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

Optimal. Leaf size=114 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b x}{c \left (b^2-4 a c\right )}+\frac{\log \left (a+b x+c x^2\right )}{2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.214693, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b x}{c \left (b^2-4 a c\right )}+\frac{\log \left (a+b x+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \left (- 6 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{x^{2} \left (2 a + b x\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{\int b\, dx}{c \left (- 4 a c + b^{2}\right )} + \frac{\log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.225686, size = 109, normalized size = 0.96 \[ \frac{\frac{2 \left (-2 a^2 c+a b (b-3 c x)+b^3 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{2 b \left (b^2-6 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\log (a+x (b+c x))}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 330, normalized size = 2.9 \[{\frac{1}{c{x}^{2}+bx+a} \left ({\frac{b \left ( 3\,ac-{b}^{2} \right ) x}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}}+{\frac{a \left ( 2\,ac-{b}^{2} \right ) }{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}} \right ) }+{\frac{\ln \left ( c \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ) }{2\,{c}^{2}}}-6\,{\frac{ab}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}\arctan \left ({\frac{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) x+c \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}} \right ) }+{\frac{{b}^{3}}{c}\arctan \left ({(2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) x+c \left ( 4\,ac-{b}^{2} \right ) b){\frac{1}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}}} \right ){\frac{1}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^2+b/x)^2/x,x)

[Out]

(b*(3*a*c-b^2)/c^2/(4*a*c-b^2)*x+a*(2*a*c-b^2)/(4*a*c-b^2)/c^2)/(c*x^2+b*x+a)+1/
2/c^2*ln(c*(4*a*c-b^2)*(c*x^2+b*x+a))-6/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-
b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x+c*(4*a*c-b^2)*b)/(64*a^3*c^5-48*a^2*b
^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*a*b+1/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^
3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x+c*(4*a*c-b^2)*b)/(64*a^3*c^5-48*a^2
*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*b^3/c

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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)^2*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261208, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a b^{3} - 6 \, a^{2} b c +{\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} +{\left (b^{4} - 6 \, a b^{2} c\right )} x\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, a b^{2} - 4 \, a^{2} c + 2 \,{\left (b^{3} - 3 \, a b c\right )} x +{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left (a b^{3} - 6 \, a^{2} b c +{\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} +{\left (b^{4} - 6 \, a b^{2} c\right )} x\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, a b^{2} - 4 \, a^{2} c + 2 \,{\left (b^{3} - 3 \, a b c\right )} x +{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((a*b^3 - 6*a^2*b*c + (b^3*c - 6*a*b*c^2)*x^2 + (b^4 - 6*a*b^2*c)*x)*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)) + (2*a*b^2 - 4*a^2*c + 2*(b^3 - 3*a*b*c)*x + (a
*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*log(c*x^2 + b*x + a)
)*sqrt(b^2 - 4*a*c))/((a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^
2 - 4*a*b*c^3)*x)*sqrt(b^2 - 4*a*c)), -1/2*(2*(a*b^3 - 6*a^2*b*c + (b^3*c - 6*a*
b*c^2)*x^2 + (b^4 - 6*a*b^2*c)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 -
4*a*c)) - (2*a*b^2 - 4*a^2*c + 2*(b^3 - 3*a*b*c)*x + (a*b^2 - 4*a^2*c + (b^2*c -
4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a
*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt(-
b^2 + 4*a*c))]

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Sympy [A]  time = 4.53166, size = 729, normalized size = 6.39 \[ \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) \log{\left (x + \frac{- 16 a^{2} c^{3} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) \log{\left (x + \frac{- 16 a^{2} c^{3} \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac{2 a^{2} c - a b^{2} + x \left (3 a b c - b^{3}\right )}{4 a^{2} c^{3} - a b^{2} c^{2} + x^{2} \left (4 a c^{4} - b^{2} c^{3}\right ) + x \left (4 a b c^{3} - b^{3} c^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2))*log(x + (-16*a**2*c**3*(-b*sqrt(-(4*a
*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b
**4*c - b**6)) + 1/(2*c**2)) + 8*a**2*c + 8*a*b**2*c**2*(-b*sqrt(-(4*a*c - b**2)
**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b*
*6)) + 1/(2*c**2)) - a*b**2 - b**4*c*(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)
/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)))
/(6*a*b*c - b**3)) + (b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3
*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2))*log(x + (-16*a**2
*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2
*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)) + 8*a**2*c + 8*a*b**2*c**2*(b*sq
rt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6)) + 1/(2*c**2)) - a*b**2 - b**4*c*(b*sqrt(-(4*a*c - b**2)**
3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6
)) + 1/(2*c**2)))/(6*a*b*c - b**3)) + (2*a**2*c - a*b**2 + x*(3*a*b*c - b**3))/(
4*a**2*c**3 - a*b**2*c**2 + x**2*(4*a*c**4 - b**2*c**3) + x*(4*a*b*c**3 - b**3*c
**2))

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GIAC/XCAS [A]  time = 0.269174, size = 169, normalized size = 1.48 \[ -\frac{{\left (b^{3} - 6 \, a b c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac{a b^{2} - 2 \, a^{2} c +{\left (b^{3} - 3 \, a b c\right )} x}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b^3 - 6*a*b*c)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^2 - 4*a*c^3)*sqr
t(-b^2 + 4*a*c)) + 1/2*ln(c*x^2 + b*x + a)/c^2 + (a*b^2 - 2*a^2*c + (b^3 - 3*a*b
*c)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^2)