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

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

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Rubi [A]  time = 0.64, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {738, 818, 800, 634, 618, 206, 628} \begin {gather*} \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 (38 a^2 b^2 c^2-35 a^3 c^3-11 a b^4 c+b^6\right )}{c^4 \left (b^2-4 a c\right )^3}-\frac {4 \left (70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4-14 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} \end {gather*}

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx &=\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 {\int \frac {x^6 (14 a+2 b x)}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=\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 (a \left (b^2-14 a c\right )+b \left (b^2-9 a c\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {\int \frac {x^4 \left (10 a \left (b^2-14 a c\right )+4 b \left (2 b^2-13 a c\right ) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{6 c \left (b^2-4 a c\right )^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 (a \left (b^2-14 a c\right )+b \left (b^2-9 a c\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (4 a \left (b^4-9 a b^2 c+35 a^2 c^2\right )+b \left (4 b^4-39 a b^2 c+122 a^2 c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\int \frac {x^2 \left (24 a \left (b^4-9 a b^2 c+35 a^2 c^2\right )+24 b \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x\right )}{a+b x+c x^2} \, dx}{6 c^2 \left (b^2-4 a c\right )^3}\\ &=\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 (a \left (b^2-14 a c\right )+b \left (b^2-9 a c\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (4 a \left (b^4-9 a b^2 c+35 a^2 c^2\right )+b \left (4 b^4-39 a b^2 c+122 a^2 c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\int \left (-\frac {24 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )}{c^2}+\frac {24 b \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x}{c}+\frac {24 \left (a \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )+b \left (b^2-4 a c\right )^3 x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{6 c^2 \left (b^2-4 a c\right )^3}\\ &=\frac {4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right ) x}{c^4 \left (b^2-4 a c\right )^3}-\frac {2 b \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x^2}{c^3 \left (b^2-4 a c\right )^3}+\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 (a \left (b^2-14 a c\right )+b \left (b^2-9 a c\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (4 a \left (b^4-9 a b^2 c+35 a^2 c^2\right )+b \left (4 b^4-39 a b^2 c+122 a^2 c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {4 \int \frac {a \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right )+b \left (b^2-4 a c\right )^3 x}{a+b x+c x^2} \, dx}{c^4 \left (b^2-4 a c\right )^3}\\ &=\frac {4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right ) x}{c^4 \left (b^2-4 a c\right )^3}-\frac {2 b \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x^2}{c^3 \left (b^2-4 a c\right )^3}+\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 (a \left (b^2-14 a c\right )+b \left (b^2-9 a c\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (4 a \left (b^4-9 a b^2 c+35 a^2 c^2\right )+b \left (4 b^4-39 a b^2 c+122 a^2 c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {(2 b) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{c^5}+\frac {\left (2 \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 )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{c^5 \left (b^2-4 a c\right )^3}\\ &=\frac {4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right ) x}{c^4 \left (b^2-4 a c\right )^3}-\frac {2 b \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x^2}{c^3 \left (b^2-4 a c\right )^3}+\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 (a \left (b^2-14 a c\right )+b \left (b^2-9 a c\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (4 a \left (b^4-9 a b^2 c+35 a^2 c^2\right )+b \left (4 b^4-39 a b^2 c+122 a^2 c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {2 b \log \left (a+b x+c x^2\right )}{c^5}-\frac {\left (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 )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^5 \left (b^2-4 a c\right )^3}\\ &=\frac {4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right ) x}{c^4 \left (b^2-4 a c\right )^3}-\frac {2 b \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x^2}{c^3 \left (b^2-4 a c\right )^3}+\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 (a \left (b^2-14 a c\right )+b \left (b^2-9 a c\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (4 a \left (b^4-9 a b^2 c+35 a^2 c^2\right )+b \left (4 b^4-39 a b^2 c+122 a^2 c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\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 ) \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 {2 b \log \left (a+b x+c x^2\right )}{c^5}\\ \end {align*}

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Mathematica [A]  time = 0.77, size = 435, normalized size = 1.25 \begin {gather*} \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} \end {gather*}

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 + 198*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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.49, size = 3314, normalized size = 9.50

