3.2197 \(\int \frac{1}{x^2 \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=239 \[ \frac{3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{3 b \log (x)}{a^4}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 x \left (b^2-4 a c\right )^2}+\frac{20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{2 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(a^3*(b^2 - 4*a*c)^2*x) + (b^2 - 2*a*c + b*c*x)
/(2*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 +
3*b*c*(b^2 - 6*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*x*(a + b*x + c*x^2)) - (3*(b^6 - 1
0*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])
/(a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[x])/a^4 + (3*b*Log[a + b*x + c*x^2])/(2*a^
4)

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Rubi [A]  time = 0.722728, antiderivative size = 239, 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 \[ \frac{3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac{3 b \log (x)}{a^4}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 x \left (b^2-4 a c\right )^2}+\frac{20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{2 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(a^3*(b^2 - 4*a*c)^2*x) + (b^2 - 2*a*c + b*c*x)
/(2*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 +
3*b*c*(b^2 - 6*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*x*(a + b*x + c*x^2)) - (3*(b^6 - 1
0*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])
/(a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[x])/a^4 + (3*b*Log[a + b*x + c*x^2])/(2*a^
4)

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Rubi in Sympy [A]  time = 145.767, size = 238, normalized size = 1. \[ \frac{- 2 a c + b^{2} + b c x}{2 a x \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} + \frac{20 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4} + 3 b c x \left (- 6 a c + b^{2}\right )}{2 a^{2} x \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{3 \left (- 5 a c + b^{2}\right ) \left (- 2 a c + b^{2}\right )}{a^{3} x \left (- 4 a c + b^{2}\right )^{2}} - \frac{3 b \log{\left (x \right )}}{a^{4}} + \frac{3 b \log{\left (a + b x + c x^{2} \right )}}{2 a^{4}} - \frac{6 \left (- 10 a^{3} c^{3} + 15 a^{2} b^{2} c^{2} - 5 a b^{4} c + \frac{b^{6}}{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{4} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*a*c + b**2 + b*c*x)/(2*a*x*(-4*a*c + b**2)*(a + b*x + c*x**2)**2) + (20*a**2
*c**2 - 20*a*b**2*c + 3*b**4 + 3*b*c*x*(-6*a*c + b**2))/(2*a**2*x*(-4*a*c + b**2
)**2*(a + b*x + c*x**2)) - 3*(-5*a*c + b**2)*(-2*a*c + b**2)/(a**3*x*(-4*a*c + b
**2)**2) - 3*b*log(x)/a**4 + 3*b*log(a + b*x + c*x**2)/(2*a**4) - 6*(-10*a**3*c*
*3 + 15*a**2*b**2*c**2 - 5*a*b**4*c + b**6/2)*atanh((b + 2*c*x)/sqrt(-4*a*c + b*
*2))/(a**4*(-4*a*c + b**2)**(5/2))

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Mathematica [A]  time = 0.826802, size = 221, normalized size = 0.92 \[ \frac{\frac{a^2 \left (-3 a b c-2 a c^2 x+b^3+b^2 c x\right )}{\left (4 a c-b^2\right ) (a+x (b+c x))^2}-\frac{a \left (46 a^2 b c^2+28 a^2 c^3 x-29 a b^3 c-26 a b^2 c^2 x+4 b^5+4 b^4 c x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{6 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+3 b \log (a+x (b+c x))-\frac{2 a}{x}-6 b \log (x)}{2 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((-2*a)/x + (a^2*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((-b^2 + 4*a*c)*(a + x*(
b + c*x))^2) - (a*(4*b^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x - 26*a*b^2*c^2*
x + 28*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (6*(b^6 - 10*a*b^4*c +
30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a
*c)^(5/2) - 6*b*Log[x] + 3*b*Log[a + x*(b + c*x)])/(2*a^4)

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Maple [B]  time = 0.029, size = 1438, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^2+b*x+a)^3,x)

[Out]

