\(\int \frac {x^7}{(a+b x+c x^2)^4} \, dx\) [2211]

Optimal result

Integrand size = 16, antiderivative size = 291 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac {x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{7/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

-b*(38*a^2*c^2-11*a*b^2*c+b^4)*x/c^3/(-4*a*c+b^2)^3+1/3*x^6*(b*x+2*a)/(-4*a*c+b^2)/(c*x^2+b*x+a)^3+1/6*x^4*(a*
(-24*a*c+b^2)+b*(-14*a*c+b^2)*x)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2+1/6*x^2*(3*a*(64*a^2*c^2-10*a*b^2*c+b^4)+b*(
140*a^2*c^2-32*a*b^2*c+3*b^4)*x)/c^2/(-4*a*c+b^2)^3/(c*x^2+b*x+a)+b*(-140*a^3*c^3+70*a^2*b^2*c^2-14*a*b^4*c+b^
6)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^4/(-4*a*c+b^2)^(7/2)+1/2*ln(c*x^2+b*x+a)/c^4

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {752, 832, 787, 648, 632, 212, 642} \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^4} \, dx=\frac {x^2 \left (b x \left (140 a^2 c^2-32 a b^2 c+3 b^4\right )+3 a \left (64 a^2 c^2-10 a b^2 c+b^4\right )\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {b x \left (38 a^2 c^2-11 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^3}+\frac {b \left (-140 a^3 c^3+70 a^2 b^2 c^2-14 a b^4 c+b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{7/2}}+\frac {x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^4 \left (b x \left (b^2-14 a c\right )+a \left (b^2-24 a c\right )\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {\log \left (a+b x+c x^2\right )}{2 c^4} \]

[In]

Int[x^7/(a + b*x + c*x^2)^4,x]

[Out]

-((b*(b^4 - 11*a*b^2*c + 38*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^3)) + (x^6*(2*a + b*x))/(3*(b^2 - 4*a*c)*(a + b*x +
c*x^2)^3) + (x^4*(a*(b^2 - 24*a*c) + b*(b^2 - 14*a*c)*x))/(6*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (x^2*(3
*a*(b^4 - 10*a*b^2*c + 64*a^2*c^2) + b*(3*b^4 - 32*a*b^2*c + 140*a^2*c^2)*x))/(6*c^2*(b^2 - 4*a*c)^3*(a + b*x
+ c*x^2)) + (b*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*
(b^2 - 4*a*c)^(7/2)) + Log[a + b*x + c*x^2]/(2*c^4)

Rule 212

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

Rule 632

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 642

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 648

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 752

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 787

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*g*(x/c
), x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(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] &&
NeQ[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*} \text {integral}& = \frac {x^6 (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^5 (12 a+b x)}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )} \\ & = \frac {x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {\int \frac {x^3 \left (4 a \left (b^2-24 a c\right )+b \left (3 b^2-22 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^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\int \frac {x \left (6 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+6 b \left (b^4-11 a b^2 c+38 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 {b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac {x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\int \frac {-6 a b \left (b^4-11 a b^2 c+38 a^2 c^2\right )+\left (-6 b^2 \left (b^4-11 a b^2 c+38 a^2 c^2\right )+6 a c \left (b^4-10 a b^2 c+64 a^2 c^2\right )\right ) x}{a+b x+c x^2} \, dx}{6 c^3 \left (b^2-4 a c\right )^3} \\ & = -\frac {b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac {x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac {\left (b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 a^3 c^3\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )^3} \\ & = -\frac {b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac {x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\log \left (a+b x+c x^2\right )}{2 c^4}+\frac {\left (b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 a^3 c^3\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4 \left (b^2-4 a c\right )^3} \\ & = -\frac {b \left (b^4-11 a b^2 c+38 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^3}+\frac {x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^4 \left (a \left (b^2-24 a c\right )+b \left (b^2-14 a c\right ) x\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^2 \left (3 a \left (b^4-10 a b^2 c+64 a^2 c^2\right )+b \left (3 b^4-32 a b^2 c+140 a^2 c^2\right ) x\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {b \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{7/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.33 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^4} \, dx=\frac {\frac {-2 b^8+29 a b^6 c-139 a^2 b^4 c^2+233 a^3 b^2 c^3-72 a^4 c^4+11 b^7 c x-98 a b^5 c^2 x+259 a^2 b^3 c^3 x-182 a^3 b c^4 x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {3 c \left (3 b^8-40 a b^6 c+191 a^2 b^4 c^2-374 a^3 b^2 c^3+192 a^4 c^4-6 b^7 c x+70 a b^5 c^2 x-266 a^2 b^3 c^3 x+308 a^3 b c^4 x\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac {2 \left (-2 a^4 c^3+b^7 x+a b^5 (b-7 c x)+a^3 b c^2 (9 b-7 c x)+2 a^2 b^3 c (-3 b+7 c x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^3}+\frac {6 b c^2 \left (b^6-14 a b^4 c+70 a^2 b^2 c^2-140 a^3 c^3\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}+3 c^2 \log (a+x (b+c x))}{6 c^6} \]

