3.2199 \(\int \frac{x^8}{\left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=349 \[ \frac{x^3 \left (b x \left (122 a^2 c^2-39 a b^2 c+4 b^4\right )+4 a \left (35 a^2 c^2-9 a b^2 c+b^4\right )\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{2 b x^2 \left (29 a^2 c^2-10 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^3}+\frac{4 x \left (-35 a^3 c^3+38 a^2 b^2 c^2-11 a b^4 c+b^6\right )}{c^4 \left (b^2-4 a c\right )^3}-\frac{4 \left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{7/2}}+\frac{x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^5 \left (b x \left (b^2-9 a c\right )+a \left (b^2-14 a c\right )\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{2 b \log \left (a+b x+c x^2\right )}{c^5} \]

[Out]

(4*(b^6 - 11*a*b^4*c + 38*a^2*b^2*c^2 - 35*a^3*c^3)*x)/(c^4*(b^2 - 4*a*c)^3) - (
2*b*(b^4 - 10*a*b^2*c + 29*a^2*c^2)*x^2)/(c^3*(b^2 - 4*a*c)^3) + (x^7*(2*a + b*x
))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (x^5*(a*(b^2 - 14*a*c) + b*(b^2 - 9*a
*c)*x))/(3*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (x^3*(4*a*(b^4 - 9*a*b^2*c +
35*a^2*c^2) + b*(4*b^4 - 39*a*b^2*c + 122*a^2*c^2)*x))/(3*c^2*(b^2 - 4*a*c)^3*(
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]])/(c^5*(b^2 - 4*a*c)^(7/2)) - (2
*b*Log[a + b*x + c*x^2])/c^5

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Rubi [A]  time = 1.45192, antiderivative size = 349, 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 \[ \frac{x^3 \left (b x \left (122 a^2 c^2-39 a b^2 c+4 b^4\right )+4 a \left (35 a^2 c^2-9 a b^2 c+b^4\right )\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{2 b x^2 \left (29 a^2 c^2-10 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^3}+\frac{4 x \left (-35 a^3 c^3+38 a^2 b^2 c^2-11 a b^4 c+b^6\right )}{c^4 \left (b^2-4 a c\right )^3}-\frac{4 \left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{7/2}}+\frac{x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^5 \left (b x \left (b^2-9 a c\right )+a \left (b^2-14 a c\right )\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{2 b \log \left (a+b x+c x^2\right )}{c^5} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(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)*x)/(c^4*(b^2 - 4*a*c)^3) - (
2*b*(b^4 - 10*a*b^2*c + 29*a^2*c^2)*x^2)/(c^3*(b^2 - 4*a*c)^3) + (x^7*(2*a + b*x
))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (x^5*(a*(b^2 - 14*a*c) + b*(b^2 - 9*a
*c)*x))/(3*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (x^3*(4*a*(b^4 - 9*a*b^2*c +
35*a^2*c^2) + b*(4*b^4 - 39*a*b^2*c + 122*a^2*c^2)*x))/(3*c^2*(b^2 - 4*a*c)^3*(
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]])/(c^5*(b^2 - 4*a*c)^(7/2)) - (2
*b*Log[a + b*x + c*x^2])/c^5

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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(x**8/(c*x**2+b*x+a)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 1.43353, size = 435, normalized size = 1.25 \[ \frac{\frac{a^4 c^3 (7 b-2 c x)-2 a^3 b^2 c^2 (7 b-8 c x)+a^2 b^4 c (7 b-20 c x)-a b^6 (b-8 c x)+b^8 (-x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^3}-\frac{12 c^2 \left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}-\frac{6 c \left (-163 a^4 b c^4+58 a^4 c^5 x+198 a^3 b^3 c^3-212 a^3 b^2 c^4 x-83 a^2 b^5 c^2+146 a^2 b^4 c^3 x+15 a b^7 c-36 a b^6 c^2 x-b^9+3 b^8 c x\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac{125 a^4 b c^4-38 a^4 c^5 x-202 a^3 b^3 c^3+220 a^3 b^2 c^4 x+95 a^2 b^5 c^2-212 a^2 b^4 c^3 x-17 a b^7 c+68 a b^6 c^2 x+b^9-7 b^8 c x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-6 b c^2 \log (a+x (b+c x))+3 c^3 x}{3 c^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x + c*x^2)^4,x]

[Out]

