3.433 \(\int \frac{1}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right )^3 x^2} \, dx\)

Optimal. Leaf size=111 \[ \frac{12 a^2 \tanh ^{-1}\left (\frac{\frac{2 a}{x}+b}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{3 a \left (\frac{2 a}{x}+b\right )}{\left (b^2-4 a c\right )^2 \left (\frac{a}{x^2}+\frac{b}{x}+c\right )}+\frac{\frac{2 a}{x}+b}{2 \left (b^2-4 a c\right ) \left (\frac{a}{x^2}+\frac{b}{x}+c\right )^2} \]

[Out]

(b + (2*a)/x)/(2*(b^2 - 4*a*c)*(c + a/x^2 + b/x)^2) - (3*a*(b + (2*a)/x))/((b^2
- 4*a*c)^2*(c + a/x^2 + b/x)) + (12*a^2*ArcTanh[(b + (2*a)/x)/Sqrt[b^2 - 4*a*c]]
)/(b^2 - 4*a*c)^(5/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.139202, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{12 a^2 \tanh ^{-1}\left (\frac{\frac{2 a}{x}+b}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{3 a \left (\frac{2 a}{x}+b\right )}{\left (b^2-4 a c\right )^2 \left (\frac{a}{x^2}+\frac{b}{x}+c\right )}+\frac{\frac{2 a}{x}+b}{2 \left (b^2-4 a c\right ) \left (\frac{a}{x^2}+\frac{b}{x}+c\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((c + a/x^2 + b/x)^3*x^2),x]

[Out]

(b + (2*a)/x)/(2*(b^2 - 4*a*c)*(c + a/x^2 + b/x)^2) - (3*a*(b + (2*a)/x))/((b^2
- 4*a*c)^2*(c + a/x^2 + b/x)) + (12*a^2*ArcTanh[(b + (2*a)/x)/Sqrt[b^2 - 4*a*c]]
)/(b^2 - 4*a*c)^(5/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.3953, size = 95, normalized size = 0.86 \[ \frac{12 a^{2} \operatorname{atanh}{\left (\frac{\frac{2 a}{x} + b}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{3 a \left (\frac{2 a}{x} + b\right )}{\left (- 4 a c + b^{2}\right )^{2} \left (\frac{a}{x^{2}} + \frac{b}{x} + c\right )} + \frac{\frac{2 a}{x} + b}{2 \left (- 4 a c + b^{2}\right ) \left (\frac{a}{x^{2}} + \frac{b}{x} + c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c+a/x**2+b/x)**3/x**2,x)

[Out]

12*a**2*atanh((2*a/x + b)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(5/2) - 3*a*(2*a
/x + b)/((-4*a*c + b**2)**2*(a/x**2 + b/x + c)) + (2*a/x + b)/(2*(-4*a*c + b**2)
*(a/x**2 + b/x + c)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.27454, size = 174, normalized size = 1.57 \[ \frac{1}{2} \left (\frac{24 a^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{a^2 c (2 c x-3 b)+a b^2 (b-4 c x)+b^4 x}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{22 a^2 b c^2-20 a^2 c^3 x-8 a b^3 c+16 a b^2 c^2 x+b^5-2 b^4 c x}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c + a/x^2 + b/x)^3*x^2),x]

[Out]

((b^5 - 8*a*b^3*c + 22*a^2*b*c^2 - 2*b^4*c*x + 16*a*b^2*c^2*x - 20*a^2*c^3*x)/(c
^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (b^4*x + a*b^2*(b - 4*c*x) + a^2*c*(-3*b
+ 2*c*x))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (24*a^2*ArcTan[(b + 2*c*x)
/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2

_______________________________________________________________________________________

Maple [B]  time = 0.015, size = 260, normalized size = 2.3 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ( -{\frac{ \left ( 10\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ){x}^{3}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) c}}+{\frac{b \left ( 2\,{a}^{2}{c}^{2}+8\,a{b}^{2}c-{b}^{4} \right ){x}^{2}}{2\,{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}-{\frac{a \left ( 6\,{a}^{2}{c}^{2}-10\,a{b}^{2}c+{b}^{4} \right ) x}{{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{{a}^{2}b \left ( 10\,ac-{b}^{2} \right ) }{2\,{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }} \right ) }+12\,{\frac{{a}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^3/x^2,x)

[Out]

(-1/c*(10*a^2*c^2-8*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*b*(2*a^2*c^2
+8*a*b^2*c-b^4)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-a*(6*a^2*c^2-10*a*b^2*c+b^4)/
(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x+1/2*a^2*b*(10*a*c-b^2)/c^2/(16*a^2*c^2-8*a*b^2*
c+b^4))/(c*x^2+b*x+a)^2+12*a^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^3*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.269721, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^3*x^2),x, algorithm="fricas")

[Out]

