3.2209 \(\int \frac{1}{x^2 \left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=352 \[ \frac{2 b \log \left (a+b x+c x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}+\frac{2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{3 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{2 \left (-70 a^3 c^3+105 a^2 b^2 c^2+3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )-32 a b^4 c+3 b^6\right )}{3 a^3 x \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{4 \left (-35 a^3 c^3+38 a^2 b^2 c^2-11 a b^4 c+b^6\right )}{a^4 x \left (b^2-4 a c\right )^3}-\frac{4 \left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{7/2}}+\frac{-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]

[Out]

(-4*(b^6 - 11*a*b^4*c + 38*a^2*b^2*c^2 - 35*a^3*c^3))/(a^4*(b^2 - 4*a*c)^3*x) +
(b^2 - 2*a*c + b*c*x)/(3*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^3) + (2*(b^4 - 7*a*
b^2*c + 7*a^2*c^2) + b*c*(2*b^2 - 13*a*c)*x)/(3*a^2*(b^2 - 4*a*c)^2*x*(a + b*x +
c*x^2)^2) + (2*(3*b^6 - 32*a*b^4*c + 105*a^2*b^2*c^2 - 70*a^3*c^3 + 3*b*c*(b^4
- 10*a*b^2*c + 29*a^2*c^2)*x))/(3*a^3*(b^2 - 4*a*c)^3*x*(a + b*x + c*x^2)) - (4*
(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*ArcTanh[(b +
2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^5*(b^2 - 4*a*c)^(7/2)) - (4*b*Log[x])/a^5 + (2*b*L
og[a + b*x + c*x^2])/a^5

_______________________________________________________________________________________

Rubi [A]  time = 1.23769, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{2 b \log \left (a+b x+c x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}+\frac{2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{3 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{2 \left (-70 a^3 c^3+105 a^2 b^2 c^2+3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )-32 a b^4 c+3 b^6\right )}{3 a^3 x \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{4 \left (-35 a^3 c^3+38 a^2 b^2 c^2-11 a b^4 c+b^6\right )}{a^4 x \left (b^2-4 a c\right )^3}-\frac{4 \left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{7/2}}+\frac{-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(b^6 - 11*a*b^4*c + 38*a^2*b^2*c^2 - 35*a^3*c^3))/(a^4*(b^2 - 4*a*c)^3*x) +
(b^2 - 2*a*c + b*c*x)/(3*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^3) + (2*(b^4 - 7*a*
b^2*c + 7*a^2*c^2) + b*c*(2*b^2 - 13*a*c)*x)/(3*a^2*(b^2 - 4*a*c)^2*x*(a + b*x +
c*x^2)^2) + (2*(3*b^6 - 32*a*b^4*c + 105*a^2*b^2*c^2 - 70*a^3*c^3 + 3*b*c*(b^4
- 10*a*b^2*c + 29*a^2*c^2)*x))/(3*a^3*(b^2 - 4*a*c)^3*x*(a + b*x + c*x^2)) - (4*
(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*ArcTanh[(b +
2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^5*(b^2 - 4*a*c)^(7/2)) - (4*b*Log[x])/a^5 + (2*b*L
og[a + b*x + c*x^2])/a^5

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 1.60509, size = 329, normalized size = 0.93 \[ \frac{\frac{a^3 \left (-3 a b c-2 a c^2 x+b^3+b^2 c x\right )}{\left (4 a c-b^2\right ) (a+x (b+c x))^3}-\frac{a^2 \left (35 a^2 b c^2+22 a^2 c^3 x-22 a b^3 c-20 a b^2 c^2 x+3 b^5+3 b^4 c x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac{3 a \left (134 a^3 b c^3+76 a^3 c^4 x-124 a^2 b^3 c^2-104 a^2 b^2 c^3 x+34 a b^5 c+32 a b^4 c^2 x-3 b^7-3 b^6 c x\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}-\frac{12 \left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}+6 b \log (a+x (b+c x))-\frac{3 a}{x}-12 b \log (x)}{3 a^5} \]

Antiderivative was successfully verified.