(3*c^3*x + (-(b^8*x) + a^2*b^4*c*(7*b - 20*c*x) - a*b^6*(b - 8*c*x) - 2*a^3*b^2*
c^2*(7*b - 8*c*x) + a^4*c^3*(7*b - 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^3) +
(b^9 - 17*a*b^7*c + 95*a^2*b^5*c^2 - 202*a^3*b^3*c^3 + 125*a^4*b*c^4 - 7*b^8*c*
x + 68*a*b^6*c^2*x - 212*a^2*b^4*c^3*x + 220*a^3*b^2*c^4*x - 38*a^4*c^5*x)/((b^2
- 4*a*c)^2*(a + x*(b + c*x))^2) - (6*c*(-b^9 + 15*a*b^7*c - 83*a^2*b^5*c^2 + 19
8*a^3*b^3*c^3 - 163*a^4*b*c^4 + 3*b^8*c*x - 36*a*b^6*c^2*x + 146*a^2*b^4*c^3*x -
212*a^3*b^2*c^4*x + 58*a^4*c^5*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) - (12*c^
2*(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) - 6*b*c^2*Log[a + x*(b + c*x)]
)/(3*c^7)

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Maple [B]  time = 0.048, size = 3283, normalized size = 9.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(c*x^2+b*x+a)^4,x)

[Out]

544/3/(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^3*a^5+56/c^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))*a*b^6+96/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))*a^2*b^3
-24/c^4/(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))*a*b^5+560/c^2/(16384*a^7*c^7-28672*a^6*b^2*c^6+2
1504*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)
)*a^3*b^2-280/c^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))*a^2*b^4-128/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))*a^3*b-94/(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^4*a^4-424/(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^5*a
^3*b^2+116*c/(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^5*a^4+
6/c^3/(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^5*b^8+1/c^4*x
+13/3/c^5/(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^3*b^10+76
/c/(c*x^2+b*x+a)^3*a^6/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x-590/3/c^2/(c
*x^2+b*x+a)^3*a^6*b/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)+535/3/c^3/(c*x^2+
b*x+a)^3*a^5*b^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)+13/3/c^5/(c*x^2+b*x+
a)^3*a^3*b^7/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)-49/c^4/(c*x^2+b*x+a)^3*a
^4*b^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)+10/c^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^4+2/c^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^7-28
0/c/(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^3-33
6*a^2*b^10*c^2+28*a*b^12*c-b^14)^(1/2))*a^4-4/c^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^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))*b^8-143/c^4/(c*x^2+b*x+a)^3*a^2*b^7/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-
b^6)*x^2-150/c^4/(c*x^2+b*x+a)^3*a^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*
x*b^6-32/c^4/(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^3*a*b^
8+292/c/(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^5*a^2*b^4-7
2/c^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^5*a*b^6-452/c
/(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^4*a^3+418/c^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^4*a^2-114/c^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^4*a-1078/c/(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^3*a^4*b^2+596/c^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^3*a^3*b^4-68/3/c^3/(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^3*a^2*b^6-304/c/(c*x^2+b*
x+a)^3*a^5*b/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-387/c^2/(c*x^2+b*x+a
)^3*a^4*b^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2+486/c^3/(c*x^2+b*x+a)
^3*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+13/c^5/(c*x^2+b*x+a)^3
*a*b^9/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-694/c^2/(c*x^2+b*x+a)^3*a^
5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*b^2+567/c^3/(c*x^2+b*x+a)^3*a^4/(
64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*b^4+13/c^5/(c*x^2+b*x+a)^3*a^2/(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x*b^8

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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(x^8/(c*x^2 + b*x + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30139, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^2 + b*x + a)^4,x, algorithm="fricas")

[Out]