-1/a^3/x-3*b*ln(x)/a^4-14/a/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1
3/a^2/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^2-2/a^3/(c*x^2+b*x+a)
^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^4-37/a/(c*x^2+b*x+a)^2*b*c^3/(16*a^2*c^2
-8*a*b^2*c+b^4)*x^2+55/2/a^2/(c*x^2+b*x+a)^2*b^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*
x^2-4/a^3/(c*x^2+b*x+a)^2*b^5*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-18/(c*x^2+b*x+a)^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c^3-7/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*x*b^2*c^2+12/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^4*c-2/a^3/(c*x
^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^6-29/(c*x^2+b*x+a)^2*b/(16*a^2*c^2-8*
a*b^2*c+b^4)*c^2+18/a/(c*x^2+b*x+a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c-5/2/a^2/(
c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)+24/a^2*c^2/(16*a^2*c^2-8*a*b^2*c+b
^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*b-12/a^3*c/(16*a^2*c^2-8*a*b^2*
c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*b^3+3/2/a^4/(16*a^2*c^2-8*a*
b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*b^5-60/a/(1024*a^5*c^5-1
280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2
*c*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280
*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*c^3+90/a^2/
(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^
(1/2)*arctan((2*c*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(10
24*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/
2))*b^2*c^2-30/a^3/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^
2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*
a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+2
0*a*b^8*c-b^10)^(1/2))*b^4*c+3/a^4/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^
3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*(16*a^2*c^2-8*a*b^2*c+b^4)*
x+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-1
60*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b^6

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.910698, 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^2 + b*x + a)^3*x^2),x, algorithm="fricas")

[Out]

