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

Optimal. Leaf size=107 \[ \frac{3 b x (2 a+b x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{x^3 (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{6 a b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

[Out]

-(x^3*(b + 2*c*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*b*x*(2*a + b*x))/(
2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (6*a*b*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 = 0.113438, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{3 b x (2 a+b x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{x^3 (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{6 a b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(x^3*(b + 2*c*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*b*x*(2*a + b*x))/(
2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) + (6*a*b*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 = 24.14, size = 102, normalized size = 0.95 \[ \frac{6 a b \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{3 b x \left (2 a + b x\right )}{2 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{x^{3} \left (b + 2 c x\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.348262, size = 126, normalized size = 1.18 \[ -\frac{8 a^3 c+a^2 \left (b^2+10 b c x+16 c^2 x^2\right )+a b x \left (2 b^2+b c x+6 c^2 x^2\right )+b^4 x^2}{2 c \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac{6 a b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(8*a^3*c + b^4*x^2 + a*b*x*(2*b^2 + b*c*x + 6*c^2*x^2) + a^2*(b^2 + 10*b*c*x +
16*c^2*x^2))/(2*c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (6*a*b*ArcTan[(b + 2*c*
x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2)

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Maple [B]  time = 0.013, size = 223, normalized size = 2.1 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ( -3\,{\frac{abc{x}^{3}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}-{\frac{ \left ( 16\,{a}^{2}{c}^{2}+a{b}^{2}c+{b}^{4} \right ){x}^{2}}{2\, \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) c}}-{\frac{ \left ( 5\,ac+{b}^{2} \right ) abx}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) c}}-{\frac{{a}^{2} \left ( 8\,ac+{b}^{2} \right ) }{2\, \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) c}} \right ) }-6\,{\frac{ab}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^3/x^3,x)

[Out]

(-3*a*b*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*(16*a^2*c^2+a*b^2*c+b^4)/c/(16*a^2*
c^2-8*a*b^2*c+b^4)*x^2-(5*a*c+b^2)*a*b/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1/2*a^2*(8
*a*c+b^2)/c/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2-6*a*b/(16*a^2*c^2-8*a*b^
2*c+b^4)/(4*a*c-b^2)^(1/2)*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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272292, size = 1, normalized size = 0.01 \[ \left [\frac{6 \,{\left (a b c^{3} x^{4} + 2 \, a b^{2} c^{2} x^{3} + 2 \, a^{2} b^{2} c x + a^{3} b c +{\left (a b^{3} c + 2 \, a^{2} b c^{2}\right )} x^{2}\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (6 \, a b c^{2} x^{3} + a^{2} b^{2} + 8 \, a^{3} c +{\left (b^{4} + a b^{2} c + 16 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} + 5 \, a^{2} b c\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3} +{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} + 2 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} +{\left (b^{6} c - 6 \, a b^{4} c^{2} + 32 \, a^{3} c^{4}\right )} x^{2} + 2 \,{\left (a b^{5} c - 8 \, a^{2} b^{3} c^{2} + 16 \, a^{3} b c^{3}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{12 \,{\left (a b c^{3} x^{4} + 2 \, a b^{2} c^{2} x^{3} + 2 \, a^{2} b^{2} c x + a^{3} b c +{\left (a b^{3} c + 2 \, a^{2} b c^{2}\right )} x^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (6 \, a b c^{2} x^{3} + a^{2} b^{2} + 8 \, a^{3} c +{\left (b^{4} + a b^{2} c + 16 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} + 5 \, a^{2} b c\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3} +{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} + 2 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} +{\left (b^{6} c - 6 \, a b^{4} c^{2} + 32 \, a^{3} c^{4}\right )} x^{2} + 2 \,{\left (a b^{5} c - 8 \, a^{2} b^{3} c^{2} + 16 \, a^{3} b c^{3}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.71233, size = 510, normalized size = 4.77 \[ 3 a b \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{- 192 a^{4} b c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{3} b^{3} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a^{2} b^{5} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{7} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{2}}{6 a b c} \right )} - 3 a b \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{192 a^{4} b c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{3} b^{3} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a^{2} b^{5} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 3 a b^{7} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{2}}{6 a b c} \right )} - \frac{8 a^{3} c + a^{2} b^{2} + 6 a b c^{2} x^{3} + x^{2} \left (16 a^{2} c^{2} + a b^{2} c + b^{4}\right ) + x \left (10 a^{2} b c + 2 a b^{3}\right )}{32 a^{4} c^{3} - 16 a^{3} b^{2} c^{2} + 2 a^{2} b^{4} c + x^{4} \left (32 a^{2} c^{5} - 16 a b^{2} c^{4} + 2 b^{4} c^{3}\right ) + x^{3} \left (64 a^{2} b c^{4} - 32 a b^{3} c^{3} + 4 b^{5} c^{2}\right ) + x^{2} \left (64 a^{3} c^{4} - 12 a b^{4} c^{2} + 2 b^{6} c\right ) + x \left (64 a^{3} b c^{3} - 32 a^{2} b^{3} c^{2} + 4 a b^{5} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a*b*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-192*a**4*b*c**3*sqrt(-1/(4*a*c - b**2
)**5) + 144*a**3*b**3*c**2*sqrt(-1/(4*a*c - b**2)**5) - 36*a**2*b**5*c*sqrt(-1/(
4*a*c - b**2)**5) + 3*a*b**7*sqrt(-1/(4*a*c - b**2)**5) + 3*a*b**2)/(6*a*b*c)) -
3*a*b*sqrt(-1/(4*a*c - b**2)**5)*log(x + (192*a**4*b*c**3*sqrt(-1/(4*a*c - b**2
)**5) - 144*a**3*b**3*c**2*sqrt(-1/(4*a*c - b**2)**5) + 36*a**2*b**5*c*sqrt(-1/(
4*a*c - b**2)**5) - 3*a*b**7*sqrt(-1/(4*a*c - b**2)**5) + 3*a*b**2)/(6*a*b*c)) -
(8*a**3*c + a**2*b**2 + 6*a*b*c**2*x**3 + x**2*(16*a**2*c**2 + a*b**2*c + b**4)
+ x*(10*a**2*b*c + 2*a*b**3))/(32*a**4*c**3 - 16*a**3*b**2*c**2 + 2*a**2*b**4*c
+ x**4*(32*a**2*c**5 - 16*a*b**2*c**4 + 2*b**4*c**3) + x**3*(64*a**2*b*c**4 - 3
2*a*b**3*c**3 + 4*b**5*c**2) + x**2*(64*a**3*c**4 - 12*a*b**4*c**2 + 2*b**6*c) +
x*(64*a**3*b*c**3 - 32*a**2*b**3*c**2 + 4*a*b**5*c))

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GIAC/XCAS [A]  time = 0.264445, size = 220, normalized size = 2.06 \[ -\frac{6 \, a b \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{6 \, a b c^{2} x^{3} + b^{4} x^{2} + a b^{2} c x^{2} + 16 \, a^{2} c^{2} x^{2} + 2 \, a b^{3} x + 10 \, a^{2} b c x + a^{2} b^{2} + 8 \, a^{3} c}{2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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