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*(13*a^3*b^9 - 199*a^4*b^7*c + 1123*a^5*b^5*c^2 - 2730*a^6*b^3*c^3 + 2360*a^7*b*c^4 - 3*(b^8*c^4 - 16*a*b
^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)*x^7 - 9*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 25
6*a^3*b^3*c^6 + 256*a^4*b*c^7)*x^6 + 3*(3*b^10*c^2 - 51*a*b^8*c^3 + 340*a^2*b^6*c^4 - 1112*a^3*b^4*c^5 + 1812*
a^4*b^2*c^6 - 1232*a^5*c^7)*x^5 + 3*(9*b^11*c - 144*a*b^9*c^2 + 874*a^2*b^7*c^3 - 2444*a^3*b^5*c^4 + 2994*a^4*
b^3*c^5 - 1160*a^5*b*c^6)*x^4 + (13*b^12 - 157*a*b^10*c + 451*a^2*b^8*c^2 + 1340*a^3*b^6*c^3 - 8946*a^4*b^4*c^
4 + 13480*a^5*b^2*c^5 - 4480*a^6*c^6)*x^3 + 3*(13*a*b^11 - 198*a^2*b^9*c + 1106*a^3*b^7*c^2 - 2619*a^4*b^5*c^3
+ 2012*a^5*b^3*c^4 + 448*a^6*b*c^5)*x^2 + 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 - 70*a^4*b^4*c^3 - 70*a^5*b^2*c^4 + 7
0*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)*sqrt(b^2 - 4*
a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 3*(13*a^2*b^
10 - 203*a^3*b^8*c + 1183*a^4*b^6*c^2 - 3058*a^5*b^4*c^3 + 3108*a^6*b^2*c^4 - 560*a^7*c^5)*x + 6*(a^3*b^9 - 16
*a^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4 + (b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256
*a^3*b^3*c^6 + 256*a^4*b*c^7)*x^6 + 3*(b^10*c^2 - 16*a*b^8*c^3 + 96*a^2*b^6*c^4 - 256*a^3*b^4*c^5 + 256*a^4*b^
2*c^6)*x^5 + 3*(b^11*c - 15*a*b^9*c^2 + 80*a^2*b^7*c^3 - 160*a^3*b^5*c^4 + 256*a^5*b*c^6)*x^4 + (b^12 - 10*a*b
^10*c + 320*a^3*b^6*c^3 - 1280*a^4*b^4*c^4 + 1536*a^5*b^2*c^5)*x^3 + 3*(a*b^11 - 15*a^2*b^9*c + 80*a^3*b^7*c^2
- 160*a^4*b^5*c^3 + 256*a^6*b*c^5)*x^2 + 3*(a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^3 + 256*
a^6*b^2*c^4)*x)*log(c*x^2 + b*x + a))/(a^3*b^8*c^5 - 16*a^4*b^6*c^6 + 96*a^5*b^4*c^7 - 256*a^6*b^2*c^8 + 256*a
^7*c^9 + (b^8*c^8 - 16*a*b^6*c^9 + 96*a^2*b^4*c^10 - 256*a^3*b^2*c^11 + 256*a^4*c^12)*x^6 + 3*(b^9*c^7 - 16*a*
b^7*c^8 + 96*a^2*b^5*c^9 - 256*a^3*b^3*c^10 + 256*a^4*b*c^11)*x^5 + 3*(b^10*c^6 - 15*a*b^8*c^7 + 80*a^2*b^6*c^
8 - 160*a^3*b^4*c^9 + 256*a^5*c^11)*x^4 + (b^11*c^5 - 10*a*b^9*c^6 + 320*a^3*b^5*c^8 - 1280*a^4*b^3*c^9 + 1536
*a^5*b*c^10)*x^3 + 3*(a*b^10*c^5 - 15*a^2*b^8*c^6 + 80*a^3*b^6*c^7 - 160*a^4*b^4*c^8 + 256*a^6*c^10)*x^2 + 3*(
