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

Optimal. Leaf size=196 \[ \frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{b x \left (3 b^2-11 a c\right )}{c^3 \left (b^2-4 a c\right )}+\frac{x^2 \left (3 b^2-8 a c\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac{b x^3}{c \left (b^2-4 a c\right )}+\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

-((b*(3*b^2 - 11*a*c)*x)/(c^3*(b^2 - 4*a*c))) + ((3*b^2 - 8*a*c)*x^2)/(2*c^2*(b^
2 - 4*a*c)) - (b*x^3)/(c*(b^2 - 4*a*c)) + (x^4*(2*a + b*x))/((b^2 - 4*a*c)*(a +
b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^
2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[a + b*x + c*x^2])/
(2*c^4)

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Rubi [A]  time = 0.439333, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac{b x \left (3 b^2-11 a c\right )}{c^3 \left (b^2-4 a c\right )}+\frac{x^2 \left (3 b^2-8 a c\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac{b x^3}{c \left (b^2-4 a c\right )}+\frac{x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x/(c + a/x^2 + b/x)^2,x]

[Out]

-((b*(3*b^2 - 11*a*c)*x)/(c^3*(b^2 - 4*a*c))) + ((3*b^2 - 8*a*c)*x^2)/(2*c^2*(b^
2 - 4*a*c)) - (b*x^3)/(c*(b^2 - 4*a*c)) + (x^4*(2*a + b*x))/((b^2 - 4*a*c)*(a +
b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^
2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[a + b*x + c*x^2])/
(2*c^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b x^{3}}{c \left (- 4 a c + b^{2}\right )} + \frac{b \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{4} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{x^{4} \left (2 a + b x\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} + \frac{\left (- 8 a c + 3 b^{2}\right ) \int x\, dx}{c^{2} \left (- 4 a c + b^{2}\right )} - \frac{\left (- 11 a c + 3 b^{2}\right ) \int b\, dx}{c^{3} \left (- 4 a c + b^{2}\right )} + \frac{\left (- 2 a c + 3 b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*x**3/(c*(-4*a*c + b**2)) + b*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)*atanh((b +
2*c*x)/sqrt(-4*a*c + b**2))/(c**4*(-4*a*c + b**2)**(3/2)) + x**4*(2*a + b*x)/((
-4*a*c + b**2)*(a + b*x + c*x**2)) + (-8*a*c + 3*b**2)*Integral(x, x)/(c**2*(-4*
a*c + b**2)) - (-11*a*c + 3*b**2)*Integral(b, x)/(c**3*(-4*a*c + b**2)) + (-2*a*
c + 3*b**2)*log(a + b*x + c*x**2)/(2*c**4)

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Mathematica [A]  time = 0.349643, size = 163, normalized size = 0.83 \[ \frac{\frac{2 b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{2 \left (2 a^3 c^2+a^2 b c (5 c x-4 b)+a b^3 (b-5 c x)+b^5 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\left (3 b^2-2 a c\right ) \log (a+x (b+c x))-4 b c x+c^2 x^2}{2 c^4} \]

Antiderivative was successfully verified.

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

[Out]

(-4*b*c*x + c^2*x^2 + (2*(2*a^3*c^2 + b^5*x + a*b^3*(b - 5*c*x) + a^2*b*c*(-4*b
+ 5*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*b*(3*b^4 - 20*a*b^2*c + 30*a^2
*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (3*b^2 - 2*
a*c)*Log[a + x*(b + c*x)])/(2*c^4)

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Maple [B]  time = 0.019, size = 662, normalized size = 3.4 \[{\frac{{x}^{2}}{2\,{c}^{2}}}-2\,{\frac{bx}{{c}^{3}}}-5\,{\frac{bx{a}^{2}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+5\,{\frac{{b}^{3}xa}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{5}x}{{c}^{4} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{{a}^{3}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+4\,{\frac{{a}^{2}{b}^{2}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{a{b}^{4}}{{c}^{4} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ){a}^{2}}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}}+7\,{\frac{\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ) a{b}^{2}}{{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{3\,\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ){b}^{4}}{2\,{c}^{4} \left ( 4\,ac-{b}^{2} \right ) }}+30\,{\frac{{a}^{2}b}{{c}^{2}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }-20\,{\frac{a{b}^{3}}{{c}^{3}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }+3\,{\frac{{b}^{5}}{{c}^{4}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c+a/x^2+b/x)^2,x)

[Out]

