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

Optimal. Leaf size=318 \[ \frac{b \left (5 b^2-17 a c\right )}{3 a^3 x^3 \left (b^2-4 a c\right )}-\frac{5 b^2-12 a c}{4 a^2 x^4 \left (b^2-4 a c\right )}-\frac{\left (3 a^2 c^2-12 a b^2 c+5 b^4\right ) \log \left (a+b x+c x^2\right )}{2 a^6}+\frac{\log (x) \left (3 a^2 c^2-12 a b^2 c+5 b^4\right )}{a^6}+\frac{b \left (29 a^2 c^2-27 a b^2 c+5 b^4\right )}{a^5 x \left (b^2-4 a c\right )}-\frac{12 a^2 c^2-22 a b^2 c+5 b^4}{2 a^4 x^2 \left (b^2-4 a c\right )}+\frac{b \left (-70 a^3 c^3+105 a^2 b^2 c^2-42 a b^4 c+5 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^6 \left (b^2-4 a c\right )^{3/2}}+\frac{-2 a c+b^2+b c x}{a x^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

-(5*b^2 - 12*a*c)/(4*a^2*(b^2 - 4*a*c)*x^4) + (b*(5*b^2 - 17*a*c))/(3*a^3*(b^2 -
4*a*c)*x^3) - (5*b^4 - 22*a*b^2*c + 12*a^2*c^2)/(2*a^4*(b^2 - 4*a*c)*x^2) + (b*
(5*b^4 - 27*a*b^2*c + 29*a^2*c^2))/(a^5*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x)
/(a*(b^2 - 4*a*c)*x^4*(a + b*x + c*x^2)) + (b*(5*b^6 - 42*a*b^4*c + 105*a^2*b^2*
c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^6*(b^2 - 4*a*c)^(3/
2)) + ((5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[x])/a^6 - ((5*b^4 - 12*a*b^2*c + 3*a
^2*c^2)*Log[a + b*x + c*x^2])/(2*a^6)

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Rubi [A]  time = 0.80403, antiderivative size = 318, 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 \[ \frac{b \left (5 b^2-17 a c\right )}{3 a^3 x^3 \left (b^2-4 a c\right )}-\frac{5 b^2-12 a c}{4 a^2 x^4 \left (b^2-4 a c\right )}-\frac{\left (3 a^2 c^2-12 a b^2 c+5 b^4\right ) \log \left (a+b x+c x^2\right )}{2 a^6}+\frac{\log (x) \left (3 a^2 c^2-12 a b^2 c+5 b^4\right )}{a^6}+\frac{b \left (29 a^2 c^2-27 a b^2 c+5 b^4\right )}{a^5 x \left (b^2-4 a c\right )}-\frac{12 a^2 c^2-22 a b^2 c+5 b^4}{2 a^4 x^2 \left (b^2-4 a c\right )}+\frac{b \left (-70 a^3 c^3+105 a^2 b^2 c^2-42 a b^4 c+5 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^6 \left (b^2-4 a c\right )^{3/2}}+\frac{-2 a c+b^2+b c x}{a x^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(5*b^2 - 12*a*c)/(4*a^2*(b^2 - 4*a*c)*x^4) + (b*(5*b^2 - 17*a*c))/(3*a^3*(b^2 -
4*a*c)*x^3) - (5*b^4 - 22*a*b^2*c + 12*a^2*c^2)/(2*a^4*(b^2 - 4*a*c)*x^2) + (b*
(5*b^4 - 27*a*b^2*c + 29*a^2*c^2))/(a^5*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*c*x)
/(a*(b^2 - 4*a*c)*x^4*(a + b*x + c*x^2)) + (b*(5*b^6 - 42*a*b^4*c + 105*a^2*b^2*
c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^6*(b^2 - 4*a*c)^(3/
2)) + ((5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[x])/a^6 - ((5*b^4 - 12*a*b^2*c + 3*a
^2*c^2)*Log[a + b*x + c*x^2])/(2*a^6)