[-1/3*(6*(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
+ (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^6
+ 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
^5 + 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^4 + (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^3 + 3*(a*b^10 - 13*a^2*b^8*c + 56*a^3*b^6*c^2 - 7
0*a^4*b^4*c^3 - 70*a^5*b^2*c^4 + 70*a^6*c^5)*x^2 + 3*(a^2*b^9 - 14*a^3*b^7*c + 7
0*a^4*b^5*c^2 - 140*a^5*b^3*c^3 + 70*a^6*b*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)) + (13*a^3*b^7 - 147*a^4*b^5*c + 535*a^5*b^3*c^2 - 590*a^6*b*c^3 - 3*(b
^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^7 - 9*(b^7*c^3 - 12*a*b^5
*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^6 + 3*(3*b^8*c^2 - 39*a*b^6*c^3 + 184*a^
2*b^4*c^4 - 376*a^3*b^2*c^5 + 308*a^4*c^6)*x^5 + 3*(9*b^9*c - 108*a*b^7*c^2 + 44
2*a^2*b^5*c^3 - 676*a^3*b^3*c^4 + 290*a^4*b*c^5)*x^4 + (13*b^10 - 105*a*b^8*c +
31*a^2*b^6*c^2 + 1464*a^3*b^4*c^3 - 3090*a^4*b^2*c^4 + 1120*a^5*c^5)*x^3 + 3*(13
*a*b^9 - 146*a^2*b^7*c + 522*a^3*b^5*c^2 - 531*a^4*b^3*c^3 - 112*a^5*b*c^4)*x^2
+ 3*(13*a^2*b^8 - 151*a^3*b^6*c + 579*a^4*b^4*c^2 - 742*a^5*b^2*c^3 + 140*a^6*c^
4)*x + 6*(a^3*b^7 - 12*a^4*b^5*c + 48*a^5*b^3*c^2 - 64*a^6*b*c^3 + (b^7*c^3 - 12
*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^6 + 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^5 + 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^4 + (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^3 + 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^2 + 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)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((a^3*b^6*c^5 - 12*a
^4*b^4*c^6 + 48*a^5*b^2*c^7 - 64*a^6*c^8 + (b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*
c^10 - 64*a^3*c^11)*x^6 + 3*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*
c^10)*x^5 + 3*(b^8*c^6 - 11*a*b^6*c^7 + 36*a^2*b^4*c^8 - 16*a^3*b^2*c^9 - 64*a^4
*c^10)*x^4 + (b^9*c^5 - 6*a*b^7*c^6 - 24*a^2*b^5*c^7 + 224*a^3*b^3*c^8 - 384*a^4
*b*c^9)*x^3 + 3*(a*b^8*c^5 - 11*a^2*b^6*c^6 + 36*a^3*b^4*c^7 - 16*a^4*b^2*c^8 -
64*a^5*c^9)*x^2 + 3*(a^2*b^7*c^5 - 12*a^3*b^5*c^6 + 48*a^4*b^3*c^7 - 64*a^5*b*c^
8)*x)*sqrt(b^2 - 4*a*c)), 1/3*(12*(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 + (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^6 + 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^5 + 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^4 + (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^3 + 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^2 + 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)*arct
an(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (13*a^3*b^7 - 147*a^4*b^5*c
+ 535*a^5*b^3*c^2 - 590*a^6*b*c^3 - 3*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 -
64*a^3*c^7)*x^7 - 9*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^
6 + 3*(3*b^8*c^2 - 39*a*b^6*c^3 + 184*a^2*b^4*c^4 - 376*a^3*b^2*c^5 + 308*a^4*c^
6)*x^5 + 3*(9*b^9*c - 108*a*b^7*c^2 + 442*a^2*b^5*c^3 - 676*a^3*b^3*c^4 + 290*a^
4*b*c^5)*x^4 + (13*b^10 - 105*a*b^8*c + 31*a^2*b^6*c^2 + 1464*a^3*b^4*c^3 - 3090
*a^4*b^2*c^4 + 1120*a^5*c^5)*x^3 + 3*(13*a*b^9 - 146*a^2*b^7*c + 522*a^3*b^5*c^2
- 531*a^4*b^3*c^3 - 112*a^5*b*c^4)*x^2 + 3*(13*a^2*b^8 - 151*a^3*b^6*c + 579*a^
4*b^4*c^2 - 742*a^5*b^2*c^3 + 140*a^6*c^4)*x + 6*(a^3*b^7 - 12*a^4*b^5*c + 48*a^
5*b^3*c^2 - 64*a^6*b*c^3 + (b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c
^6)*x^6 + 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^5 + 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^4 + (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^3 + 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^2 + 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)*log(c*x^2 + b*x + a))
*sqrt(-b^2 + 4*a*c))/((a^3*b^6*c^5 - 12*a^4*b^4*c^6 + 48*a^5*b^2*c^7 - 64*a^6*c^
8 + (b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*x^6 + 3*(b^7*c^7 -
12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*x^5 + 3*(b^8*c^6 - 11*a*b^6*c^7 +
36*a^2*b^4*c^8 - 16*a^3*b^2*c^9 - 64*a^4*c^10)*x^4 + (b^9*c^5 - 6*a*b^7*c^6 - 2
4*a^2*b^5*c^7 + 224*a^3*b^3*c^8 - 384*a^4*b*c^9)*x^3 + 3*(a*b^8*c^5 - 11*a^2*b^6
*c^6 + 36*a^3*b^4*c^7 - 16*a^4*b^2*c^8 - 64*a^5*c^9)*x^2 + 3*(a^2*b^7*c^5 - 12*a
^3*b^5*c^6 + 48*a^4*b^3*c^7 - 64*a^5*b*c^8)*x)*sqrt(-b^2 + 4*a*c))]

