3.2618 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=152 \[ \frac{95783075 \sqrt{1-2 x}}{15065589 \sqrt{5 x+3}}-\frac{985525 \sqrt{1-2 x}}{1369599 (5 x+3)^{3/2}}-\frac{1090}{41503 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac{190}{1617 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{14985 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

[Out]

-190/(1617*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - 1090/(41503*Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2)) - (985525*Sqrt[1 - 2*x])/(1369599*(3 + 5*x)^(3/2)) + 3/(7*(1 - 2*x)^(3/
2)*(2 + 3*x)*(3 + 5*x)^(3/2)) + (95783075*Sqrt[1 - 2*x])/(15065589*Sqrt[3 + 5*x]
) - (14985*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

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Rubi [A]  time = 0.406552, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{95783075 \sqrt{1-2 x}}{15065589 \sqrt{5 x+3}}-\frac{985525 \sqrt{1-2 x}}{1369599 (5 x+3)^{3/2}}-\frac{1090}{41503 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac{190}{1617 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{14985 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-190/(1617*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - 1090/(41503*Sqrt[1 - 2*x]*(3 + 5*x
)^(3/2)) - (985525*Sqrt[1 - 2*x])/(1369599*(3 + 5*x)^(3/2)) + 3/(7*(1 - 2*x)^(3/
2)*(2 + 3*x)*(3 + 5*x)^(3/2)) + (95783075*Sqrt[1 - 2*x])/(15065589*Sqrt[3 + 5*x]
) - (14985*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

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Rubi in Sympy [A]  time = 38.642, size = 139, normalized size = 0.91 \[ \frac{95783075 \sqrt{- 2 x + 1}}{15065589 \sqrt{5 x + 3}} - \frac{985525 \sqrt{- 2 x + 1}}{1369599 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{14985 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2401} - \frac{1090}{41503 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{190}{1617 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{3}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

95783075*sqrt(-2*x + 1)/(15065589*sqrt(5*x + 3)) - 985525*sqrt(-2*x + 1)/(136959
9*(5*x + 3)**(3/2)) - 14985*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)
))/2401 - 1090/(41503*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) - 190/(1617*(-2*x + 1)**(
3/2)*(5*x + 3)**(3/2)) + 3/(7*(-2*x + 1)**(3/2)*(3*x + 2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.144564, size = 87, normalized size = 0.57 \[ \frac{5746984500 x^4+1402439900 x^3-3498236655 x^2-429626520 x+555141781}{15065589 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac{14985 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{686 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(555141781 - 429626520*x - 3498236655*x^2 + 1402439900*x^3 + 5746984500*x^4)/(15
065589*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2)) - (14985*ArcTan[(-20 - 37*x)/(
2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(686*Sqrt[7])

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Maple [B]  time = 0.026, size = 305, normalized size = 2. \[{\frac{1}{ \left ( 421836492+632754738\,x \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 197455846500\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-90171503235\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-89513317080\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-48975313170\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -6014771280\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/210918246*(1-2*x)^(1/2)*(197455846500*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x^5+171128400300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))*x^4-90171503235*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))*x^3+80457783000*x^4*(-10*x^2-x+3)^(1/2)-89513317080*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+19634158600*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-4897531
3170*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))-6014771280*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)^(3/2)

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Maxima [A]  time = 1.51228, size = 163, normalized size = 1.07 \[ \frac{14985}{4802} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{191566150 \, x}{15065589 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{100119385}{15065589 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{57250 \, x}{17787 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{3}{7 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{30715}{17787 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14985/4802*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 191566150
/15065589*x/sqrt(-10*x^2 - x + 3) + 100119385/15065589/sqrt(-10*x^2 - x + 3) + 5
7250/17787*x/(-10*x^2 - x + 3)^(3/2) + 3/7/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10
*x^2 - x + 3)^(3/2)) - 30715/17787/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.226336, size = 167, normalized size = 1.1 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (5746984500 \, x^{4} + 1402439900 \, x^{3} - 3498236655 \, x^{2} - 429626520 \, x + 555141781\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 658186155 \,{\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{210918246 \,{\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} \]

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)^(5/2)),x, algorithm="fricas")

[Out]

1/210918246*sqrt(7)*(2*sqrt(7)*(5746984500*x^4 + 1402439900*x^3 - 3498236655*x^2
- 429626520*x + 555141781)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 658186155*(300*x^5 +
260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5
*x + 3)*sqrt(-2*x + 1))))/(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*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)**(5/2)/(2+3*x)**2/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.424971, size = 475, normalized size = 3.12 \[ -\frac{125}{702768} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{2997}{9604} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{3750}{14641} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{5346 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{343 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} - \frac{32 \,{\left (956 \, \sqrt{5}{\left (5 \, x + 3\right )} - 5643 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{376639725 \,{\left (2 \, x - 1\right )}^{2}} \]

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)^(5/2)),x, algorithm="giac")

[Out]

-125/702768*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 + 2997/9604*sqrt(70)*sqrt(10)
*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 3750/14641*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))) + 5346/343*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(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)*sq
rt(-10*x + 5) - sqrt(22)))^2 + 280) - 32/376639725*(956*sqrt(5)*(5*x + 3) - 5643
*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2