[-1/2*(3*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^5 + 2*(b^7*c
- 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^4 + (b^8 - 8*a*b^6*c + 10*a^2*
b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^3 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^
3*c^2 - 20*a^4*b*c^3)*x^2 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^
3)*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^3*b^4 - 16*a^4*b^2*c + 32*a^
5*c^2 + 6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*x^4 + 3*(4*a*b^5*c - 29*a^2*b
^3*c^2 + 46*a^3*b*c^3)*x^3 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*
c^3)*x^2 + (9*a^2*b^5 - 68*a^3*b^3*c + 122*a^4*b*c^2)*x - 3*((b^5*c^2 - 8*a*b^3*
c^3 + 16*a^2*b*c^4)*x^5 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*x^4 + (b^7 -
6*a*b^5*c + 32*a^3*b*c^3)*x^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*x^2 + (
a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x)*log(c*x^2 + b*x + a) + 6*((b^5*c^2 - 8*
a*b^3*c^3 + 16*a^2*b*c^4)*x^5 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*x^4 + (
b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*x
^2 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x)*log(x))*sqrt(b^2 - 4*a*c))/(((a^4
*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*x^5 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a
^6*b*c^3)*x^4 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*x^3 + 2*(a^5*b^5 - 8*a^6*b^
3*c + 16*a^7*b*c^2)*x^2 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*x)*sqrt(b^2 - 4*a
*c)), 1/2*(6*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^5 + 2*(b^
7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^4 + (b^8 - 8*a*b^6*c + 10*
a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^3 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^
3*b^3*c^2 - 20*a^4*b*c^3)*x^2 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^
5*c^3)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (2*a^3*b^4 - 1
6*a^4*b^2*c + 32*a^5*c^2 + 6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*x^4 + 3*(4
*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*x^3 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^
3*b^2*c^2 + 50*a^4*c^3)*x^2 + (9*a^2*b^5 - 68*a^3*b^3*c + 122*a^4*b*c^2)*x - 3*(
(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^5 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2
*c^3)*x^4 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a
^3*b^2*c^2)*x^2 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x)*log(c*x^2 + b*x + a)
+ 6*((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^5 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a
^2*b^2*c^3)*x^4 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^3 + 2*(a*b^6 - 8*a^2*b^4*c
+ 16*a^3*b^2*c^2)*x^2 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x)*log(x))*sqrt(-
b^2 + 4*a*c))/(((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*x^5 + 2*(a^4*b^5*c -
8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^4 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*x^3 + 2
*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^2 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^
2)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 152.205, size = 5722, normalized size = 23.94 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 1
0*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4
*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))*log(x + (-108544*a**16*b*c**
8*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 +
10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b*
*4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 224768*a**15*b**3*c**
7*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 +
10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b*
*4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 202752*a**14*b**5*c**
6*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 +
10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b*
*4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 104128*a**13*b**7*c**
5*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 +
10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b*
*4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 19200*a**13*c**9*(3*b
/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*
b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**
3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 33320*a**12*b**9*c**4*(3*b/(2*
a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4
*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 -
160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 44736*a**12*b**2*c**8*(3*b/(2*a
**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*
c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 1
60*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 6806*a**11*b**11*c**3*(3*b/(2*a**4)
- 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c -
b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a
**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 101232*a**11*b**4*c**7*(3*b/(2*a**4)
- 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c -
b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a
**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 867*a**10*b**13*c**2*(3*b/(2*a**4) - 3*
sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)
/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b
**6*c**2 + 20*a*b**8*c - b**10)))**2 - 77268*a**10*b**6*c**6*(3*b/(2*a**4) - 3*s
qrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/
(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b*
*6*c**2 + 20*a*b**8*c - b**10))) + 63*a**9*b**15*c*(3*b/(2*a**4) - 3*sqrt(-(4*a*
c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1
024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 +
20*a*b**8*c - b**10)))**2 + 31368*a**9*b**8*c**5*(3*b/(2*a**4) - 3*sqrt(-(4*a*c
- b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(102
4*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20
*a*b**8*c - b**10))) - 57600*a**9*b*c**9 - 2*a**8*b**17*(3*b/(2*a**4) - 3*sqrt(-
(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a*
*4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c*
*2 + 20*a*b**8*c - b**10)))**2 - 7545*a**8*b**10*c**4*(3*b/(2*a**4) - 3*sqrt(-(4
*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4
*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2
+ 20*a*b**8*c - b**10))) + 842688*a**8*b**3*c**8 + 1086*a**7*b**12*c**3*(3*b/(2
*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**
4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 -
160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 1719216*a**7*b**5*c**7 - 87*a**6*
b**14*c**2*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b*
*2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 64
0*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 1592964*a**6*b*
*7*c**6 + 3*a**5*b**16*c*(3*b/(2*a**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**
3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*
b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 8
43048*a**5*b**9*c**5 + 277245*a**4*b**11*c**4 - 57996*a**3*b**13*c**3 + 7542*a**
2*b**15*c**2 - 558*a*b**17*c + 18*b**19)/(18000*a**9*c**10 + 333720*a**8*b**2*c*
*9 - 991980*a**7*b**4*c**8 + 1099710*a**6*b**6*c**7 - 651186*a**5*b**8*c**6 + 23
1795*a**4*b**10*c**5 - 51480*a**3*b**12*c**4 + 7020*a**2*b**14*c**3 - 540*a*b**1
6*c**2 + 18*b**18*c)) + (3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3
- 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b
**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))*log(
x + (-108544*a**16*b*c**8*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c*
*3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4
*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2
+ 224768*a**15*b**3*c**7*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c*
*3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4
*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2
- 202752*a**14*b**5*c**6*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c*
*3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4
*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2
+ 104128*a**13*b**7*c**5*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c*
*3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4
*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2
- 19200*a**13*c**9*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 3
0*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*
c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 33320*
a**12*b**9*c**4*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a*
*2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 44736*a
**12*b**2*c**8*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**
2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 6806*a**11*
b**11*c**3*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b*
*2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 64
0*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 101232*a**11
*b**4*c**7*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b*
*2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 64
0*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 867*a**10*b**13
*c**2*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c*
*2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**
3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 77268*a**10*b**6*
c**6*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**
2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3
*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 63*a**9*b**15*c*(3*b/
(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b
**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3
- 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 + 31368*a**9*b**8*c**5*(3*b/(2
*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**
4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 -
160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 57600*a**9*b*c**9 - 2*a**8*b**17*
(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 1
0*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4
*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))**2 - 7545*a**8*b**10*c**4*(3
*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*
a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c
**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 842688*a**8*b**3*c**8 + 1086
*a**7*b**12*c**3*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a
**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**
4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 1719216*a
**7*b**5*c**7 - 87*a**6*b**14*c**2*(3*b/(2*a**4) + 3*sqrt(-(4*a*c - b**2)**5)*(2
0*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(1024*a**5*c**5 -
1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b*
*10))) + 1592964*a**6*b**7*c**6 + 3*a**5*b**16*c*(3*b/(2*a**4) + 3*sqrt(-(4*a*c
- b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*a**4*(102
4*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20
*a*b**8*c - b**10))) - 843048*a**5*b**9*c**5 + 277245*a**4*b**11*c**4 - 57996*a*
*3*b**13*c**3 + 7542*a**2*b**15*c**2 - 558*a*b**17*c + 18*b**19)/(18000*a**9*c**
10 + 333720*a**8*b**2*c**9 - 991980*a**7*b**4*c**8 + 1099710*a**6*b**6*c**7 - 65
1186*a**5*b**8*c**6 + 231795*a**4*b**10*c**5 - 51480*a**3*b**12*c**4 + 7020*a**2
*b**14*c**3 - 540*a*b**16*c**2 + 18*b**18*c)) - (32*a**4*c**2 - 16*a**3*b**2*c +
2*a**2*b**4 + x**4*(60*a**2*c**4 - 42*a*b**2*c**3 + 6*b**4*c**2) + x**3*(138*a*
*2*b*c**3 - 87*a*b**3*c**2 + 12*b**5*c) + x**2*(100*a**3*c**3 + 14*a**2*b**2*c**
2 - 36*a*b**4*c + 6*b**6) + x*(122*a**3*b*c**2 - 68*a**2*b**3*c + 9*a*b**5))/(x*
*5*(32*a**5*c**4 - 16*a**4*b**2*c**3 + 2*a**3*b**4*c**2) + x**4*(64*a**5*b*c**3
- 32*a**4*b**3*c**2 + 4*a**3*b**5*c) + x**3*(64*a**6*c**3 - 12*a**4*b**4*c + 2*a
**3*b**6) + x**2*(64*a**6*b*c**2 - 32*a**5*b**3*c + 4*a**4*b**5) + x*(32*a**7*c*
*2 - 16*a**6*b**2*c + 2*a**5*b**4)) - 3*b*log(x)/a**4