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Rubi in Sympy [A]  time = 127.985, size = 308, normalized size = 0.97 \[ \frac{- 2 a c + b^{2} + b c x}{a x^{4} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{- 12 a c + 5 b^{2}}{4 a^{2} x^{4} \left (- 4 a c + b^{2}\right )} + \frac{b \left (- 17 a c + 5 b^{2}\right )}{3 a^{3} x^{3} \left (- 4 a c + b^{2}\right )} - \frac{12 a^{2} c^{2} - 22 a b^{2} c + 5 b^{4}}{2 a^{4} x^{2} \left (- 4 a c + b^{2}\right )} + \frac{b \left (29 a^{2} c^{2} - 27 a b^{2} c + 5 b^{4}\right )}{a^{5} x \left (- 4 a c + b^{2}\right )} + \frac{b \left (- 70 a^{3} c^{3} + 105 a^{2} b^{2} c^{2} - 42 a b^{4} c + 5 b^{6}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{6} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\left (3 a^{2} c^{2} - 12 a b^{2} c + 5 b^{4}\right ) \log{\left (x \right )}}{a^{6}} - \frac{\left (3 a^{2} c^{2} - 12 a b^{2} c + 5 b^{4}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*a*c + b**2 + b*c*x)/(a*x**4*(-4*a*c + b**2)*(a + b*x + c*x**2)) - (-12*a*c +
5*b**2)/(4*a**2*x**4*(-4*a*c + b**2)) + b*(-17*a*c + 5*b**2)/(3*a**3*x**3*(-4*a
*c + b**2)) - (12*a**2*c**2 - 22*a*b**2*c + 5*b**4)/(2*a**4*x**2*(-4*a*c + b**2)
) + b*(29*a**2*c**2 - 27*a*b**2*c + 5*b**4)/(a**5*x*(-4*a*c + b**2)) + b*(-70*a*
*3*c**3 + 105*a**2*b**2*c**2 - 42*a*b**4*c + 5*b**6)*atanh((b + 2*c*x)/sqrt(-4*a
*c + b**2))/(a**6*(-4*a*c + b**2)**(3/2)) + (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)
*log(x)/a**6 - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)*log(a + b*x + c*x**2)/(2*a**
6)

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Mathematica [A]  time = 0.642513, size = 272, normalized size = 0.86 \[ \frac{-\frac{3 a^4}{x^4}+\frac{8 a^3 b}{x^3}+\frac{6 a^2 \left (2 a c-3 b^2\right )}{x^2}+12 \log (x) \left (3 a^2 c^2-12 a b^2 c+5 b^4\right )-6 \left (3 a^2 c^2-12 a b^2 c+5 b^4\right ) \log (a+x (b+c x))+\frac{12 b \left (-70 a^3 c^3+105 a^2 b^2 c^2-42 a b^4 c+5 b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}-\frac{12 a \left (2 a^3 c^3-9 a^2 b^2 c^2-5 a^2 b c^3 x+6 a b^4 c+5 a b^3 c^2 x-b^6-b^5 c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{24 a b \left (3 a c-2 b^2\right )}{x}}{12 a^6} \]

Antiderivative was successfully verified.

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

[Out]

((-3*a^4)/x^4 + (8*a^3*b)/x^3 + (6*a^2*(-3*b^2 + 2*a*c))/x^2 - (24*a*b*(-2*b^2 +
3*a*c))/x - (12*a*(-b^6 + 6*a*b^4*c - 9*a^2*b^2*c^2 + 2*a^3*c^3 - b^5*c*x + 5*a
*b^3*c^2*x - 5*a^2*b*c^3*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (12*b*(5*b^6 -
42*a*b^4*c + 105*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]
])/(-b^2 + 4*a*c)^(3/2) + 12*(5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*Log[x] - 6*(5*b^4
- 12*a*b^2*c + 3*a^2*c^2)*Log[a + x*(b + c*x)])/(12*a^6)