[In]

Integrate[x^7/(a + b*x + c*x^2)^4,x]

[Out]

((-2*b^8 + 29*a*b^6*c - 139*a^2*b^4*c^2 + 233*a^3*b^2*c^3 - 72*a^4*c^4 + 11*b^7*c*x - 98*a*b^5*c^2*x + 259*a^2
*b^3*c^3*x - 182*a^3*b*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (3*c*(3*b^8 - 40*a*b^6*c + 191*a^2*b^4*c
^2 - 374*a^3*b^2*c^3 + 192*a^4*c^4 - 6*b^7*c*x + 70*a*b^5*c^2*x - 266*a^2*b^3*c^3*x + 308*a^3*b*c^4*x))/((b^2
- 4*a*c)^3*(a + x*(b + c*x))) + (2*(-2*a^4*c^3 + b^7*x + a*b^5*(b - 7*c*x) + a^3*b*c^2*(9*b - 7*c*x) + 2*a^2*b
^3*c*(-3*b + 7*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^3) + (6*b*c^2*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*
a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) + 3*c^2*Log[a + x*(b + c*x)])/(6*c^6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(680\) vs. \(2(279)=558\).

Time = 16.88 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.34

method	result	size
default	\(\frac {\frac {\left (154 c^{3} a^{3}-133 a^{2} b^{2} c^{2}+35 a \,b^{4} c -3 b^{6}\right ) b \,x^{5}}{c^{2} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {\left (192 a^{4} c^{4}+242 a^{3} b^{2} c^{3}-341 a^{2} b^{4} c^{2}+100 a \,b^{6} c -9 b^{8}\right ) x^{4}}{2 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}+\frac {b \left (2272 a^{4} c^{4}-1698 a^{3} b^{2} c^{3}+117 a^{2} b^{4} c^{2}+76 a \,b^{6} c -11 b^{8}\right ) x^{3}}{6 c^{4} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {a \left (288 a^{4} c^{4}+152 a^{3} b^{2} c^{3}-381 a^{2} b^{4} c^{2}+119 a \,b^{6} c -11 b^{8}\right ) x^{2}}{2 c^{4} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {a^{2} b \left (428 c^{3} a^{3}-460 a^{2} b^{2} c^{2}+126 a \,b^{4} c -11 b^{6}\right ) x}{2 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{4}}+\frac {\left (352 c^{3} a^{3}-438 a^{2} b^{2} c^{2}+124 a \,b^{4} c -11 b^{6}\right ) a^{3}}{6 c^{4} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {\frac {\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-38 a^{3} b \,c^{2}+11 a^{2} c \,b^{3}-a \,b^{5}-\frac {\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}\)	\(681\)
risch	\(\text {Expression too large to display}\)	\(2683\)

[In]

int(x^7/(c*x^2+b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

((154*a^3*c^3-133*a^2*b^2*c^2+35*a*b^4*c-3*b^6)*b/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5+1/2*(192*
a^4*c^4+242*a^3*b^2*c^3-341*a^2*b^4*c^2+100*a*b^6*c-9*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^3*x^4+
1/6*b/c^4*(2272*a^4*c^4-1698*a^3*b^2*c^3+117*a^2*b^4*c^2+76*a*b^6*c-11*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^
4*c-b^6)*x^3+1/2/c^4*a*(288*a^4*c^4+152*a^3*b^2*c^3-381*a^2*b^4*c^2+119*a*b^6*c-11*b^8)/(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)*x^2+1/2*a^2*b*(428*a^3*c^3-460*a^2*b^2*c^2+126*a*b^4*c-11*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2
+12*a*b^4*c-b^6)/c^4*x+1/6*(352*a^3*c^3-438*a^2*b^2*c^2+124*a*b^4*c-11*b^6)/c^4*a^3/(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)^3+1/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^3*(1/2*(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)/c*ln(c*x^2+b*x+a)+2*(-38*a^3*b*c^2+11*a^2*c*b^3-a*b^5-1/2*(64*a^3*c^3-48*a^2*b^2*c^2+12*a
*b^4*c-b^6)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1432 vs. \(2 (279) = 558\).

