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

Optimal. Leaf size=104 \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{b}{a^2 x}-\frac{1}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.30891, antiderivative size = 104, 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 \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{b}{a^2 x}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.2823, size = 97, normalized size = 0.93 \[ - \frac{1}{2 a x^{2}} + \frac{b}{a^{2} x} + \frac{b \left (- 3 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{3} \sqrt{- 4 a c + b^{2}}} + \frac{\left (- a c + b^{2}\right ) \log{\left (x \right )}}{a^{3}} - \frac{\left (- a c + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.254644, size = 102, normalized size = 0.98 \[ \frac{-\frac{a^2}{x^2}+2 \log (x) \left (b^2-a c\right )+\left (a c-b^2\right ) \log (a+x (b+c x))-\frac{2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{2 a b}{x}}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 150, normalized size = 1.4 \[ -{\frac{1}{2\,a{x}^{2}}}-{\frac{\ln \left ( x \right ) c}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}}}+{\frac{b}{{a}^{2}x}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) }{2\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}}{2\,{a}^{3}}}+3\,{\frac{bc}{{a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}}{{a}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+b*x^3+a*x^2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320145, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{3} - 3 \, a b c\right )} x^{2} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left ({\left (b^{2} - a c\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (b^{2} - a c\right )} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} a^{3} x^{2}}, -\frac{2 \,{\left (b^{3} - 3 \, a b c\right )} x^{2} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (b^{2} - a c\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (b^{2} - a c\right )} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.2672, size = 1525, normalized size = 14.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2
*a**3))*log(x + (24*a**9*c**3*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*
a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 - 42*a**8*b**2*c**2*(-b*sqrt(-4*a*c + b
**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 + 17*a**
7*b**4*c*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c -
b**2)/(2*a**3))**2 + 12*a**7*c**4*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**
3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) - 2*a**6*b**6*(-b*sqrt(-4*a*c + b**2)
*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 - 15*a**6*b*
*2*c**3*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c -
b**2)/(2*a**3)) + 7*a**5*b**4*c**2*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**
3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) - 12*a**5*c**5 - a**4*b**6*c*(-b*sqrt
(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))
+ 63*a**4*b**2*c**4 - 103*a**3*b**4*c**3 + 70*a**2*b**6*c**2 - 20*a*b**8*c + 2*b
**10)/(27*a**4*b*c**5 - 63*a**3*b**3*c**4 + 54*a**2*b**5*c**3 - 18*a*b**7*c**2 +
2*b**9*c)) + (b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a
*c - b**2)/(2*a**3))*log(x + (24*a**9*c**3*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)
/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 - 42*a**8*b**2*c**2*(b*sqrt
(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))*
*2 + 17*a**7*b**4*c*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)
) + (a*c - b**2)/(2*a**3))**2 + 12*a**7*c**4*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**
2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) - 2*a**6*b**6*(b*sqrt(-4*a*c
+ b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 - 15
*a**6*b**2*c**3*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) +
(a*c - b**2)/(2*a**3)) + 7*a**5*b**4*c**2*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/
(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) - 12*a**5*c**5 - a**4*b**6*c*(b
*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a*
*3)) + 63*a**4*b**2*c**4 - 103*a**3*b**4*c**3 + 70*a**2*b**6*c**2 - 20*a*b**8*c
+ 2*b**10)/(27*a**4*b*c**5 - 63*a**3*b**3*c**4 + 54*a**2*b**5*c**3 - 18*a*b**7*c
**2 + 2*b**9*c)) + (-a + 2*b*x)/(2*a**2*x**2) - (a*c - b**2)*log(x + (-12*a**5*c
**5 + 63*a**4*b**2*c**4 - 12*a**4*c**4*(a*c - b**2) - 103*a**3*b**4*c**3 + 15*a*
*3*b**2*c**3*(a*c - b**2) + 24*a**3*c**3*(a*c - b**2)**2 + 70*a**2*b**6*c**2 - 7
*a**2*b**4*c**2*(a*c - b**2) - 42*a**2*b**2*c**2*(a*c - b**2)**2 - 20*a*b**8*c +
a*b**6*c*(a*c - b**2) + 17*a*b**4*c*(a*c - b**2)**2 + 2*b**10 - 2*b**6*(a*c - b
**2)**2)/(27*a**4*b*c**5 - 63*a**3*b**3*c**4 + 54*a**2*b**5*c**3 - 18*a*b**7*c**
2 + 2*b**9*c))/a**3

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GIAC/XCAS [A]  time = 0.264168, size = 142, normalized size = 1.37 \[ -\frac{{\left (b^{2} - a c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac{{\left (b^{2} - a c\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{3}} + \frac{2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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