a^2*b^9*c^5 - 16*a^3*b^7*c^6 + 96*a^4*b^5*c^7 - 256*a^5*b^3*c^8 + 256*a^6*b*c^9)*x), -1/3*(13*a^3*b^9 - 199*a^
4*b^7*c + 1123*a^5*b^5*c^2 - 2730*a^6*b^3*c^3 + 2360*a^7*b*c^4 - 3*(b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 -
256*a^3*b^2*c^7 + 256*a^4*c^8)*x^7 - 9*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*
c^7)*x^6 + 3*(3*b^10*c^2 - 51*a*b^8*c^3 + 340*a^2*b^6*c^4 - 1112*a^3*b^4*c^5 + 1812*a^4*b^2*c^6 - 1232*a^5*c^7
)*x^5 + 3*(9*b^11*c - 144*a*b^9*c^2 + 874*a^2*b^7*c^3 - 2444*a^3*b^5*c^4 + 2994*a^4*b^3*c^5 - 1160*a^5*b*c^6)*
x^4 + (13*b^12 - 157*a*b^10*c + 451*a^2*b^8*c^2 + 1340*a^3*b^6*c^3 - 8946*a^4*b^4*c^4 + 13480*a^5*b^2*c^5 - 44
80*a^6*c^6)*x^3 + 3*(13*a*b^11 - 198*a^2*b^9*c + 1106*a^3*b^7*c^2 - 2619*a^4*b^5*c^3 + 2012*a^5*b^3*c^4 + 448*
a^6*b*c^5)*x^2 + 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 - 14
0*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)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 +
4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 3*(13*a^2*b^10 - 203*a^3*b^8*c + 1183*a^4*b^6*c^2 - 3058*a^5*b^4*c^3 + 31
08*a^6*b^2*c^4 - 560*a^7*c^5)*x + 6*(a^3*b^9 - 16*a^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4
+ (b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*x^6 + 3*(b^10*c^2 - 16*a*b^8*c^
3 + 96*a^2*b^6*c^4 - 256*a^3*b^4*c^5 + 256*a^4*b^2*c^6)*x^5 + 3*(b^11*c - 15*a*b^9*c^2 + 80*a^2*b^7*c^3 - 160*
a^3*b^5*c^4 + 256*a^5*b*c^6)*x^4 + (b^12 - 10*a*b^10*c + 320*a^3*b^6*c^3 - 1280*a^4*b^4*c^4 + 1536*a^5*b^2*c^5
)*x^3 + 3*(a*b^11 - 15*a^2*b^9*c + 80*a^3*b^7*c^2 - 160*a^4*b^5*c^3 + 256*a^6*b*c^5)*x^2 + 3*(a^2*b^10 - 16*a^
3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^3 + 256*a^6*b^2*c^4)*x)*log(c*x^2 + b*x + a))/(a^3*b^8*c^5 - 16*a^4*b
^6*c^6 + 96*a^5*b^4*c^7 - 256*a^6*b^2*c^8 + 256*a^7*c^9 + (b^8*c^8 - 16*a*b^6*c^9 + 96*a^2*b^4*c^10 - 256*a^3*
b^2*c^11 + 256*a^4*c^12)*x^6 + 3*(b^9*c^7 - 16*a*b^7*c^8 + 96*a^2*b^5*c^9 - 256*a^3*b^3*c^10 + 256*a^4*b*c^11)
*x^5 + 3*(b^10*c^6 - 15*a*b^8*c^7 + 80*a^2*b^6*c^8 - 160*a^3*b^4*c^9 + 256*a^5*c^11)*x^4 + (b^11*c^5 - 10*a*b^
9*c^6 + 320*a^3*b^5*c^8 - 1280*a^4*b^3*c^9 + 1536*a^5*b*c^10)*x^3 + 3*(a*b^10*c^5 - 15*a^2*b^8*c^6 + 80*a^3*b^
6*c^7 - 160*a^4*b^4*c^8 + 256*a^6*c^10)*x^2 + 3*(a^2*b^9*c^5 - 16*a^3*b^7*c^6 + 96*a^4*b^5*c^7 - 256*a^5*b^3*c
^8 + 256*a^6*b*c^9)*x)]

