3.2561 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=130 \[ \frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{5 x+3}}-\frac{28705 \sqrt{1-2 x}}{17787 (5 x+3)^{3/2}}-\frac{58}{539 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac{4887 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

-58/(539*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (28705*Sqrt[1 - 2*x])/(17787*(3 + 5*x)
^(3/2)) + 3/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2)) + (2841815*Sqrt[1 - 2*x]
)/(195657*Sqrt[3 + 5*x]) - (4887*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
(49*Sqrt[7])

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Rubi [A]  time = 0.324932, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{5 x+3}}-\frac{28705 \sqrt{1-2 x}}{17787 (5 x+3)^{3/2}}-\frac{58}{539 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac{4887 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-58/(539*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (28705*Sqrt[1 - 2*x])/(17787*(3 + 5*x)
^(3/2)) + 3/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(3/2)) + (2841815*Sqrt[1 - 2*x]
)/(195657*Sqrt[3 + 5*x]) - (4887*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
(49*Sqrt[7])

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Rubi in Sympy [A]  time = 29.324, size = 119, normalized size = 0.92 \[ \frac{2841815 \sqrt{- 2 x + 1}}{195657 \sqrt{5 x + 3}} - \frac{28705 \sqrt{- 2 x + 1}}{17787 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{4887 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} - \frac{58}{539 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{3}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2841815*sqrt(-2*x + 1)/(195657*sqrt(5*x + 3)) - 28705*sqrt(-2*x + 1)/(17787*(5*x
+ 3)**(3/2)) - 4887*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/343
- 58/(539*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 3/(7*sqrt(-2*x + 1)*(3*x + 2)*(5*x
+ 3)**(3/2))

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Mathematica [A]  time = 0.11028, size = 85, normalized size = 0.65 \[ \frac{\sqrt{1-2 x} \left (85254450 x^3+63467215 x^2-20145298 x-16461125\right )}{195657 (5 x+3)^{3/2} \left (6 x^2+x-2\right )}-\frac{4887 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{98 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-16461125 - 20145298*x + 63467215*x^2 + 85254450*x^3))/(195657*(
3 + 5*x)^(3/2)*(-2 + x + 6*x^2)) - (4887*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*S
qrt[3 + 5*x])])/(98*Sqrt[7])

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Maple [B]  time = 0.024, size = 257, normalized size = 2. \[{\frac{1}{ \left ( 5478396+8217594\,x \right ) \left ( -1+2\,x \right ) }\sqrt{1-2\,x} \left ( 2927068650\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-995203341\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+888541010\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-351248238\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -282034172\,x\sqrt{-10\,{x}^{2}-x+3}-230455750\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2739198*(1-2*x)^(1/2)*(2927068650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))*x^4+4000327155*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))*x^3+663468894*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^2+1193562300*x^3*(-10*x^2-x+3)^(1/2)-995203341*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+888541010*x^2*(-10*x^2-x+3)^(1/2)-351248238*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-282034172*x*(-10*x^2-x+3)
^(1/2)-230455750*(-10*x^2-x+3)^(1/2))/(2+3*x)/(-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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236565, size = 147, normalized size = 1.13 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (85254450 \, x^{3} + 63467215 \, x^{2} - 20145298 \, x - 16461125\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 19513791 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2739198 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2739198*sqrt(7)*(2*sqrt(7)*(85254450*x^3 + 63467215*x^2 - 20145298*x - 1646112
5)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 19513791*(150*x^4 + 205*x^3 + 34*x^2 - 51*x -
18)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(150*x^4 +
205*x^3 + 34*x^2 - 51*x - 18)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.366384, size = 458, normalized size = 3.52 \[ -\frac{25}{63888} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{4887}{6860} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{775}{1331} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{32 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{326095 \,{\left (2 \, x - 1\right )}} + \frac{1782 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{49 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/63888*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 + 4887/6860*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)))) + 775/1331*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))) - 32/326095*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x
- 1) + 1782/49*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)