[1/2*(12*(a^2*c^4*x^4 + 2*a^2*b*c^3*x^3 + 2*a^3*b*c^2*x + a^4*c^2 + (a^2*b^2*c^2
+ 2*a^3*c^3)*x^2)*log(-(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x - (2*c^2*x^2 + 2*
b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) - (a^2*b^3 - 10*a^3*b
*c + 2*(b^4*c - 8*a*b^2*c^2 + 10*a^2*c^3)*x^3 + (b^5 - 8*a*b^3*c - 2*a^2*b*c^2)*
x^2 + 2*(a*b^4 - 10*a^2*b^2*c + 6*a^3*c^2)*x)*sqrt(b^2 - 4*a*c))/((a^2*b^4*c^2 -
8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*
c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2
+ 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x)*sqrt(b^2 - 4*a*c)), 1/2*(24*(
a^2*c^4*x^4 + 2*a^2*b*c^3*x^3 + 2*a^3*b*c^2*x + a^4*c^2 + (a^2*b^2*c^2 + 2*a^3*c
^3)*x^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (a^2*b^3 - 10*a
^3*b*c + 2*(b^4*c - 8*a*b^2*c^2 + 10*a^2*c^3)*x^3 + (b^5 - 8*a*b^3*c - 2*a^2*b*c
^2)*x^2 + 2*(a*b^4 - 10*a^2*b^2*c + 6*a^3*c^2)*x)*sqrt(-b^2 + 4*a*c))/((a^2*b^4*
c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*
(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5
)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x)*sqrt(-b^2 + 4*a*c))]

_______________________________________________________________________________________

Sympy [A]  time = 6.66454, size = 547, normalized size = 4.93 \[ - 6 a^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{- 384 a^{5} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 288 a^{4} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 72 a^{3} b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 a^{2} b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 a^{2} b}{12 a^{2} c} \right )} + 6 a^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{384 a^{5} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 288 a^{4} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 72 a^{3} b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 6 a^{2} b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 a^{2} b}{12 a^{2} c} \right )} - \frac{- 10 a^{3} b c + a^{2} b^{3} + x^{3} \left (20 a^{2} c^{3} - 16 a b^{2} c^{2} + 2 b^{4} c\right ) + x^{2} \left (- 2 a^{2} b c^{2} - 8 a b^{3} c + b^{5}\right ) + x \left (12 a^{3} c^{2} - 20 a^{2} b^{2} c + 2 a b^{4}\right )}{32 a^{4} c^{4} - 16 a^{3} b^{2} c^{3} + 2 a^{2} b^{4} c^{2} + x^{4} \left (32 a^{2} c^{6} - 16 a b^{2} c^{5} + 2 b^{4} c^{4}\right ) + x^{3} \left (64 a^{2} b c^{5} - 32 a b^{3} c^{4} + 4 b^{5} c^{3}\right ) + x^{2} \left (64 a^{3} c^{5} - 12 a b^{4} c^{3} + 2 b^{6} c^{2}\right ) + x \left (64 a^{3} b c^{4} - 32 a^{2} b^{3} c^{3} + 4 a b^{5} c^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c+a/x**2+b/x)**3/x**2,x)

[Out]

-6*a**2*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-384*a**5*c**3*sqrt(-1/(4*a*c - b**2
)**5) + 288*a**4*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) - 72*a**3*b**4*c*sqrt(-1/(
4*a*c - b**2)**5) + 6*a**2*b**6*sqrt(-1/(4*a*c - b**2)**5) + 6*a**2*b)/(12*a**2*
c)) + 6*a**2*sqrt(-1/(4*a*c - b**2)**5)*log(x + (384*a**5*c**3*sqrt(-1/(4*a*c -
b**2)**5) - 288*a**4*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) + 72*a**3*b**4*c*sqrt(
-1/(4*a*c - b**2)**5) - 6*a**2*b**6*sqrt(-1/(4*a*c - b**2)**5) + 6*a**2*b)/(12*a
**2*c)) - (-10*a**3*b*c + a**2*b**3 + x**3*(20*a**2*c**3 - 16*a*b**2*c**2 + 2*b*
*4*c) + x**2*(-2*a**2*b*c**2 - 8*a*b**3*c + b**5) + x*(12*a**3*c**2 - 20*a**2*b*
*2*c + 2*a*b**4))/(32*a**4*c**4 - 16*a**3*b**2*c**3 + 2*a**2*b**4*c**2 + x**4*(3
2*a**2*c**6 - 16*a*b**2*c**5 + 2*b**4*c**4) + x**3*(64*a**2*b*c**5 - 32*a*b**3*c
**4 + 4*b**5*c**3) + x**2*(64*a**3*c**5 - 12*a*b**4*c**3 + 2*b**6*c**2) + x*(64*
a**3*b*c**4 - 32*a**2*b**3*c**3 + 4*a*b**5*c**2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.295476, size = 273, normalized size = 2.46 \[ \frac{12 \, a^{2} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{4} c x^{3} - 16 \, a b^{2} c^{2} x^{3} + 20 \, a^{2} c^{3} x^{3} + b^{5} x^{2} - 8 \, a b^{3} c x^{2} - 2 \, a^{2} b c^{2} x^{2} + 2 \, a b^{4} x - 20 \, a^{2} b^{2} c x + 12 \, a^{3} c^{2} x + a^{2} b^{3} - 10 \, a^{3} b c}{2 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^3*x^2),x, algorithm="giac")

[Out]

12*a^2*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sq
rt(-b^2 + 4*a*c)) - 1/2*(2*b^4*c*x^3 - 16*a*b^2*c^2*x^3 + 20*a^2*c^3*x^3 + b^5*x
^2 - 8*a*b^3*c*x^2 - 2*a^2*b*c^2*x^2 + 2*a*b^4*x - 20*a^2*b^2*c*x + 12*a^3*c^2*x
+ a^2*b^3 - 10*a^3*b*c)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^2 + b*x + a)
^2)