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Maple [B]  time = 0.027, size = 923, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/a^2/x^4+1/a^3/x^2*c-3/2/a^4/x^2*b^2+3/a^4*ln(x)*c^2-12/a^5*ln(x)*b^2*c+5/a^
6*ln(x)*b^4+2/3/a^3*b/x^3-6/a^4*b/x*c+4/a^5*b^3/x-5/a^3/(c*x^2+b*x+a)*b*c^3/(4*a
*c-b^2)*x+5/a^4/(c*x^2+b*x+a)*b^3*c^2/(4*a*c-b^2)*x-1/a^5/(c*x^2+b*x+a)*b^5*c/(4
*a*c-b^2)*x+2/a^2/(c*x^2+b*x+a)/(4*a*c-b^2)*c^3-9/a^3/(c*x^2+b*x+a)/(4*a*c-b^2)*
b^2*c^2+6/a^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^4*c-1/a^5/(c*x^2+b*x+a)/(4*a*c-b^2)*b^
6-6/a^3/(4*a*c-b^2)*c^3*ln((4*a*c-b^2)*(c*x^2+b*x+a))+51/2/a^4/(4*a*c-b^2)*c^2*l
n((4*a*c-b^2)*(c*x^2+b*x+a))*b^2-16/a^5/(4*a*c-b^2)*c*ln((4*a*c-b^2)*(c*x^2+b*x+
a))*b^4+5/2/a^6/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^6-70/a^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))*b*c^3+105/a^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^3*c^2-42/a^5/(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*c+5/a^6/(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^7