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giac [A]  time = 0.18, size = 467, normalized size = 1.34 \begin {gather*} \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 \log \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}} \end {gather*}

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*log(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)

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maple [B]  time = 0.09, size = 2336, normalized size = 6.69

result 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]

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+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-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^6-694/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^4-304/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+2/c^5/(64*
a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*ln(c*x^2+b*x+a)*b^7+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+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+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^2-114/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^4-150/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^6-143/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+1/c^4*x-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)-72/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^3-94/(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^2-280/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c
-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^4+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^4-32/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+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+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+560/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*ar
ctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^3*b^2-280/c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)
*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b^4+56/c^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^6-590/3/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)-4/c^5/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(4*a*c-b
^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^8+96/c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*ln(c*x^2+b
*x+a)*a^2*b^3-128/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*ln(c*x^2+b*x+a)*a^3*b-24/c^4/(64*a^3*c^3-48*a
^2*b^2*c^2+12*a*b^4*c-b^6)*ln(c*x^2+b*x+a)*a*b^5

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 1.79, size = 1159, normalized size = 3.32 \begin {gather*} \frac {x}{c^4}-\frac {\frac {2\,x^5\,\left (58\,a^4\,c^5-212\,a^3\,b^2\,c^4+146\,a^2\,b^4\,c^3-36\,a\,b^6\,c^2+3\,b^8\,c\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-\frac {2\,x^4\,\left (47\,a^4\,b\,c^4+226\,a^3\,b^3\,c^3-209\,a^2\,b^5\,c^2+57\,a\,b^7\,c-5\,b^9\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {x^3\,\left (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}\right )}{3\,c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {x^2\,\left (304\,a^5\,b\,c^4+387\,a^4\,b^3\,c^3-486\,a^3\,b^5\,c^2+143\,a^2\,b^7\,c-13\,a\,b^9\right )}{c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a^2\,\left (-590\,a^4\,b\,c^3+535\,a^3\,b^3\,c^2-147\,a^2\,b^5\,c+13\,a\,b^7\right )}{3\,c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a\,x\,\left (76\,a^5\,c^4-694\,a^4\,b^2\,c^3+567\,a^3\,b^4\,c^2-150\,a^2\,b^6\,c+13\,a\,b^8\right )}{c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}}{x^3\,\left (b^3\,c^4+6\,a\,b\,c^5\right )+x^2\,\left (3\,a^2\,c^5+3\,a\,b^2\,c^4\right )+a^3\,c^4+c^7\,x^6+x^4\,\left (3\,b^2\,c^5+3\,a\,c^6\right )+3\,b\,c^6\,x^5+3\,a^2\,b\,c^4\,x}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-65536\,a^7\,b\,c^7+114688\,a^6\,b^3\,c^6-86016\,a^5\,b^5\,c^5+35840\,a^4\,b^7\,c^4-8960\,a^3\,b^9\,c^3+1344\,a^2\,b^{11}\,c^2-112\,a\,b^{13}\,c+4\,b^{15}\right )}{2\,\left (16384\,a^7\,c^{12}-28672\,a^6\,b^2\,c^{11}+21504\,a^5\,b^4\,c^{10}-8960\,a^4\,b^6\,c^9+2240\,a^3\,b^8\,c^8-336\,a^2\,b^{10}\,c^7+28\,a\,b^{12}\,c^6-b^{14}\,c^5\right )}-\frac {4\,\mathrm {atan}\left (\frac {\left (\frac {4\,x\,\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 )}{c^4\,{\left (4\,a\,c-b^2\right )}^7}+\frac {2\,\left (-64\,a^3\,b\,c^7+48\,a^2\,b^3\,c^6-12\,a\,b^5\,c^5+b^7\,c^4\right )\,\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 )}{c^9\,{\left (4\,a\,c-b^2\right )}^7\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}\right )\,\left (64\,a^3\,c^8\,{\left (4\,a\,c-b^2\right )}^{7/2}-b^6\,c^5\,{\left (4\,a\,c-b^2\right )}^{7/2}+12\,a\,b^4\,c^6\,{\left (4\,a\,c-b^2\right )}^{7/2}-48\,a^2\,b^2\,c^7\,{\left (4\,a\,c-b^2\right )}^{7/2}\right )}{140\,a^4\,c^4-280\,a^3\,b^2\,c^3+140\,a^2\,b^4\,c^2-28\,a\,b^6\,c+2\,b^8}\right )\,\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 )}{c^5\,{\left (4\,a\,c-b^2\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a + b*x + c*x^2)^4,x)

