3.2552 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{3125575 \sqrt{1-2 x}}{166012 \sqrt{5 x+3}}-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{555}{196 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{177255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

-6205/(7546*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (3125575*Sqrt[1 - 2*x])/(166012*Sqrt[
3 + 5*x]) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 555/(196*Sqrt[1 - 2
*x]*(2 + 3*x)*Sqrt[3 + 5*x]) + (177255*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.335411, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3125575 \sqrt{1-2 x}}{166012 \sqrt{5 x+3}}-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{555}{196 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{177255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-6205/(7546*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (3125575*Sqrt[1 - 2*x])/(166012*Sqrt[
3 + 5*x]) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 555/(196*Sqrt[1 - 2
*x]*(2 + 3*x)*Sqrt[3 + 5*x]) + (177255*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(1372*Sqrt[7])

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Rubi in Sympy [A]  time = 29.3628, size = 126, normalized size = 0.92 \[ - \frac{3125575 \sqrt{- 2 x + 1}}{166012 \sqrt{5 x + 3}} + \frac{177255 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} - \frac{6205}{7546 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{555}{196 \sqrt{- 2 x + 1} \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{3}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} \]

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)**(3/2),x)

[Out]

-3125575*sqrt(-2*x + 1)/(166012*sqrt(5*x + 3)) + 177255*sqrt(7)*atan(sqrt(7)*sqr
t(-2*x + 1)/(7*sqrt(5*x + 3)))/9604 - 6205/(7546*sqrt(-2*x + 1)*sqrt(5*x + 3)) +
555/(196*sqrt(-2*x + 1)*(3*x + 2)*sqrt(5*x + 3)) + 3/(14*sqrt(-2*x + 1)*(3*x +
2)**2*sqrt(5*x + 3))

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Mathematica [A]  time = 0.113021, size = 82, normalized size = 0.6 \[ \frac{56260350 x^3+45655035 x^2-12730165 x-12072596}{166012 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{177255 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-12072596 - 12730165*x + 45655035*x^2 + 56260350*x^3)/(166012*Sqrt[1 - 2*x]*(2
+ 3*x)^2*Sqrt[3 + 5*x]) + (177255*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 = 257, normalized size = 1.9 \[ -{\frac{1}{2324168\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) }\sqrt{1-2\,x} \left ( 1930306950\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-686331360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+639170490\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-257374260\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -178222310\,x\sqrt{-10\,{x}^{2}-x+3}-169016344\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2324168*(1-2*x)^(1/2)*(1930306950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x^4+2766773295*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))*x^3+536196375*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^2+787644900*x^3*(-10*x^2-x+3)^(1/2)-686331360*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+639170490*x^2*(-10*x^2-x+3)^(1/2)-257374260*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-178222310*x*(-10*x^2-x+3)
^(1/2)-169016344*(-10*x^2-x+3)^(1/2))/(2+3*x)^2/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+
5*x)^(1/2)

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Maxima [A]  time = 1.50601, size = 193, normalized size = 1.41 \[ -\frac{177255}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3125575 \, x}{83006 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3262085}{166012 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3}{14 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{555}{196 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-177255/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 312557
5/83006*x/sqrt(-10*x^2 - x + 3) - 3262085/166012/sqrt(-10*x^2 - x + 3) + 3/14/(9
*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3
)) + 555/196/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.241395, size = 147, normalized size = 1.07 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (56260350 \, x^{3} + 45655035 \, x^{2} - 12730165 \, x - 12072596\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 21447855 \,{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2324168 \,{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2324168*sqrt(7)*(2*sqrt(7)*(56260350*x^3 + 45655035*x^2 - 12730165*x - 120725
96)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 21447855*(90*x^4 + 129*x^3 + 25*x^2 - 32*x -
12)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(90*x^4 + 1
29*x^3 + 25*x^2 - 32*x - 12)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)**(3/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.40021, size = 462, normalized size = 3.37 \[ -\frac{35451}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{125}{242} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{32 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{207515 \,{\left (2 \, x - 1\right )}} - \frac{297 \,{\left (47 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 10520 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-35451/38416*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))) - 125/242*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))) - 32/207515*sqrt(5)*sqr
t(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 297/98*(47*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 + 10520*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