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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)^2*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.04124, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(6*((5*b^7*c - 42*a*b^5*c^2 + 105*a^2*b^3*c^3 - 70*a^3*b*c^4)*x^6 + (5*b^8
- 42*a*b^6*c + 105*a^2*b^4*c^2 - 70*a^3*b^2*c^3)*x^5 + (5*a*b^7 - 42*a^2*b^5*c
+ 105*a^3*b^3*c^2 - 70*a^4*b*c^3)*x^4)*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*a^5*b^2 - 12*a^6*c - 12*(5*a*b^5*c - 27*a^2*b^3*c^2 + 29*a^3*b*c^3)*x^5 - 6*(
10*a*b^6 - 59*a^2*b^4*c + 80*a^3*b^2*c^2 - 12*a^4*c^3)*x^4 - 2*(15*a^2*b^5 - 86*
a^3*b^3*c + 104*a^4*b*c^2)*x^3 + (10*a^3*b^4 - 49*a^4*b^2*c + 36*a^5*c^2)*x^2 -
5*(a^4*b^3 - 4*a^5*b*c)*x + 6*((5*b^6*c - 32*a*b^4*c^2 + 51*a^2*b^2*c^3 - 12*a^3
*c^4)*x^6 + (5*b^7 - 32*a*b^5*c + 51*a^2*b^3*c^2 - 12*a^3*b*c^3)*x^5 + (5*a*b^6
- 32*a^2*b^4*c + 51*a^3*b^2*c^2 - 12*a^4*c^3)*x^4)*log(c*x^2 + b*x + a) - 12*((5
*b^6*c - 32*a*b^4*c^2 + 51*a^2*b^2*c^3 - 12*a^3*c^4)*x^6 + (5*b^7 - 32*a*b^5*c +
51*a^2*b^3*c^2 - 12*a^3*b*c^3)*x^5 + (5*a*b^6 - 32*a^2*b^4*c + 51*a^3*b^2*c^2 -
12*a^4*c^3)*x^4)*log(x))*sqrt(b^2 - 4*a*c))/(((a^6*b^2*c - 4*a^7*c^2)*x^6 + (a^
6*b^3 - 4*a^7*b*c)*x^5 + (a^7*b^2 - 4*a^8*c)*x^4)*sqrt(b^2 - 4*a*c)), -1/12*(12*
((5*b^7*c - 42*a*b^5*c^2 + 105*a^2*b^3*c^3 - 70*a^3*b*c^4)*x^6 + (5*b^8 - 42*a*b
^6*c + 105*a^2*b^4*c^2 - 70*a^3*b^2*c^3)*x^5 + (5*a*b^7 - 42*a^2*b^5*c + 105*a^3
*b^3*c^2 - 70*a^4*b*c^3)*x^4)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*
c)) + (3*a^5*b^2 - 12*a^6*c - 12*(5*a*b^5*c - 27*a^2*b^3*c^2 + 29*a^3*b*c^3)*x^5
- 6*(10*a*b^6 - 59*a^2*b^4*c + 80*a^3*b^2*c^2 - 12*a^4*c^3)*x^4 - 2*(15*a^2*b^5
- 86*a^3*b^3*c + 104*a^4*b*c^2)*x^3 + (10*a^3*b^4 - 49*a^4*b^2*c + 36*a^5*c^2)*
x^2 - 5*(a^4*b^3 - 4*a^5*b*c)*x + 6*((5*b^6*c - 32*a*b^4*c^2 + 51*a^2*b^2*c^3 -
12*a^3*c^4)*x^6 + (5*b^7 - 32*a*b^5*c + 51*a^2*b^3*c^2 - 12*a^3*b*c^3)*x^5 + (5*
a*b^6 - 32*a^2*b^4*c + 51*a^3*b^2*c^2 - 12*a^4*c^3)*x^4)*log(c*x^2 + b*x + a) -
12*((5*b^6*c - 32*a*b^4*c^2 + 51*a^2*b^2*c^3 - 12*a^3*c^4)*x^6 + (5*b^7 - 32*a*b
^5*c + 51*a^2*b^3*c^2 - 12*a^3*b*c^3)*x^5 + (5*a*b^6 - 32*a^2*b^4*c + 51*a^3*b^2
*c^2 - 12*a^4*c^3)*x^4)*log(x))*sqrt(-b^2 + 4*a*c))/(((a^6*b^2*c - 4*a^7*c^2)*x^
6 + (a^6*b^3 - 4*a^7*b*c)*x^5 + (a^7*b^2 - 4*a^8*c)*x^4)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 137.52, size = 6181, normalized size = 19.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c -
5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a*
*2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))*log(x + (4608*a**19*c**7*(-b*sqrt(-(4*
a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a*
*6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*
a*b**2*c + 5*b**4)/(2*a**6))**2 - 26432*a**18*b**2*c**6*(-b*sqrt(-(4*a*c - b**2)
**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3
*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c +
5*b**4)/(2*a**6))**2 + 38640*a**17*b**4*c**5*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a*
*3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*
a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*
a**6))**2 - 26124*a**16*b**6*c**4*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 1
05*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c
**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 +
9603*a**15*b**8*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2
*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b
**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 - 6912*a**15*
c**9*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4
*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) -
(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 1989*a**14*b**10*c**2*(-b*sqrt(
-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(
2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 -
12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 37616*a**14*b**2*c**8*(-b*sqrt(-(4*a*c - b
**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*
a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*
c + 5*b**4)/(2*a**6)) + 219*a**13*b**12*c*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*
c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**
2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**
6))**2 - 96472*a**13*b**4*c**7*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*
a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2
+ 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 10*a*
*12*b**14*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a
*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6
)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 112063*a**12*b**6*c**6*
(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c -
5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a*
*2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 69023*a**11*b**8*c**5*(-b*sqrt(-(4*a
*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**
6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a
*b**2*c + 5*b**4)/(2*a**6)) - 20736*a**11*c**11 + 24355*a**10*b**10*c**4*(-b*sqr
t(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)
/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2
- 12*a*b**2*c + 5*b**4)/(2*a**6)) + 373872*a**10*b**2*c**10 - 4964*a**9*b**12*c
**3*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*
c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (
3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 2277288*a**9*b**4*c**9 + 545*a**
8*b**14*c**2*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 4
2*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b
**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 6487391*a**8*b**6*c**8
- 25*a**7*b**16*c*(-b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**
2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*
c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 9943570*a**7*b**8*
c**7 + 9090837*a**6*b**10*c**6 - 5264714*a**5*b**12*c**5 + 1984426*a**4*b**14*c*
*4 - 486146*a**3*b**16*c**3 + 74720*a**2*b**18*c**2 - 6550*a*b**20*c + 250*b**22
)/(90720*a**10*b*c**11 - 844130*a**9*b**3*c**10 + 3174507*a**8*b**5*c**9 - 58850
10*a**7*b**7*c**8 + 6168225*a**6*b**9*c**7 - 3960180*a**5*b**11*c**6 + 1618470*a