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

[Out]

((-3*a)/x + (a^3*(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))^3) - (a^2*(3*b^5 - 22*a*b^3*c + 35*a^2*b*c^2 + 3*b^4*c*x - 20*a*b^2*c^
2*x + 22*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + (3*a*(-3*b^7 + 34*a
*b^5*c - 124*a^2*b^3*c^2 + 134*a^3*b*c^3 - 3*b^6*c*x + 32*a*b^4*c^2*x - 104*a^2*
b^2*c^3*x + 76*a^3*c^4*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) - (12*(b^8 - 14*a
*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*ArcTan[(b + 2*c*x)/Sqrt[
-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) - 12*b*Log[x] + 6*b*Log[a + x*(b + c*x)])/(
3*a^5)

_______________________________________________________________________________________

Maple [B]  time = 0.037, size = 3109, normalized size = 8.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^2+b*x+a)^4,x)

[Out]

24/a^4*c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*ln((64*a^3*c^3-48*a^2*b^2*c^
2+12*a*b^4*c-b^6)*(c*x^2+b*x+a))*b^5+560/a^2/(16384*a^7*c^7-28672*a^6*b^2*c^6+21
504*a^5*b^4*c^5-8960*a^4*b^6*c^4+2240*a^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b
^14)^(1/2)*arctan((2*c*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+b*(64*a^3*c^
3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(16384*a^7*c^7-28672*a^6*b^2*c^6+21504*a^5*b^4
*c^5-8960*a^4*b^6*c^4+2240*a^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b^14)^(1/2))
*b^2*c^3-280/a^3/(16384*a^7*c^7-28672*a^6*b^2*c^6+21504*a^5*b^4*c^5-8960*a^4*b^6
*c^4+2240*a^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b^14)^(1/2)*arctan((2*c*(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+b*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-
b^6))/(16384*a^7*c^7-28672*a^6*b^2*c^6+21504*a^5*b^4*c^5-8960*a^4*b^6*c^4+2240*a
^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b^14)^(1/2))*b^4*c^2+3/a^4/(c*x^2+b*x+a)
^3*b^9/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-116*a/(c*x^2+b*x+a)^3/(64*
a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*c^4+7/a^3/(c*x^2+b*x+a)^3/(64*a^3*c^3-4
8*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*b^8-590/3*a/(c*x^2+b*x+a)^3*b/(64*a^3*c^3-48*a^2
*b^2*c^2+12*a*b^4*c-b^6)*c^3-49/a/(c*x^2+b*x+a)^3*b^5/(64*a^3*c^3-48*a^2*b^2*c^2
+12*a*b^4*c-b^6)*c-76/a/(c*x^2+b*x+a)^3*c^6/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*
c-b^6)*x^5+56/a^4/(16384*a^7*c^7-28672*a^6*b^2*c^6+21504*a^5*b^4*c^5-8960*a^4*b^
6*c^4+2240*a^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b^14)^(1/2)*arctan((2*c*(64*
a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+b*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c
-b^6))/(16384*a^7*c^7-28672*a^6*b^2*c^6+21504*a^5*b^4*c^5-8960*a^4*b^6*c^4+2240*
a^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b^14)^(1/2))*b^6*c+128/a^2*c^3/(64*a^3*
c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*ln((64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)
*(c*x^2+b*x+a))*b-496/(c*x^2+b*x+a)^3*b/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^
6)*x^2*c^4-166/(c*x^2+b*x+a)^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*b^2*
c^3-4*b*ln(x)/a^5+13/3/a^2/(c*x^2+b*x+a)^3*b^7/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b
^4*c-b^6)-2/a^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*ln((64*a^3*c^3-48*a^2
*b^2*c^2+12*a*b^4*c-b^6)*(c*x^2+b*x+a))*b^7-280/a/(16384*a^7*c^7-28672*a^6*b^2*c
^6+21504*a^5*b^4*c^5-8960*a^4*b^6*c^4+2240*a^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^1
2*c-b^14)^(1/2)*arctan((2*c*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+b*(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(16384*a^7*c^7-28672*a^6*b^2*c^6+21504*a^
5*b^4*c^5-8960*a^4*b^6*c^4+2240*a^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b^14)^(
1/2))*c^4-4/a^5/(16384*a^7*c^7-28672*a^6*b^2*c^6+21504*a^5*b^4*c^5-8960*a^4*b^6*
c^4+2240*a^3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b^14)^(1/2)*arctan((2*c*(64*a^
3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+b*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b
^6))/(16384*a^7*c^7-28672*a^6*b^2*c^6+21504*a^5*b^4*c^5-8960*a^4*b^6*c^4+2240*a^
3*b^8*c^3-336*a^2*b^10*c^2+28*a*b^12*c-b^14)^(1/2))*b^8+535/3/(c*x^2+b*x+a)^3*b^
3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^2-544/3/(c*x^2+b*x+a)^3*c^5/(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3+397/a/(c*x^2+b*x+a)^3*b^3/(64*a^3*c^3-
48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2*c^3-30/a^2/(c*x^2+b*x+a)^3*b^5/(64*a^3*c^3-48
*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2*c^2+104/a^2/(c*x^2+b*x+a)^3*c^5/(64*a^3*c^3-48*
a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5*b^2-32/a^3/(c*x^2+b*x+a)^3*c^4/(64*a^3*c^3-48*a^
2*b^2*c^2+12*a*b^4*c-b^6)*x^5*b^4+3/a^4/(c*x^2+b*x+a)^3*c^3/(64*a^3*c^3-48*a^2*b
^2*c^2+12*a*b^4*c-b^6)*x^5*b^6-20/a^3/(c*x^2+b*x+a)^3*b^7/(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)*x^2*c+243/a/(c*x^2+b*x+a)^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a
*b^4*c-b^6)*x*b^4*c^2-75/a^2/(c*x^2+b*x+a)^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4
*c-b^6)*x*b^6*c-286/a/(c*x^2+b*x+a)^3*b*c^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*
c-b^6)*x^4+332/a^2/(c*x^2+b*x+a)^3*b^3*c^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c
-b^6)*x^4-98/a^3/(c*x^2+b*x+a)^3*b^5*c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b
^6)*x^4+9/a^4/(c*x^2+b*x+a)^3*b^7*c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)
*x^4-102/a/(c*x^2+b*x+a)^3*c^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3*b^
2+832/3/a^2/(c*x^2+b*x+a)^3*c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3*b
^4-93/a^3/(c*x^2+b*x+a)^3*c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3*b^6
+9/a^4/(c*x^2+b*x+a)^3*c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3*b^8-96/a
^3*c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*ln((64*a^3*c^3-48*a^2*b^2*c^2+
12*a*b^4*c-b^6)*(c*x^2+b*x+a))*b^3-1/a^4/x

