3.2608 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=130 \[ -\frac{1840225 \sqrt{1-2 x}}{1369599 \sqrt{5 x+3}}-\frac{3830}{124509 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}-\frac{190}{1617 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{3105 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

[Out]

-190/(1617*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - 3830/(124509*Sqrt[1 - 2*x]*Sqrt[3 +
5*x]) - (1840225*Sqrt[1 - 2*x])/(1369599*Sqrt[3 + 5*x]) + 3/(7*(1 - 2*x)^(3/2)*(
2 + 3*x)*Sqrt[3 + 5*x]) + (3105*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(
343*Sqrt[7])

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Rubi [A]  time = 0.333581, 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 \[ -\frac{1840225 \sqrt{1-2 x}}{1369599 \sqrt{5 x+3}}-\frac{3830}{124509 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}-\frac{190}{1617 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{3105 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-190/(1617*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - 3830/(124509*Sqrt[1 - 2*x]*Sqrt[3 +
5*x]) - (1840225*Sqrt[1 - 2*x])/(1369599*Sqrt[3 + 5*x]) + 3/(7*(1 - 2*x)^(3/2)*(
2 + 3*x)*Sqrt[3 + 5*x]) + (3105*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(
343*Sqrt[7])

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Rubi in Sympy [A]  time = 28.3811, size = 119, normalized size = 0.92 \[ - \frac{1840225 \sqrt{- 2 x + 1}}{1369599 \sqrt{5 x + 3}} + \frac{3105 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2401} - \frac{3830}{124509 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{190}{1617 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{3}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1840225*sqrt(-2*x + 1)/(1369599*sqrt(5*x + 3)) + 3105*sqrt(7)*atan(sqrt(7)*sqrt
(-2*x + 1)/(7*sqrt(5*x + 3)))/2401 - 3830/(124509*sqrt(-2*x + 1)*sqrt(5*x + 3))
- 190/(1617*(-2*x + 1)**(3/2)*sqrt(5*x + 3)) + 3/(7*(-2*x + 1)**(3/2)*(3*x + 2)*
sqrt(5*x + 3))

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Mathematica [A]  time = 0.110323, size = 82, normalized size = 0.63 \[ \frac{-22082700 x^3+7613680 x^2+8760465 x-3499599}{1369599 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}+\frac{3105 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{686 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-3499599 + 8760465*x + 7613680*x^2 - 22082700*x^3)/(1369599*(1 - 2*x)^(3/2)*(2
+ 3*x)*Sqrt[3 + 5*x]) + (3105*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x
])])/(686*Sqrt[7])

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Maple [B]  time = 0.025, size = 257, normalized size = 2. \[ -{\frac{1}{ \left ( 38348772+57523158\,x \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 743895900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-458735805\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-61991325\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-106591520\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -122646510\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/19174386*(1-2*x)^(1/2)*(743895900*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^4+198372240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))*x^3-458735805*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^2+309157800*x^3*(-10*x^2-x+3)^(1/2)-61991325*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x-106591520*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))-122646510*x*(-10*x^2-x+3)^(1
/2)[phone]*(-10*x^2-x+3)^(1/2))/(2+3*x)/(-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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229766, size = 147, normalized size = 1.13 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (22082700 \, x^{3} - 7613680 \, x^{2} - 8760465 \, x + 3499599\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 12398265 \,{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{19174386 \,{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/19174386*sqrt(7)*(2*sqrt(7)*(22082700*x^3 - 7613680*x^2 - 8760465*x + 3499599
)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 12398265*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*a
rctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(60*x^4 + 16*x^3
- 37*x^2 - 5*x + 6)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.337768, size = 393, normalized size = 3.02 \[ -\frac{621}{9604} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{125}{2662} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{1782 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{343 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} - \frac{32 \,{\left (373 \, \sqrt{5}{\left (5 \, x + 3\right )} - 2244 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{34239975 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-621/9604*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)))) - 125/2662*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))) - 1782/343*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)))/(((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) - 32/34239975*(373*
sqrt(5)*(5*x + 3) - 2244*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2