**4*b**13*c**5 - 423276*a**3*b**15*c**4 + 68670*a**2*b**17*c**3 - 6300*a*b**19*c
**2 + 250*b**21*c)) + (b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*
c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b*
*4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))*log(x + (4608*a**
19*c**7*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b*
*4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))
- (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 - 26432*a**18*b**2*c**6*(b*s
qrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**
6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c*
*2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 38640*a**17*b**4*c**5*(b*sqrt(-(4*a*c
- b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(
64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b*
*2*c + 5*b**4)/(2*a**6))**2 - 26124*a**16*b**6*c**4*(b*sqrt(-(4*a*c - b**2)**3)*
(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3
- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**
4)/(2*a**6))**2 + 9603*a**15*b**8*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3
- 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b*
*2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))*
*2 - 6912*a**15*c**9*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c
**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**
4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 1989*a**14*b**10
*c**2*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4
*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) -
(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 37616*a**14*b**2*c**8*(b*sqr
t(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)
/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2
- 12*a*b**2*c + 5*b**4)/(2*a**6)) + 219*a**13*b**12*c*(b*sqrt(-(4*a*c - b**2)**
3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c
**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*
b**4)/(2*a**6))**2 - 96472*a**13*b**4*c**7*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*
c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**
2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**
6)) - 10*a**12*b**14*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c
**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**
4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6))**2 + 112063*a**12*
b**6*c**6*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*
b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)
) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 69023*a**11*b**8*c**5*(b*sq
rt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c - 5*b**6
)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a**2*c**
2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 20736*a**11*c**11 + 24355*a**10*b**10*c**4
*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a*b**4*c -
5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (3*a*
*2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 373872*a**10*b**2*c**10 - 4964*a**9*
b**12*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**2 + 42*a
*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6
)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 2277288*a**9*b**4*c**9 + 5
45*a**8*b**14*c**2*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2*c**
2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*
c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) + 6487391*a**8*b**6*
c**8 - 25*a**7*b**16*c*(b*sqrt(-(4*a*c - b**2)**3)*(70*a**3*c**3 - 105*a**2*b**2
*c**2 + 42*a*b**4*c - 5*b**6)/(2*a**6*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b
**4*c - b**6)) - (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)/(2*a**6)) - 9943570*a**7*b
**8*c**7 + 9090837*a**6*b**10*c**6 - 5264714*a**5*b**12*c**5 + 1984426*a**4*b**1
4*c**4 - 486146*a**3*b**16*c**3 + 74720*a**2*b**18*c**2 - 6550*a*b**20*c + 250*b
**22)/(90720*a**10*b*c**11 - 844130*a**9*b**3*c**10 + 3174507*a**8*b**5*c**9 - 5
885010*a**7*b**7*c**8 + 6168225*a**6*b**9*c**7 - 3960180*a**5*b**11*c**6 + 16184
70*a**4*b**13*c**5 - 423276*a**3*b**15*c**4 + 68670*a**2*b**17*c**3 - 6300*a*b**
19*c**2 + 250*b**21*c)) - (12*a**5*c - 3*a**4*b**2 + x**5*(348*a**2*b*c**3 - 324
*a*b**3*c**2 + 60*b**5*c) + x**4*(-72*a**3*c**3 + 480*a**2*b**2*c**2 - 354*a*b**
4*c + 60*b**6) + x**3*(208*a**3*b*c**2 - 172*a**2*b**3*c + 30*a*b**5) + x**2*(-3
6*a**4*c**2 + 49*a**3*b**2*c - 10*a**2*b**4) + x*(-20*a**4*b*c + 5*a**3*b**3))/(
x**6*(48*a**6*c**2 - 12*a**5*b**2*c) + x**5*(48*a**6*b*c - 12*a**5*b**3) + x**4*
(48*a**7*c - 12*a**6*b**2)) + (3*a**2*c**2 - 12*a*b**2*c + 5*b**4)*log(x + (-207
36*a**11*c**11 + 373872*a**10*b**2*c**10 - 2277288*a**9*b**4*c**9 - 6912*a**9*c*
*9*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) + 6487391*a**8*b**6*c**8 + 37616*a**8*b*
*2*c**8*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) - 9943570*a**7*b**8*c**7 - 96472*a*
*7*b**4*c**7*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) + 4608*a**7*c**7*(3*a**2*c**2
- 12*a*b**2*c + 5*b**4)**2 + 9090837*a**6*b**10*c**6 + 112063*a**6*b**6*c**6*(3*
a**2*c**2 - 12*a*b**2*c + 5*b**4) - 26432*a**6*b**2*c**6*(3*a**2*c**2 - 12*a*b**
2*c + 5*b**4)**2 - 5264714*a**5*b**12*c**5 - 69023*a**5*b**8*c**5*(3*a**2*c**2 -
12*a*b**2*c + 5*b**4) + 38640*a**5*b**4*c**5*(3*a**2*c**2 - 12*a*b**2*c + 5*b**
4)**2 + 1984426*a**4*b**14*c**4 + 24355*a**4*b**10*c**4*(3*a**2*c**2 - 12*a*b**2
*c + 5*b**4) - 26124*a**4*b**6*c**4*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2 - 48
6146*a**3*b**16*c**3 - 4964*a**3*b**12*c**3*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)
+ 9603*a**3*b**8*c**3*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2 + 74720*a**2*b**1
8*c**2 + 545*a**2*b**14*c**2*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) - 1989*a**2*b*
*10*c**2*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2 - 6550*a*b**20*c - 25*a*b**16*c
*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4) + 219*a*b**12*c*(3*a**2*c**2 - 12*a*b**2*c
+ 5*b**4)**2 + 250*b**22 - 10*b**14*(3*a**2*c**2 - 12*a*b**2*c + 5*b**4)**2)/(9
0720*a**10*b*c**11 - 844130*a**9*b**3*c**10 + 3174507*a**8*b**5*c**9 - 5885010*a
**7*b**7*c**8 + 6168225*a**6*b**9*c**7 - 3960180*a**5*b**11*c**6 + 1618470*a**4*
b**13*c**5 - 423276*a**3*b**15*c**4 + 68670*a**2*b**17*c**3 - 6300*a*b**19*c**2
+ 250*b**21*c))/a**6