_______________________________________________________________________________________

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)^4*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 2.51817, 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)^4*x^2),x, algorithm="fricas")

[Out]

[-1/3*(6*((b^8*c^3 - 14*a*b^6*c^4 + 70*a^2*b^4*c^5 - 140*a^3*b^2*c^6 + 70*a^4*c^
7)*x^7 + 3*(b^9*c^2 - 14*a*b^7*c^3 + 70*a^2*b^5*c^4 - 140*a^3*b^3*c^5 + 70*a^4*b
*c^6)*x^6 + 3*(b^10*c - 13*a*b^8*c^2 + 56*a^2*b^6*c^3 - 70*a^3*b^4*c^4 - 70*a^4*
b^2*c^5 + 70*a^5*c^6)*x^5 + (b^11 - 8*a*b^9*c - 14*a^2*b^7*c^2 + 280*a^3*b^5*c^3
- 770*a^4*b^3*c^4 + 420*a^5*b*c^5)*x^4 + 3*(a*b^10 - 13*a^2*b^8*c + 56*a^3*b^6*
c^2 - 70*a^4*b^4*c^3 - 70*a^5*b^2*c^4 + 70*a^6*c^5)*x^3 + 3*(a^2*b^9 - 14*a^3*b^
7*c + 70*a^4*b^5*c^2 - 140*a^5*b^3*c^3 + 70*a^6*b*c^4)*x^2 + (a^3*b^8 - 14*a^4*b
^6*c + 70*a^5*b^4*c^2 - 140*a^6*b^2*c^3 + 70*a^7*c^4)*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)) + (3*a^4*b^6 - 36*a^5*b^4*c + 144*a^6*b^2*c^2 - 192*a^7*c^3 + 12
*(a*b^6*c^3 - 11*a^2*b^4*c^4 + 38*a^3*b^2*c^5 - 35*a^4*c^6)*x^6 + 6*(6*a*b^7*c^2
- 67*a^2*b^5*c^3 + 238*a^3*b^3*c^4 - 239*a^4*b*c^5)*x^5 + 2*(18*a*b^8*c - 189*a
^2*b^6*c^2 + 578*a^3*b^4*c^3 - 225*a^4*b^2*c^4 - 560*a^5*c^5)*x^4 + 3*(4*a*b^9 -
26*a^2*b^7*c - 54*a^3*b^5*c^2 + 621*a^4*b^3*c^3 - 880*a^5*b*c^4)*x^3 + 3*(10*a^
2*b^8 - 108*a^3*b^6*c + 351*a^4*b^4*c^2 - 214*a^5*b^2*c^3 - 308*a^6*c^4)*x^2 + (
22*a^3*b^7 - 255*a^4*b^5*c + 967*a^5*b^3*c^2 - 1166*a^6*b*c^3)*x - 6*((b^7*c^3 -
12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^7 + 3*(b^8*c^2 - 12*a*b^6*c^3 +
48*a^2*b^4*c^4 - 64*a^3*b^2*c^5)*x^6 + 3*(b^9*c - 11*a*b^7*c^2 + 36*a^2*b^5*c^3
- 16*a^3*b^3*c^4 - 64*a^4*b*c^5)*x^5 + (b^10 - 6*a*b^8*c - 24*a^2*b^6*c^2 + 224
*a^3*b^4*c^3 - 384*a^4*b^2*c^4)*x^4 + 3*(a*b^9 - 11*a^2*b^7*c + 36*a^3*b^5*c^2 -
16*a^4*b^3*c^3 - 64*a^5*b*c^4)*x^3 + 3*(a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2
- 64*a^5*b^2*c^3)*x^2 + (a^3*b^7 - 12*a^4*b^5*c + 48*a^5*b^3*c^2 - 64*a^6*b*c^3
