3.18 \(\int \frac{1}{x^2 \left (a x^2+b x^3+c x^4\right )} \, dx\)

Optimal. Leaf size=137 \[ \frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{b \log (x) \left (b^2-2 a c\right )}{a^4}-\frac{b^2-a c}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c}}-\frac{1}{3 a x^3} \]

[Out]

-1/(3*a*x^3) + b/(2*a^2*x^2) - (b^2 - a*c)/(a^3*x) - ((b^4 - 4*a*b^2*c + 2*a^2*c
^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*a*c]) - (b*(b^2 -
2*a*c)*Log[x])/a^4 + (b*(b^2 - 2*a*c)*Log[a + b*x + c*x^2])/(2*a^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.390557, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{b \log (x) \left (b^2-2 a c\right )}{a^4}-\frac{b^2-a c}{a^3 x}+\frac{b}{2 a^2 x^2}-\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \sqrt{b^2-4 a c}}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a*x^2 + b*x^3 + c*x^4)),x]

[Out]

-1/(3*a*x^3) + b/(2*a^2*x^2) - (b^2 - a*c)/(a^3*x) - ((b^4 - 4*a*b^2*c + 2*a^2*c
^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*a*c]) - (b*(b^2 -
2*a*c)*Log[x])/a^4 + (b*(b^2 - 2*a*c)*Log[a + b*x + c*x^2])/(2*a^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 57.3996, size = 129, normalized size = 0.94 \[ - \frac{1}{3 a x^{3}} + \frac{b}{2 a^{2} x^{2}} - \frac{- a c + b^{2}}{a^{3} x} - \frac{b \left (- 2 a c + b^{2}\right ) \log{\left (x \right )}}{a^{4}} + \frac{b \left (- 2 a c + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 a^{4}} - \frac{\left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{4} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(c*x**4+b*x**3+a*x**2),x)

[Out]

-1/(3*a*x**3) + b/(2*a**2*x**2) - (-a*c + b**2)/(a**3*x) - b*(-2*a*c + b**2)*log
(x)/a**4 + b*(-2*a*c + b**2)*log(a + b*x + c*x**2)/(2*a**4) - (2*a**2*c**2 - 4*a
*b**2*c + b**4)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(a**4*sqrt(-4*a*c + b**2)
)

_______________________________________________________________________________________

Mathematica [A]  time = 0.173341, size = 131, normalized size = 0.96 \[ \frac{-\frac{2 a^3}{x^3}+\frac{6 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{3 a^2 b}{x^2}-6 \log (x) \left (b^3-2 a b c\right )+3 \left (b^3-2 a b c\right ) \log (a+x (b+c x))+\frac{6 a \left (a c-b^2\right )}{x}}{6 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a*x^2 + b*x^3 + c*x^4)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 214, normalized size = 1.6 \[ -{\frac{1}{3\,a{x}^{3}}}+{\frac{c}{{a}^{2}x}}-{\frac{{b}^{2}}{{a}^{3}x}}+2\,{\frac{b\ln \left ( x \right ) c}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}-{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) b}{{a}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{2\,{a}^{4}}}+2\,{\frac{{c}^{2}}{{a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{2}c}{{a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}}{{a}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^4+b*x^3+a*x^2),x)

[Out]

-1/3/a/x^3+1/a^2/x*c-1/a^3/x*b^2+2*b/a^3*ln(x)*c-b^3/a^4*ln(x)+1/2*b/a^2/x^2-1/a
^3*c*ln(c*x^2+b*x+a)*b+1/2/a^4*ln(c*x^2+b*x+a)*b^3+2/a^2/(4*a*c-b^2)^(1/2)*arcta
n((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2-4/a^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*b^2*c+1/a^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
*b^4

_______________________________________________________________________________________

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*x^4 + b*x^3 + a*x^2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.367374, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x^{3} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (3 \,{\left (b^{3} - 2 \, a b c\right )} x^{3} \log \left (c x^{2} + b x + a\right ) - 6 \,{\left (b^{3} - 2 \, a b c\right )} x^{3} \log \left (x\right ) + 3 \, a^{2} b x - 2 \, a^{3} - 6 \,{\left (a b^{2} - a^{2} c\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{6 \, \sqrt{b^{2} - 4 \, a c} a^{4} x^{3}}, \frac{6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x^{3} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (3 \,{\left (b^{3} - 2 \, a b c\right )} x^{3} \log \left (c x^{2} + b x + a\right ) - 6 \,{\left (b^{3} - 2 \, a b c\right )} x^{3} \log \left (x\right ) + 3 \, a^{2} b x - 2 \, a^{3} - 6 \,{\left (a b^{2} - a^{2} c\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{6 \, \sqrt{-b^{2} + 4 \, a c} a^{4} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^3 + a*x^2)*x^2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 28.0519, size = 2105, normalized size = 15.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(c*x**4+b*x**3+a*x**2),x)