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GIAC/XCAS [A]  time = 0.274683, size = 468, normalized size = 1.47 \[ -\frac{{\left (5 \, b^{7} - 42 \, a b^{5} c + 105 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{6} b^{2} - 4 \, a^{7} c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (5 \, b^{4} - 12 \, a b^{2} c + 3 \, a^{2} c^{2}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{6}} + \frac{{\left (5 \, b^{4} - 12 \, a b^{2} c + 3 \, a^{2} c^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{6}} - \frac{3 \, a^{5} b^{2} - 12 \, a^{6} c - 12 \,{\left (5 \, a b^{5} c - 27 \, a^{2} b^{3} c^{2} + 29 \, a^{3} b c^{3}\right )} x^{5} - 6 \,{\left (10 \, a b^{6} - 59 \, a^{2} b^{4} c + 80 \, a^{3} b^{2} c^{2} - 12 \, a^{4} c^{3}\right )} x^{4} - 2 \,{\left (15 \, a^{2} b^{5} - 86 \, a^{3} b^{3} c + 104 \, a^{4} b c^{2}\right )} x^{3} +{\left (10 \, a^{3} b^{4} - 49 \, a^{4} b^{2} c + 36 \, a^{5} c^{2}\right )} x^{2} - 5 \,{\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x}{12 \,{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} a^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(5*b^7 - 42*a*b^5*c + 105*a^2*b^3*c^2 - 70*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(-
b^2 + 4*a*c))/((a^6*b^2 - 4*a^7*c)*sqrt(-b^2 + 4*a*c)) - 1/2*(5*b^4 - 12*a*b^2*c
+ 3*a^2*c^2)*ln(c*x^2 + b*x + a)/a^6 + (5*b^4 - 12*a*b^2*c + 3*a^2*c^2)*ln(abs(
x))/a^6 - 1/12*(3*a^5*b^2 - 12*a^6*c - 12*(5*a*b^5*c - 27*a^2*b^3*c^2 + 29*a^3*b
*c^3)*x^5 - 6*(10*a*b^6 - 59*a^2*b^4*c + 80*a^3*b^2*c^2 - 12*a^4*c^3)*x^4 - 2*(1
5*a^2*b^5 - 86*a^3*b^3*c + 104*a^4*b*c^2)*x^3 + (10*a^3*b^4 - 49*a^4*b^2*c + 36*
a^5*c^2)*x^2 - 5*(a^4*b^3 - 4*a^5*b*c)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*a^6*x
^4)