1/2*x^2/c^2-2/c^3*x*b-5/c^2/(c*x^2+b*x+a)*b/(4*a*c-b^2)*x*a^2+5/c^3/(c*x^2+b*x+a
)*b^3/(4*a*c-b^2)*x*a-1/c^4/(c*x^2+b*x+a)*b^5/(4*a*c-b^2)*x-2/c^2/(c*x^2+b*x+a)*
a^3/(4*a*c-b^2)+4/c^3/(c*x^2+b*x+a)*a^2/(4*a*c-b^2)*b^2-1/c^4/(c*x^2+b*x+a)*a/(4
*a*c-b^2)*b^4-4/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a^2+7/c^3/(4*a*c-b
^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*b^2-3/2/c^4/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^
2+b*x+a))*b^4+30/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*
(4*a*c-b^2)*c*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))
*a^2*b-20/c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-
b^2)*c*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b^3+
3/c^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x
+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^5

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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(x/(c + b/x + a/x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270392, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2
*b*c^3)*x^2 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*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^4 - 8*a^2*b^2*c + 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 -
3*(b^3*c^2 - 4*a*b*c^3)*x^3 - (4*b^4*c - 17*a*b^2*c^2 + 4*a^2*c^3)*x^2 + 2*(b^5
- 7*a*b^3*c + 13*a^2*b*c^2)*x + (3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2 + (3*b^4*c -
14*a*b^2*c^2 + 8*a^2*c^3)*x^2 + (3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*x)*log(c*x^2
+ b*x + a))*sqrt(b^2 - 4*a*c))/((a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^
2 + (b^3*c^4 - 4*a*b*c^5)*x)*sqrt(b^2 - 4*a*c)), -1/2*(2*(3*a*b^5 - 20*a^2*b^3*c
+ 30*a^3*b*c^2 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^2 + (3*b^6 - 20*a*b^
4*c + 30*a^2*b^2*c^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) -
(2*a*b^4 - 8*a^2*b^2*c + 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 - 3*(b^3*c^2 - 4*a
*b*c^3)*x^3 - (4*b^4*c - 17*a*b^2*c^2 + 4*a^2*c^3)*x^2 + 2*(b^5 - 7*a*b^3*c + 13
*a^2*b*c^2)*x + (3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2 + (3*b^4*c - 14*a*b^2*c^2 +
8*a^2*c^3)*x^2 + (3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*x)*log(c*x^2 + b*x + a))*sqr
t(-b^2 + 4*a*c))/((a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4 -
4*a*b*c^5)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 7.77928, size = 1012, normalized size = 5.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b*x/c**3 + (-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)
/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b
**2)/(2*c**4))*log(x + (16*a**3*c**2 - 17*a**2*b**2*c + 16*a**2*c**5*(-b*sqrt(-(
4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 -
48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + 3*a*b**4
- 8*a*b**2*c**4*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b*
*4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c -
3*b**2)/(2*c**4)) + b**4*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*
b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)
) - (2*a*c - 3*b**2)/(2*c**4)))/(30*a**2*b*c**2 - 20*a*b**3*c + 3*b**5)) + (b*sq
rt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c*
*3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4))*log(x
+ (16*a**3*c**2 - 17*a**2*b**2*c + 16*a**2*c**5*(b*sqrt(-(4*a*c - b**2)**3)*(30
*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 1
2*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + 3*a*b**4 - 8*a*b**2*c**4*(b*s
qrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c
**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + b*
*4*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**
4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2
*c**4)))/(30*a**2*b*c**2 - 20*a*b**3*c + 3*b**5)) - (2*a**3*c**2 - 4*a**2*b**2*c
+ a*b**4 + x*(5*a**2*b*c**2 - 5*a*b**3*c + b**5))/(4*a**2*c**5 - a*b**2*c**4 +
x**2*(4*a*c**6 - b**2*c**5) + x*(4*a*b*c**5 - b**3*c**4)) + x**2/(2*c**2)

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GIAC/XCAS [A]  time = 0.291716, size = 254, normalized size = 1.3 \[ -\frac{{\left (3 \, b^{5} - 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (3 \, b^{2} - 2 \, a c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{c^{2} x^{2} - 4 \, b c x}{2 \, c^{4}} + \frac{a b^{4} - 4 \, a^{2} b^{2} c + 2 \, a^{3} c^{2} +{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} x}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^
2*c^4 - 4*a*c^5)*sqrt(-b^2 + 4*a*c)) + 1/2*(3*b^2 - 2*a*c)*ln(c*x^2 + b*x + a)/c
^4 + 1/2*(c^2*x^2 - 4*b*c*x)/c^4 + (a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2 + (b^5 - 5*a
*b^3*c + 5*a^2*b*c^2)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^4)