Time = 0.80 (sec) , antiderivative size = 2884, normalized size of antiderivative = 9.91 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]

[In]

integrate(x^7/(c*x^2+b*x+a)^4,x, algorithm="fricas")

[Out]

[1/6*(11*a^3*b^8 - 168*a^4*b^6*c + 934*a^5*b^4*c^2 - 2104*a^6*b^2*c^3 + 1408*a^7*c^4 + 6*(3*b^9*c^2 - 47*a*b^7
*c^3 + 273*a^2*b^5*c^4 - 686*a^3*b^3*c^5 + 616*a^4*b*c^6)*x^5 + 3*(9*b^10*c - 136*a*b^8*c^2 + 741*a^2*b^6*c^3
- 1606*a^3*b^4*c^4 + 776*a^4*b^2*c^5 + 768*a^5*c^6)*x^4 + (11*b^11 - 120*a*b^9*c + 187*a^2*b^7*c^2 + 2166*a^3*
b^5*c^3 - 9064*a^4*b^3*c^4 + 9088*a^5*b*c^5)*x^3 + 3*(11*a*b^10 - 163*a^2*b^8*c + 857*a^3*b^6*c^2 - 1676*a^4*b
^4*c^3 + 320*a^5*b^2*c^4 + 1152*a^6*c^5)*x^2 + 3*(a^3*b^7 - 14*a^4*b^5*c + 70*a^5*b^3*c^2 - 140*a^6*b*c^3 + (b
^7*c^3 - 14*a*b^5*c^4 + 70*a^2*b^3*c^5 - 140*a^3*b*c^6)*x^6 + 3*(b^8*c^2 - 14*a*b^6*c^3 + 70*a^2*b^4*c^4 - 140
*a^3*b^2*c^5)*x^5 + 3*(b^9*c - 13*a*b^7*c^2 + 56*a^2*b^5*c^3 - 70*a^3*b^3*c^4 - 140*a^4*b*c^5)*x^4 + (b^10 - 8
*a*b^8*c - 14*a^2*b^6*c^2 + 280*a^3*b^4*c^3 - 840*a^4*b^2*c^4)*x^3 + 3*(a*b^9 - 13*a^2*b^7*c + 56*a^3*b^5*c^2
- 70*a^4*b^3*c^3 - 140*a^5*b*c^4)*x^2 + 3*(a^2*b^8 - 14*a^3*b^6*c + 70*a^4*b^4*c^2 - 140*a^5*b^2*c^3)*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*(1
1*a^2*b^9 - 170*a^3*b^7*c + 964*a^4*b^5*c^2 - 2268*a^5*b^3*c^3 + 1712*a^6*b*c^4)*x + 3*(a^3*b^8 - 16*a^4*b^6*c
+ 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3*b^2*c^6
+ 256*a^4*c^7)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 + 256*a^4*b*c^6)*x^5 + 3*(b
^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^11 - 10*a*b^9*c + 320*a^3*b^5*
c^3 - 1280*a^4*b^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6*c^2 - 160*a^4*b^4*c^3 + 2
56*a^6*c^5)*x^2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 + 256*a^6*b*c^4)*x)*log(c*x^2 +
b*x + a))/(a^3*b^8*c^4 - 16*a^4*b^6*c^5 + 96*a^5*b^4*c^6 - 256*a^6*b^2*c^7 + 256*a^7*c^8 + (b^8*c^7 - 16*a*b^
6*c^8 + 96*a^2*b^4*c^9 - 256*a^3*b^2*c^10 + 256*a^4*c^11)*x^6 + 3*(b^9*c^6 - 16*a*b^7*c^7 + 96*a^2*b^5*c^8 - 2
56*a^3*b^3*c^9 + 256*a^4*b*c^10)*x^5 + 3*(b^10*c^5 - 15*a*b^8*c^6 + 80*a^2*b^6*c^7 - 160*a^3*b^4*c^8 + 256*a^5
*c^10)*x^4 + (b^11*c^4 - 10*a*b^9*c^5 + 320*a^3*b^5*c^7 - 1280*a^4*b^3*c^8 + 1536*a^5*b*c^9)*x^3 + 3*(a*b^10*c
