(-2*x + 1)**(3/2)*sqrt(5*x + 3)) + 81/(28*(-2*x + 1)**(3/2)*(3*x
+ 2)*sqrt(5*x + 3)) + 3/(14*(-2*x + 1)**(3/2)*(3*x + 2)**2*sqrt(5*x + 3))

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Mathematica [A]  time = 0.12052, size = 90, normalized size = 0.57 \[ \frac{79515 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}}-\frac{\sqrt{1-2 x} \left (1667076300 x^4+520073880 x^3-1053213025 x^2-169466391 x+178740084\right )}{5478396 \sqrt{5 x+3} \left (6 x^2+x-2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(178740084 - 169466391*x - 1053213025*x^2 + 520073880*x^3 + 1667
076300*x^4))/(5478396*Sqrt[3 + 5*x]*(-2 + x + 6*x^2)^2) + (79515*ArcTan[(-20 - 3
7*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Maple [B]  time = 0.026, size = 305, normalized size = 1.9 \[ -{\frac{1}{76697544\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 57150611100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-25082768205\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-28257802155\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-14744982350\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2372529474\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/76697544*(1-2*x)^(1/2)*(57150611100*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^5+53340570360*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^4-25082768205*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))*x^3+23339068200*x^4*(-10*x^2-x+3)^(1/2)-28257802155*7^(1/2)*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+7281034320*x^3*(-10*x^2-x+3)^(1/2)+25
40027160*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-1474498235
0*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))-2372529474*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)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22703, size = 167, normalized size = 1.05 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1667076300 \, x^{4} + 520073880 \, x^{3} - 1053213025 \, x^{2} - 169466391 \, x + 178740084\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 317503395 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{76697544 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/76697544*sqrt(7)*(2*sqrt(7)*(1667076300*x^4 + 520073880*x^3 - 1053213025*x^2
- 169466391*x + 178740084)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 317503395*(180*x^5 + 1
68*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x +
3)*sqrt(-2*x + 1))))/(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.449638, size = 479, normalized size = 3.01 \[ -\frac{15903}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{625}{2662} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{32 \,{\left (944 \, \sqrt{5}{\left (5 \, x + 3\right )} - 5577 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{239679825 \,{\left (2 \, x - 1\right )}^{2}} - \frac{891 \,{\left (337 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 75880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4802 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15903/38416*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 625/2662*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))) - 32/239679825*(944*sq
rt(5)*(5*x + 3) - 5577*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 891/
4802*(337*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 + 75880*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)^2