3.437 \(\int \frac{1}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right )^3 x^6} \, dx\)

Optimal. Leaf size=101 \[ -\frac{12 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 c (b+2 c x)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{b+2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0912883, antiderivative size = 101, 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 \[ -\frac{12 c^2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 c (b+2 c x)}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{b+2 c x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((c + a/x^2 + b/x)^3*x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.6955, size = 95, normalized size = 0.94 \[ - \frac{12 c^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{3 c \left (b + 2 c x\right )}{\left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{b + 2 c x}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c+a/x**2+b/x)**3/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.159161, size = 97, normalized size = 0.96 \[ \frac{\frac{24 c^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{(b+2 c x) \left (-2 c \left (5 a+3 c x^2\right )+b^2-6 b c x\right )}{(a+x (b+c x))^2}}{2 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c + a/x^2 + b/x)^3*x^6),x]

[Out]

(-(((b + 2*c*x)*(b^2 - 6*b*c*x - 2*c*(5*a + 3*c*x^2)))/(a + x*(b + c*x))^2) + (2
4*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*
c)^2)

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Maple [A]  time = 0.005, size = 129, normalized size = 1.3 \[{\frac{2\,cx+b}{ \left ( 8\,ac-2\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{2}}}+6\,{\frac{{c}^{2}x}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+3\,{\frac{bc}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+12\,{\frac{{c}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^3/x^6,x)

[Out]

1/2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^2+6*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)*x+3*
c/(4*a*c-b^2)^2/(c*x^2+b*x+a)*b+12*c^2/(4*a*c-b^2)^(5/2)*arctan((2*c*x+b)/(4*a*c
-b^2)^(1/2))

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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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262293, size = 1, normalized size = 0.01 \[ \left [\frac{12 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{3} + 2 \, a b c^{2} x + a^{2} c^{2} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} - b^{3} + 10 \, a b c + 4 \,{\left (b^{2} c + 5 \, a c^{2}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{24 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{3} + 2 \, a b c^{2} x + a^{2} c^{2} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} - b^{3} + 10 \, a b c + 4 \,{\left (b^{2} c + 5 \, a c^{2}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^3*x^6),x, algorithm="fricas")

[Out]

[1/2*(12*(c^4*x^4 + 2*b*c^3*x^3 + 2*a*b*c^2*x + a^2*c^2 + (b^2*c^2 + 2*a*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)) + (12*c^3*x^3 + 18*b*c^2*x^2 - b^3 +
10*a*b*c + 4*(b^2*c + 5*a*c^2)*x)*sqrt(b^2 - 4*a*c))/((a^2*b^4 - 8*a^3*b^2*c + 1
6*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c
+ 16*a^3*b*c^2)*x)*sqrt(b^2 - 4*a*c)), 1/2*(24*(c^4*x^4 + 2*b*c^3*x^3 + 2*a*b*c^
2*x + a^2*c^2 + (b^2*c^2 + 2*a*c^3)*x^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/
(b^2 - 4*a*c)) + (12*c^3*x^3 + 18*b*c^2*x^2 - b^3 + 10*a*b*c + 4*(b^2*c + 5*a*c^
2)*x)*sqrt(-b^2 + 4*a*c))/((a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*
b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 -
6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt(-b^
2 + 4*a*c))]

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Sympy [A]  time = 5.19414, size = 474, normalized size = 4.69 \[ - 6 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{- 384 a^{3} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 288 a^{2} b^{2} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 72 a b^{4} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b^{6} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c^{2}}{12 c^{3}} \right )} + 6 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{384 a^{3} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 288 a^{2} b^{2} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 72 a b^{4} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 6 b^{6} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c^{2}}{12 c^{3}} \right )} + \frac{10 a b c - b^{3} + 18 b c^{2} x^{2} + 12 c^{3} x^{3} + x \left (20 a c^{2} + 4 b^{2} c\right )}{32 a^{4} c^{2} - 16 a^{3} b^{2} c + 2 a^{2} b^{4} + x^{4} \left (32 a^{2} c^{4} - 16 a b^{2} c^{3} + 2 b^{4} c^{2}\right ) + x^{3} \left (64 a^{2} b c^{3} - 32 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \left (64 a^{3} c^{3} - 12 a b^{4} c + 2 b^{6}\right ) + x \left (64 a^{3} b c^{2} - 32 a^{2} b^{3} c + 4 a b^{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c+a/x**2+b/x)**3/x**6,x)

[Out]

-6*c**2*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-384*a**3*c**5*sqrt(-1/(4*a*c - b**2
)**5) + 288*a**2*b**2*c**4*sqrt(-1/(4*a*c - b**2)**5) - 72*a*b**4*c**3*sqrt(-1/(
4*a*c - b**2)**5) + 6*b**6*c**2*sqrt(-1/(4*a*c - b**2)**5) + 6*b*c**2)/(12*c**3)
) + 6*c**2*sqrt(-1/(4*a*c - b**2)**5)*log(x + (384*a**3*c**5*sqrt(-1/(4*a*c - b*
*2)**5) - 288*a**2*b**2*c**4*sqrt(-1/(4*a*c - b**2)**5) + 72*a*b**4*c**3*sqrt(-1
/(4*a*c - b**2)**5) - 6*b**6*c**2*sqrt(-1/(4*a*c - b**2)**5) + 6*b*c**2)/(12*c**
3)) + (10*a*b*c - b**3 + 18*b*c**2*x**2 + 12*c**3*x**3 + x*(20*a*c**2 + 4*b**2*c
))/(32*a**4*c**2 - 16*a**3*b**2*c + 2*a**2*b**4 + x**4*(32*a**2*c**4 - 16*a*b**2
*c**3 + 2*b**4*c**2) + x**3*(64*a**2*b*c**3 - 32*a*b**3*c**2 + 4*b**5*c) + x**2*
(64*a**3*c**3 - 12*a*b**4*c + 2*b**6) + x*(64*a**3*b*c**2 - 32*a**2*b**3*c + 4*a
*b**5))

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GIAC/XCAS [A]  time = 0.313823, size = 184, normalized size = 1.82 \[ \frac{12 \, c^{2} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{3} x^{3} + 18 \, b c^{2} x^{2} + 4 \, b^{2} c x + 20 \, a c^{2} x - b^{3} + 10 \, a b c}{2 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c + b/x + a/x^2)^3*x^6),x, algorithm="giac")

[Out]

12*c^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*(12*c^3*x^3 + 18*b*c^2*x^2 + 4*b^2*c*x + 20*a*c^2*x - b^
3 + 10*a*b*c)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*(c*x^2 + b*x + a)^2)