)*x)*log(c*x^2 + b*x + a) + 12*((b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^
3*b*c^6)*x^7 + 3*(b^8*c^2 - 12*a*b^6*c^3 + 48*a^2*b^4*c^4 - 64*a^3*b^2*c^5)*x^6
+ 3*(b^9*c - 11*a*b^7*c^2 + 36*a^2*b^5*c^3 - 16*a^3*b^3*c^4 - 64*a^4*b*c^5)*x^5
+ (b^10 - 6*a*b^8*c - 24*a^2*b^6*c^2 + 224*a^3*b^4*c^3 - 384*a^4*b^2*c^4)*x^4 +
3*(a*b^9 - 11*a^2*b^7*c + 36*a^3*b^5*c^2 - 16*a^4*b^3*c^3 - 64*a^5*b*c^4)*x^3 +
3*(a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*x^2 + (a^3*b^7 - 12
*a^4*b^5*c + 48*a^5*b^3*c^2 - 64*a^6*b*c^3)*x)*log(x))*sqrt(b^2 - 4*a*c))/(((a^5
*b^6*c^3 - 12*a^6*b^4*c^4 + 48*a^7*b^2*c^5 - 64*a^8*c^6)*x^7 + 3*(a^5*b^7*c^2 -
12*a^6*b^5*c^3 + 48*a^7*b^3*c^4 - 64*a^8*b*c^5)*x^6 + 3*(a^5*b^8*c - 11*a^6*b^6*
c^2 + 36*a^7*b^4*c^3 - 16*a^8*b^2*c^4 - 64*a^9*c^5)*x^5 + (a^5*b^9 - 6*a^6*b^7*c
- 24*a^7*b^5*c^2 + 224*a^8*b^3*c^3 - 384*a^9*b*c^4)*x^4 + 3*(a^6*b^8 - 11*a^7*b
^6*c + 36*a^8*b^4*c^2 - 16*a^9*b^2*c^3 - 64*a^10*c^4)*x^3 + 3*(a^7*b^7 - 12*a^8*
b^5*c + 48*a^9*b^3*c^2 - 64*a^10*b*c^3)*x^2 + (a^8*b^6 - 12*a^9*b^4*c + 48*a^10*
b^2*c^2 - 64*a^11*c^3)*x)*sqrt(b^2 - 4*a*c)), 1/3*(12*((b^8*c^3 - 14*a*b^6*c^4 +
70*a^2*b^4*c^5 - 140*a^3*b^2*c^6 + 70*a^4*c^7)*x^7 + 3*(b^9*c^2 - 14*a*b^7*c^3
+ 70*a^2*b^5*c^4 - 140*a^3*b^3*c^5 + 70*a^4*b*c^6)*x^6 + 3*(b^10*c - 13*a*b^8*c^
2 + 56*a^2*b^6*c^3 - 70*a^3*b^4*c^4 - 70*a^4*b^2*c^5 + 70*a^5*c^6)*x^5 + (b^11 -
8*a*b^9*c - 14*a^2*b^7*c^2 + 280*a^3*b^5*c^3 - 770*a^4*b^3*c^4 + 420*a^5*b*c^5)
*x^4 + 3*(a*b^10 - 13*a^2*b^8*c + 56*a^3*b^6*c^2 - 70*a^4*b^4*c^3 - 70*a^5*b^2*c
^4 + 70*a^6*c^5)*x^3 + 3*(a^2*b^9 - 14*a^3*b^7*c + 70*a^4*b^5*c^2 - 140*a^5*b^3*
c^3 + 70*a^6*b*c^4)*x^2 + (a^3*b^8 - 14*a^4*b^6*c + 70*a^5*b^4*c^2 - 140*a^6*b^2
*c^3 + 70*a^7*c^4)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (3
*a^4*b^6 - 36*a^5*b^4*c + 144*a^6*b^2*c^2 - 192*a^7*c^3 + 12*(a*b^6*c^3 - 11*a^2
*b^4*c^4 + 38*a^3*b^2*c^5 - 35*a^4*c^6)*x^6 + 6*(6*a*b^7*c^2 - 67*a^2*b^5*c^3 +
238*a^3*b^3*c^4 - 239*a^4*b*c^5)*x^5 + 2*(18*a*b^8*c - 189*a^2*b^6*c^2 + 578*a^3
*b^4*c^3 - 225*a^4*b^2*c^4 - 560*a^5*c^5)*x^4 + 3*(4*a*b^9 - 26*a^2*b^7*c - 54*a