[Out]

x/c^4 - ((2*x^5*(3*b^8*c + 58*a^4*c^5 - 36*a*b^6*c^2 + 146*a^2*b^4*c^3 - 212*a^3*b^2*c^4))/(b^6 - 64*a^3*c^3 +
48*a^2*b^2*c^2 - 12*a*b^4*c) - (2*x^4*(47*a^4*b*c^4 - 5*b^9 - 209*a^2*b^5*c^2 + 226*a^3*b^3*c^3 + 57*a*b^7*c)
)/(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c) + (x^3*(13*b^10 + 544*a^5*c^5 - 68*a^2*b^6*c^2 + 1788*a^3*b
^4*c^3 - 3234*a^4*b^2*c^4 - 96*a*b^8*c))/(3*c*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (x^2*(143*a^
2*b^7*c - 13*a*b^9 + 304*a^5*b*c^4 - 486*a^3*b^5*c^2 + 387*a^4*b^3*c^3))/(c*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2
- 12*a*b^4*c)) + (a^2*(13*a*b^7 - 147*a^2*b^5*c - 590*a^4*b*c^3 + 535*a^3*b^3*c^2))/(3*c*(b^6 - 64*a^3*c^3 +
48*a^2*b^2*c^2 - 12*a*b^4*c)) + (a*x*(13*a*b^8 + 76*a^5*c^4 - 150*a^2*b^6*c + 567*a^3*b^4*c^2 - 694*a^4*b^2*c^
3))/(c*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)))/(x^3*(b^3*c^4 + 6*a*b*c^5) + x^2*(3*a^2*c^5 + 3*a*b^
2*c^4) + a^3*c^4 + c^7*x^6 + x^4*(3*a*c^6 + 3*b^2*c^5) + 3*b*c^6*x^5 + 3*a^2*b*c^4*x) + (log(a + b*x + c*x^2)*
(4*b^15 - 65536*a^7*b*c^7 + 1344*a^2*b^11*c^2 - 8960*a^3*b^9*c^3 + 35840*a^4*b^7*c^4 - 86016*a^5*b^5*c^5 + 114
688*a^6*b^3*c^6 - 112*a*b^13*c))/(2*(16384*a^7*c^12 - b^14*c^5 + 28*a*b^12*c^6 - 336*a^2*b^10*c^7 + 2240*a^3*b
^8*c^8 - 8960*a^4*b^6*c^9 + 21504*a^5*b^4*c^10 - 28672*a^6*b^2*c^11)) - (4*atan((((4*x*(b^8 + 70*a^4*c^4 + 70*
a^2*b^4*c^2 - 140*a^3*b^2*c^3 - 14*a*b^6*c))/(c^4*(4*a*c - b^2)^7) + (2*(b^7*c^4 - 12*a*b^5*c^5 - 64*a^3*b*c^7
+ 48*a^2*b^3*c^6)*(b^8 + 70*a^4*c^4 + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 - 14*a*b^6*c))/(c^9*(4*a*c - b^2)^7*(b
^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)))*(64*a^3*c^8*(4*a*c - b^2)^(7/2) - b^6*c^5*(4*a*c - b^2)^(7/2)
+ 12*a*b^4*c^6*(4*a*c - b^2)^(7/2) - 48*a^2*b^2*c^7*(4*a*c - b^2)^(7/2)))/(2*b^8 + 140*a^4*c^4 + 140*a^2*b^4*
c^2 - 280*a^3*b^2*c^3 - 28*a*b^6*c))*(b^8 + 70*a^4*c^4 + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 - 14*a*b^6*c))/(c^5*
(4*a*c - b^2)^(7/2))

________________________________________________________________________________________

sympy [B]  time = 11.23, size = 2769, normalized size = 7.93

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 - 286
72*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 + 2
8*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 + 2150
4*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 + 28
0*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 + 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 + 2
240*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*(1
6384*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 + 53
5*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 - 1161*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 - 1
44*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 - 432*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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