_______________________________________________________________________________________

Sympy [A]  time = 43.291, size = 2769, normalized size = 7.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(c*x**2+b*x+a)**4,x)

[Out]

(-2*b/c**5 - 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**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)))*log(x + (-372*a**4*b*c**3 - 256*a**4*c**8*
(-2*b/c**5 - 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**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))) + 232*a**3*b**3*c**2 + 256*a**3*b**2*c**7*
(-2*b/c**5 - 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**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))) - 52*a**2*b**5*c - 96*a**2*b**4*c**6*(-2*b
/c**5 - 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**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**1
0*c**2 + 28*a*b**12*c - b**14))) + 4*a*b**7 + 16*a*b**6*c**5*(-2*b/c**5 - 2*sqrt
(-(4*a*c - b**2)**7)*(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)/(c**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))) - b**8*c**4*(-2*b/c**5 - 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**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))))/(280*a**4*
c**4 - 560*a**3*b**2*c**3 + 280*a**2*b**4*c**2 - 56*a*b**6*c + 4*b**8)) + (-2*b/
c**5 + 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 2
1504*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)))*log(x + (-372*a**4*b*c**3 - 256*a**4*c**8*(-2*b/
c**5 + 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 2
1504*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))) + 232*a**3*b**3*c**2 + 256*a**3*b**2*c**7*(-2*b/
c**5 + 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**5*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 2
1504*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))) - 52*a**2*b**5*c - 96*a**2*b**4*c**6*(-2*b/c**5
+ 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**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))) + 4*a*b**7 + 16*a*b**6*c**5*(-2*b/c**5 + 2*sqrt(-(4*a
*c - b**2)**7)*(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)/(c**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))) - b**8*c**4*(-2*b/c**5 + 2*sqrt(-(4*a*c - b**2)**7)*(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)/(c**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))))/(280*a**4*c**4 -
560*a**3*b**2*c**3 + 280*a**2*b**4*c**2 - 56*a*b**6*c + 4*b**8)) + (-590*a**6*b
*c**3 + 535*a**5*b**3*c**2 - 147*a**4*b**5*c + 13*a**3*b**7 + x**5*(348*a**4*c**
6 - 1272*a**3*b**2*c**5 + 876*a**2*b**4*c**4 - 216*a*b**6*c**3 + 18*b**8*c**2) +
x**4*(-282*a**4*b*c**5 - 1356*a**3*b**3*c**4 + 1254*a**2*b**5*c**3 - 342*a*b**7
*c**2 + 30*b**9*c) + x**3*(544*a**5*c**5 - 3234*a**4*b**2*c**4 + 1788*a**3*b**4*
c**3 - 68*a**2*b**6*c**2 - 96*a*b**8*c + 13*b**10) + x**2*(-912*a**5*b*c**4 - 11
61*a**4*b**3*c**3 + 1458*a**3*b**5*c**2 - 429*a**2*b**7*c + 39*a*b**9) + x*(228*
a**6*c**4 - 2082*a**5*b**2*c**3 + 1701*a**4*b**4*c**2 - 450*a**3*b**6*c + 39*a**
2*b**8))/(192*a**6*c**8 - 144*a**5*b**2*c**7 + 36*a**4*b**4*c**6 - 3*a**3*b**6*c
**5 + x**6*(192*a**3*c**11 - 144*a**2*b**2*c**10 + 36*a*b**4*c**9 - 3*b**6*c**8)
+ x**5*(576*a**3*b*c**10 - 432*a**2*b**3*c**9 + 108*a*b**5*c**8 - 9*b**7*c**7)
+ x**4*(576*a**4*c**10 + 144*a**3*b**2*c**9 - 324*a**2*b**4*c**8 + 99*a*b**6*c**
7 - 9*b**8*c**6) + x**3*(1152*a**4*b*c**9 - 672*a**3*b**3*c**8 + 72*a**2*b**5*c*
*7 + 18*a*b**7*c**6 - 3*b**9*c**5) + x**2*(576*a**5*c**9 + 144*a**4*b**2*c**8 -
324*a**3*b**4*c**7 + 99*a**2*b**6*c**6 - 9*a*b**8*c**5) + x*(576*a**5*b*c**8 - 4
32*a**4*b**3*c**7 + 108*a**3*b**5*c**6 - 9*a**2*b**7*c**5)) + x/c**4