^3*b^5*c^2 + 621*a^4*b^3*c^3 - 880*a^5*b*c^4)*x^3 + 3*(10*a^2*b^8 - 108*a^3*b^6*
c + 351*a^4*b^4*c^2 - 214*a^5*b^2*c^3 - 308*a^6*c^4)*x^2 + (22*a^3*b^7 - 255*a^4
*b^5*c + 967*a^5*b^3*c^2 - 1166*a^6*b*c^3)*x - 6*((b^7*c^3 - 12*a*b^5*c^4 + 48*a
^2*b^3*c^5 - 64*a^3*b*c^6)*x^7 + 3*(b^8*c^2 - 12*a*b^6*c^3 + 48*a^2*b^4*c^4 - 64
*a^3*b^2*c^5)*x^6 + 3*(b^9*c - 11*a*b^7*c^2 + 36*a^2*b^5*c^3 - 16*a^3*b^3*c^4 -
64*a^4*b*c^5)*x^5 + (b^10 - 6*a*b^8*c - 24*a^2*b^6*c^2 + 224*a^3*b^4*c^3 - 384*a
^4*b^2*c^4)*x^4 + 3*(a*b^9 - 11*a^2*b^7*c + 36*a^3*b^5*c^2 - 16*a^4*b^3*c^3 - 64
*a^5*b*c^4)*x^3 + 3*(a^2*b^8 - 12*a^3*b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*x
^2 + (a^3*b^7 - 12*a^4*b^5*c + 48*a^5*b^3*c^2 - 64*a^6*b*c^3)*x)*log(c*x^2 + b*x
+ a) + 12*((b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^7 + 3*(b^
8*c^2 - 12*a*b^6*c^3 + 48*a^2*b^4*c^4 - 64*a^3*b^2*c^5)*x^6 + 3*(b^9*c - 11*a*b^
7*c^2 + 36*a^2*b^5*c^3 - 16*a^3*b^3*c^4 - 64*a^4*b*c^5)*x^5 + (b^10 - 6*a*b^8*c
- 24*a^2*b^6*c^2 + 224*a^3*b^4*c^3 - 384*a^4*b^2*c^4)*x^4 + 3*(a*b^9 - 11*a^2*b^
7*c + 36*a^3*b^5*c^2 - 16*a^4*b^3*c^3 - 64*a^5*b*c^4)*x^3 + 3*(a^2*b^8 - 12*a^3*
b^6*c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*x^2 + (a^3*b^7 - 12*a^4*b^5*c + 48*a^5*
b^3*c^2 - 64*a^6*b*c^3)*x)*log(x))*sqrt(-b^2 + 4*a*c))/(((a^5*b^6*c^3 - 12*a^6*b
^4*c^4 + 48*a^7*b^2*c^5 - 64*a^8*c^6)*x^7 + 3*(a^5*b^7*c^2 - 12*a^6*b^5*c^3 + 48
*a^7*b^3*c^4 - 64*a^8*b*c^5)*x^6 + 3*(a^5*b^8*c - 11*a^6*b^6*c^2 + 36*a^7*b^4*c^
3 - 16*a^8*b^2*c^4 - 64*a^9*c^5)*x^5 + (a^5*b^9 - 6*a^6*b^7*c - 24*a^7*b^5*c^2 +
224*a^8*b^3*c^3 - 384*a^9*b*c^4)*x^4 + 3*(a^6*b^8 - 11*a^7*b^6*c + 36*a^8*b^4*c
^2 - 16*a^9*b^2*c^3 - 64*a^10*c^4)*x^3 + 3*(a^7*b^7 - 12*a^8*b^5*c + 48*a^9*b^3*
c^2 - 64*a^10*b*c^3)*x^2 + (a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c