[Out]

(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b*
*4)/(2*a**4*(4*a*c - b**2)))*log(x + (-52*a**11*b*c**3*(-b*(2*a*c - b**2)/(2*a**
4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2
)))**2 + 57*a**10*b**3*c**2*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2
*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))**2 - 19*a**9*b**5*c*(-b
*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)
/(2*a**4*(4*a*c - b**2)))**2 + 4*a**9*c**5*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4
*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) + 2*a**8
*b**7*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*
c + b**4)/(2*a**4*(4*a*c - b**2)))**2 + 23*a**8*b**2*c**4*(-b*(2*a*c - b**2)/(2*
a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b
**2))) - 26*a**7*b**4*c**3*(-b*(2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*
a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) + 9*a**6*b**6*c**2*(-b*(
2*a*c - b**2)/(2*a**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(
2*a**4*(4*a*c - b**2))) - 8*a**6*b*c**6 - a**5*b**8*c*(-b*(2*a*c - b**2)/(2*a**4
) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)
)) + 166*a**5*b**3*c**5 - 361*a**4*b**5*c**4 + 312*a**3*b**7*c**3 - 130*a**2*b**
9*c**2 + 26*a*b**11*c - 2*b**13)/(2*a**6*c**7 + 60*a**5*b**2*c**6 - 207*a**4*b**
4*c**5 + 224*a**3*b**6*c**4 - 108*a**2*b**8*c**3 + 24*a*b**10*c**2 - 2*b**12*c))
+ (-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c +
b**4)/(2*a**4*(4*a*c - b**2)))*log(x + (-52*a**11*b*c**3*(-b*(2*a*c - b**2)/(2*
a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b
**2)))**2 + 57*a**10*b**3*c**2*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)
*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2)))**2 - 19*a**9*b**5*c*
(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b*
*4)/(2*a**4*(4*a*c - b**2)))**2 + 4*a**9*c**5*(-b*(2*a*c - b**2)/(2*a**4) + sqrt
(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) + 2*a
**8*b**7*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b*
*2*c + b**4)/(2*a**4*(4*a*c - b**2)))**2 + 23*a**8*b**2*c**4*(-b*(2*a*c - b**2)/
(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c
- b**2))) - 26*a**7*b**4*c**3*(-b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*
(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b**2))) + 9*a**6*b**6*c**2*(-
b*(2*a*c - b**2)/(2*a**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4
)/(2*a**4*(4*a*c - b**2))) - 8*a**6*b*c**6 - a**5*b**8*c*(-b*(2*a*c - b**2)/(2*a
**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*a**4*(4*a*c - b*
*2))) + 166*a**5*b**3*c**5 - 361*a**4*b**5*c**4 + 312*a**3*b**7*c**3 - 130*a**2*
b**9*c**2 + 26*a*b**11*c - 2*b**13)/(2*a**6*c**7 + 60*a**5*b**2*c**6 - 207*a**4*
b**4*c**5 + 224*a**3*b**6*c**4 - 108*a**2*b**8*c**3 + 24*a*b**10*c**2 - 2*b**12*
c)) + (-2*a**2 + 3*a*b*x + x**2*(6*a*c - 6*b**2))/(6*a**3*x**3) + b*(2*a*c - b**
2)*log(x + (-8*a**6*b*c**6 + 166*a**5*b**3*c**5 + 4*a**5*b*c**5*(2*a*c - b**2) -
361*a**4*b**5*c**4 + 23*a**4*b**3*c**4*(2*a*c - b**2) + 312*a**3*b**7*c**3 - 26
*a**3*b**5*c**3*(2*a*c - b**2) - 52*a**3*b**3*c**3*(2*a*c - b**2)**2 - 130*a**2*
b**9*c**2 + 9*a**2*b**7*c**2*(2*a*c - b**2) + 57*a**2*b**5*c**2*(2*a*c - b**2)**
2 + 26*a*b**11*c - a*b**9*c*(2*a*c - b**2) - 19*a*b**7*c*(2*a*c - b**2)**2 - 2*b
**13 + 2*b**9*(2*a*c - b**2)**2)/(2*a**6*c**7 + 60*a**5*b**2*c**6 - 207*a**4*b**
4*c**5 + 224*a**3*b**6*c**4 - 108*a**2*b**8*c**3 + 24*a*b**10*c**2 - 2*b**12*c))
/a**4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.263691, size = 184, normalized size = 1.34 \[ \frac{{\left (b^{3} - 2 \, a b c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac{{\left (b^{3} - 2 \, a b c\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{4}} + \frac{3 \, a^{2} b x - 2 \, a^{3} - 6 \,{\left (a b^{2} - a^{2} c\right )} x^{2}}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^3 + a*x^2)*x^2),x, algorithm="giac")

[Out]

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