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GIAC/XCAS [A]  time = 0.213731, size = 630, normalized size = 1.81 \[ \frac{4 \,{\left (b^{8} - 14 \, a b^{6} c + 70 \, a^{2} b^{4} c^{2} - 140 \, a^{3} b^{2} c^{3} + 70 \, a^{4} c^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} c^{5} - 12 \, a b^{4} c^{6} + 48 \, a^{2} b^{2} c^{7} - 64 \, a^{3} c^{8}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x}{c^{4}} - \frac{2 \, b{\rm ln}\left (c x^{2} + b x + a\right )}{c^{5}} - \frac{13 \, a^{3} b^{7} - 147 \, a^{4} b^{5} c + 535 \, a^{5} b^{3} c^{2} - 590 \, a^{6} b c^{3} + 6 \,{\left (3 \, b^{8} c^{2} - 36 \, a b^{6} c^{3} + 146 \, a^{2} b^{4} c^{4} - 212 \, a^{3} b^{2} c^{5} + 58 \, a^{4} c^{6}\right )} x^{5} + 6 \,{\left (5 \, b^{9} c - 57 \, a b^{7} c^{2} + 209 \, a^{2} b^{5} c^{3} - 226 \, a^{3} b^{3} c^{4} - 47 \, a^{4} b c^{5}\right )} x^{4} +{\left (13 \, b^{10} - 96 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 1788 \, a^{3} b^{4} c^{3} - 3234 \, a^{4} b^{2} c^{4} + 544 \, a^{5} c^{5}\right )} x^{3} + 3 \,{\left (13 \, a b^{9} - 143 \, a^{2} b^{7} c + 486 \, a^{3} b^{5} c^{2} - 387 \, a^{4} b^{3} c^{3} - 304 \, a^{5} b c^{4}\right )} x^{2} + 3 \,{\left (13 \, a^{2} b^{8} - 150 \, a^{3} b^{6} c + 567 \, a^{4} b^{4} c^{2} - 694 \, a^{5} b^{2} c^{3} + 76 \, a^{6} c^{4}\right )} x}{3 \,{\left (c x^{2} + b x + a\right )}^{3}{\left (b^{2} - 4 \, a c\right )}^{3} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^2 + b*x + a)^4,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))/((b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c
^8)*sqrt(-b^2 + 4*a*c)) + x/c^4 - 2*b*ln(c*x^2 + b*x + a)/c^5 - 1/3*(13*a^3*b^7
- 147*a^4*b^5*c + 535*a^5*b^3*c^2 - 590*a^6*b*c^3 + 6*(3*b^8*c^2 - 36*a*b^6*c^3
+ 146*a^2*b^4*c^4 - 212*a^3*b^2*c^5 + 58*a^4*c^6)*x^5 + 6*(5*b^9*c - 57*a*b^7*c^
2 + 209*a^2*b^5*c^3 - 226*a^3*b^3*c^4 - 47*a^4*b*c^5)*x^4 + (13*b^10 - 96*a*b^8*
c - 68*a^2*b^6*c^2 + 1788*a^3*b^4*c^3 - 3234*a^4*b^2*c^4 + 544*a^5*c^5)*x^3 + 3*
(13*a*b^9 - 143*a^2*b^7*c + 486*a^3*b^5*c^2 - 387*a^4*b^3*c^3 - 304*a^5*b*c^4)*x
^2 + 3*(13*a^2*b^8 - 150*a^3*b^6*c + 567*a^4*b^4*c^2 - 694*a^5*b^2*c^3 + 76*a^6*
c^4)*x)/((c*x^2 + b*x + a)^3*(b^2 - 4*a*c)^3*c^5)