^3)*x)*sqrt(-b^2 + 4*a*c))]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215567, size = 670, normalized size = 1.9 \[ \frac{4 \,{\left (b^{8} - 14 \, a b^{6} c + 70 \, a^{2} b^{4} c^{2} - 140 \, a^{3} b^{2} c^{3} + 70 \, a^{4} c^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{6} - 12 \, a^{6} b^{4} c + 48 \, a^{7} b^{2} c^{2} - 64 \, a^{8} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{2 \, b{\rm ln}\left (c x^{2} + b x + a\right )}{a^{5}} - \frac{4 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} - \frac{3 \, a^{4} b^{6} - 36 \, a^{5} b^{4} c + 144 \, a^{6} b^{2} c^{2} - 192 \, a^{7} c^{3} + 12 \,{\left (a b^{6} c^{3} - 11 \, a^{2} b^{4} c^{4} + 38 \, a^{3} b^{2} c^{5} - 35 \, a^{4} c^{6}\right )} x^{6} + 6 \,{\left (6 \, a b^{7} c^{2} - 67 \, a^{2} b^{5} c^{3} + 238 \, a^{3} b^{3} c^{4} - 239 \, a^{4} b c^{5}\right )} x^{5} + 2 \,{\left (18 \, a b^{8} c - 189 \, a^{2} b^{6} c^{2} + 578 \, a^{3} b^{4} c^{3} - 225 \, a^{4} b^{2} c^{4} - 560 \, a^{5} c^{5}\right )} x^{4} + 3 \,{\left (4 \, a b^{9} - 26 \, a^{2} b^{7} c - 54 \, a^{3} b^{5} c^{2} + 621 \, a^{4} b^{3} c^{3} - 880 \, a^{5} b c^{4}\right )} x^{3} + 3 \,{\left (10 \, a^{2} b^{8} - 108 \, a^{3} b^{6} c + 351 \, a^{4} b^{4} c^{2} - 214 \, a^{5} b^{2} c^{3} - 308 \, a^{6} c^{4}\right )} x^{2} +{\left (22 \, a^{3} b^{7} - 255 \, a^{4} b^{5} c + 967 \, a^{5} b^{3} c^{2} - 1166 \, a^{6} b c^{3}\right )} x}{3 \,{\left (c x^{2} + b x + a\right )}^{3}{\left (b^{2} - 4 \, a c\right )}^{3} a^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*arctan((2*c
*x + b)/sqrt(-b^2 + 4*a*c))/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c
^3)*sqrt(-b^2 + 4*a*c)) + 2*b*ln(c*x^2 + b*x + a)/a^5 - 4*b*ln(abs(x))/a^5 - 1/3
*(3*a^4*b^6 - 36*a^5*b^4*c + 144*a^6*b^2*c^2 - 192*a^7*c^3 + 12*(a*b^6*c^3 - 11*
a^2*b^4*c^4 + 38*a^3*b^2*c^5 - 35*a^4*c^6)*x^6 + 6*(6*a*b^7*c^2 - 67*a^2*b^5*c^3
+ 238*a^3*b^3*c^4 - 239*a^4*b*c^5)*x^5 + 2*(18*a*b^8*c - 189*a^2*b^6*c^2 + 578*
a^3*b^4*c^3 - 225*a^4*b^2*c^4 - 560*a^5*c^5)*x^4 + 3*(4*a*b^9 - 26*a^2*b^7*c - 5
4*a^3*b^5*c^2 + 621*a^4*b^3*c^3 - 880*a^5*b*c^4)*x^3 + 3*(10*a^2*b^8 - 108*a^3*b
^6*c + 351*a^4*b^4*c^2 - 214*a^5*b^2*c^3 - 308*a^6*c^4)*x^2 + (22*a^3*b^7 - 255*
a^4*b^5*c + 967*a^5*b^3*c^2 - 1166*a^6*b*c^3)*x)/((c*x^2 + b*x + a)^3*(b^2 - 4*a
*c)^3*a^5*x)