3.2562 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=159 \[ \frac{707286025 \sqrt{1-2 x}}{5478396 \sqrt{5 x+3}}-\frac{7090175 \sqrt{1-2 x}}{498036 (5 x+3)^{3/2}}-\frac{8515}{7546 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{765}{196 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}-\frac{1215945 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

-8515/(7546*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (7090175*Sqrt[1 - 2*x])/(498036*(3
+ 5*x)^(3/2)) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + 765/(196*Sqrt
[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2)) + (707286025*Sqrt[1 - 2*x])/(5478396*Sqrt[3
+ 5*x]) - (1215945*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7]
)

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Rubi [A]  time = 0.412832, 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 \[ \frac{707286025 \sqrt{1-2 x}}{5478396 \sqrt{5 x+3}}-\frac{7090175 \sqrt{1-2 x}}{498036 (5 x+3)^{3/2}}-\frac{8515}{7546 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{765}{196 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}-\frac{1215945 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-8515/(7546*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (7090175*Sqrt[1 - 2*x])/(498036*(3
+ 5*x)^(3/2)) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + 765/(196*Sqrt
[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2)) + (707286025*Sqrt[1 - 2*x])/(5478396*Sqrt[3
+ 5*x]) - (1215945*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7]
)

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Rubi in Sympy [A]  time = 36.5257, size = 146, normalized size = 0.92 \[ \frac{707286025 \sqrt{- 2 x + 1}}{5478396 \sqrt{5 x + 3}} - \frac{7090175 \sqrt{- 2 x + 1}}{498036 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{1215945 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} - \frac{8515}{7546 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{765}{196 \sqrt{- 2 x + 1} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{3}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

707286025*sqrt(-2*x + 1)/(5478396*sqrt(5*x + 3)) - 7090175*sqrt(-2*x + 1)/(49803
6*(5*x + 3)**(3/2)) - 1215945*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x +
3)))/9604 - 8515/(7546*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 765/(196*sqrt(-2*x + 1
)*(3*x + 2)*(5*x + 3)**(3/2)) + 3/(14*sqrt(-2*x + 1)*(3*x + 2)**2*(5*x + 3)**(3/
2))

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Mathematica [A]  time = 0.12058, size = 87, normalized size = 0.55 \[ \frac{-63655742250 x^4-89836042575 x^3-16567908760 x^2+22311149965 x+8194676012}{5478396 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}-\frac{1215945 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(8194676012 + 22311149965*x - 16567908760*x^2 - 89836042575*x^3 - 63655742250*x^
4)/(5478396*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (1215945*ArcTan[(-20 -
37*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 \[{\frac{1}{76697544\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) }\sqrt{1-2\,x} \left ( 2184870773250\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-412697812725\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-757421868060\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+231950722640\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-174789661860\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -312356099510\,x\sqrt{-10\,{x}^{2}-x+3}-114725464168\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/76697544*(1-2*x)^(1/2)*(2184870773250*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x^5+4442570572275*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^4+2485897413120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^3+891180391500*x^4*(-10*x^2-x+3)^(1/2)-412697812725*7^(1/2)*arcta
n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1257704596050*x^3*(-10*x^2-x+3
)^(1/2)-757421868060*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x+231950722640*x^2*(-10*x^2-x+3)^(1/2)-174789661860*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))-312356099510*x*(-10*x^2-x+3)^(1/2)-114725464168*(
-10*x^2-x+3)^(1/2))/(2+3*x)^2/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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

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)^(3/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.237201, size = 167, normalized size = 1.05 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (63655742250 \, x^{4} + 89836042575 \, x^{3} + 16567908760 \, x^{2} - 22311149965 \, x - 8194676012\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 4855268385 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{76697544 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} \]

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)^(3/2)),x, algorithm="fricas")

[Out]

1/76697544*sqrt(7)*(2*sqrt(7)*(63655742250*x^4 + 89836042575*x^3 + 16567908760*x
^2 - 22311149965*x - 8194676012)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 4855268385*(450*
x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*arctan(1/14*sqrt(7)*(37*x + 20)/(
sqrt(5*x + 3)*sqrt(-2*x + 1))))/(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*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)**(3/2)/(2+3*x)**3/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.48209, size = 544, normalized size = 3.42 \[ -\frac{125}{63888} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{243189}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{11875}{2662} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{64 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{2282665 \,{\left (2 \, x - 1\right )}} + \frac{891 \,{\left (67 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 16120 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

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)^(3/2)),x, algorithm="giac")

[Out]

-125/63888*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 + 243189/38416*sqrt(70)*sqrt(1
0)*(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)))) + 11875/2662*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))) - 64/2282665*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1) + 891/98*(67*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 + 16120*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