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GIAC/XCAS [A]  time = 0.210666, size = 417, normalized size = 1.74 \[ \frac{3 \,{\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \, b{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac{3 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{2 \, a^{3} b^{4} - 16 \, a^{4} b^{2} c + 32 \, a^{5} c^{2} + 6 \,{\left (a b^{4} c^{2} - 7 \, a^{2} b^{2} c^{3} + 10 \, a^{3} c^{4}\right )} x^{4} + 3 \,{\left (4 \, a b^{5} c - 29 \, a^{2} b^{3} c^{2} + 46 \, a^{3} b c^{3}\right )} x^{3} + 2 \,{\left (3 \, a b^{6} - 18 \, a^{2} b^{4} c + 7 \, a^{3} b^{2} c^{2} + 50 \, a^{4} c^{3}\right )} x^{2} +{\left (9 \, a^{2} b^{5} - 68 \, a^{3} b^{3} c + 122 \, a^{4} b c^{2}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}^{2} a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x + b)/sqrt(-b^2
+ 4*a*c))/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) + 3/2*b*ln(c
*x^2 + b*x + a)/a^4 - 3*b*ln(abs(x))/a^4 - 1/2*(2*a^3*b^4 - 16*a^4*b^2*c + 32*a^
5*c^2 + 6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*x^4 + 3*(4*a*b^5*c - 29*a^2*b
^3*c^2 + 46*a^3*b*c^3)*x^3 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*
c^3)*x^2 + (9*a^2*b^5 - 68*a^3*b^3*c + 122*a^4*b*c^2)*x)/((c*x^2 + b*x + a)^2*(b
^2 - 4*a*c)^2*a^4*x)