3.2617 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{5 x+3}}-\frac{19130 \sqrt{1-2 x}}{195657 (5 x+3)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{162 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 412/(5929*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
)) - (19130*Sqrt[1 - 2*x])/(195657*(3 + 5*x)^(3/2)) + (1001590*Sqrt[1 - 2*x])/(2
152227*Sqrt[3 + 5*x]) - (162*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*
Sqrt[7])

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Rubi [A]  time = 0.320598, antiderivative size = 123, 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 \[ \frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{5 x+3}}-\frac{19130 \sqrt{1-2 x}}{195657 (5 x+3)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{162 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

4/(231*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 412/(5929*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
)) - (19130*Sqrt[1 - 2*x])/(195657*(3 + 5*x)^(3/2)) + (1001590*Sqrt[1 - 2*x])/(2
152227*Sqrt[3 + 5*x]) - (162*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*
Sqrt[7])

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Rubi in Sympy [A]  time = 29.6887, size = 114, normalized size = 0.93 \[ - \frac{162 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} - \frac{400636 \sqrt{5 x + 3}}{2152227 \sqrt{- 2 x + 1}} + \frac{29710}{27951 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{370}{2541 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-162*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/343 - 400636*sqrt(5*
x + 3)/(2152227*sqrt(-2*x + 1)) + 29710/(27951*sqrt(-2*x + 1)*sqrt(5*x + 3)) - 3
70/(2541*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 4/(231*(-2*x + 1)**(3/2)*(5*x + 3)**
(3/2))

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Mathematica [A]  time = 0.152722, size = 75, normalized size = 0.61 \[ \frac{20031800 x^3-8854440 x^2-6468522 x+2981164}{2152227 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{81 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(2981164 - 6468522*x - 8854440*x^2 + 20031800*x^3)/(2152227*(1 - 2*x)^(3/2)*(3 +
5*x)^(3/2)) - (81*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(49*Sq
rt[7])

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Maple [B]  time = 0.023, size = 250, normalized size = 2. \[{\frac{1}{15065589\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 355776300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-209908017\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-21346578\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-61981080\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -45279654\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15065589*(1-2*x)^(1/2)*(355776300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))*x^4+71155260*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^
(1/2))*x^3-209908017*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x^2+140222600*x^3*(-10*x^2-x+3)^(1/2)-21346578*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x-61981080*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-45279654*x*(-10*x^2-x+3)^(1/2)+
20868148*(-10*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.50277, size = 117, normalized size = 0.95 \[ \frac{81}{343} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2003180 \, x}{2152227 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1085762}{2152227 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{740 \, x}{2541 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{326}{2541 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/343*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2003180/21522
27*x/sqrt(-10*x^2 - x + 3) + 1085762/2152227/sqrt(-10*x^2 - x + 3) + 740/2541*x/
(-10*x^2 - x + 3)^(3/2) - 326/2541/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.234832, size = 147, normalized size = 1.2 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (10015900 \, x^{3} - 4427220 \, x^{2} - 3234261 \, x + 1490582\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3557763 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{15065589 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15065589*sqrt(7)*(2*sqrt(7)*(10015900*x^3 - 4427220*x^2 - 3234261*x + 1490582)
*sqrt(5*x + 3)*sqrt(-2*x + 1) + 3557763*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*ar
ctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(100*x^4 + 20*x^3
- 59*x^2 - 6*x + 9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279605, size = 315, normalized size = 2.56 \[ -\frac{25}{702768} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{81}{3430} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{675}{29282} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{32 \,{\left (379 \, \sqrt{5}{\left (5 \, x + 3\right )} - 2277 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{53805675 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/702768*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 + 81/3430*sqrt(70)*sqrt(10)*(p
i + 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)))) + 675/29282*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))) - 32/53805675*(379*sqrt(5)*(5*x + 3) - 2277*sqrt(5))*s
qrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2