^4 - 15*a^2*b^8*c^5 + 80*a^3*b^6*c^6 - 160*a^4*b^4*c^7 + 256*a^6*c^9)*x^2 + 3*(a^2*b^9*c^4 - 16*a^3*b^7*c^5 +
96*a^4*b^5*c^6 - 256*a^5*b^3*c^7 + 256*a^6*b*c^8)*x), 1/6*(11*a^3*b^8 - 168*a^4*b^6*c + 934*a^5*b^4*c^2 - 2104
*a^6*b^2*c^3 + 1408*a^7*c^4 + 6*(3*b^9*c^2 - 47*a*b^7*c^3 + 273*a^2*b^5*c^4 - 686*a^3*b^3*c^5 + 616*a^4*b*c^6)
*x^5 + 3*(9*b^10*c - 136*a*b^8*c^2 + 741*a^2*b^6*c^3 - 1606*a^3*b^4*c^4 + 776*a^4*b^2*c^5 + 768*a^5*c^6)*x^4 +
(11*b^11 - 120*a*b^9*c + 187*a^2*b^7*c^2 + 2166*a^3*b^5*c^3 - 9064*a^4*b^3*c^4 + 9088*a^5*b*c^5)*x^3 + 3*(11*
a*b^10 - 163*a^2*b^8*c + 857*a^3*b^6*c^2 - 1676*a^4*b^4*c^3 + 320*a^5*b^2*c^4 + 1152*a^6*c^5)*x^2 + 6*(a^3*b^7
- 14*a^4*b^5*c + 70*a^5*b^3*c^2 - 140*a^6*b*c^3 + (b^7*c^3 - 14*a*b^5*c^4 + 70*a^2*b^3*c^5 - 140*a^3*b*c^6)*x
^6 + 3*(b^8*c^2 - 14*a*b^6*c^3 + 70*a^2*b^4*c^4 - 140*a^3*b^2*c^5)*x^5 + 3*(b^9*c - 13*a*b^7*c^2 + 56*a^2*b^5*
c^3 - 70*a^3*b^3*c^4 - 140*a^4*b*c^5)*x^4 + (b^10 - 8*a*b^8*c - 14*a^2*b^6*c^2 + 280*a^3*b^4*c^3 - 840*a^4*b^2
*c^4)*x^3 + 3*(a*b^9 - 13*a^2*b^7*c + 56*a^3*b^5*c^2 - 70*a^4*b^3*c^3 - 140*a^5*b*c^4)*x^2 + 3*(a^2*b^8 - 14*a
^3*b^6*c + 70*a^4*b^4*c^2 - 140*a^5*b^2*c^3)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2
- 4*a*c)) + 3*(11*a^2*b^9 - 170*a^3*b^7*c + 964*a^4*b^5*c^2 - 2268*a^5*b^3*c^3 + 1712*a^6*b*c^4)*x + 3*(a^3*b
^8 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5
- 256*a^3*b^2*c^6 + 256*a^4*c^7)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 + 256*a^4*
b*c^6)*x^5 + 3*(b^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^11 - 10*a*b^9
*c + 320*a^3*b^5*c^3 - 1280*a^4*b^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6*c^2 - 16
0*a^4*b^4*c^3 + 256*a^6*c^5)*x^2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 + 256*a^6*b*c^
4)*x)*log(c*x^2 + b*x + a))/(a^3*b^8*c^4 - 16*a^4*b^6*c^5 + 96*a^5*b^4*c^6 - 256*a^6*b^2*c^7 + 256*a^7*c^8 + (
b^8*c^7 - 16*a*b^6*c^8 + 96*a^2*b^4*c^9 - 256*a^3*b^2*c^10 + 256*a^4*c^11)*x^6 + 3*(b^9*c^6 - 16*a*b^7*c^7 + 9
6*a^2*b^5*c^8 - 256*a^3*b^3*c^9 + 256*a^4*b*c^10)*x^5 + 3*(b^10*c^5 - 15*a*b^8*c^6 + 80*a^2*b^6*c^7 - 160*a^3*
b^4*c^8 + 256*a^5*c^10)*x^4 + (b^11*c^4 - 10*a*b^9*c^5 + 320*a^3*b^5*c^7 - 1280*a^4*b^3*c^8 + 1536*a^5*b*c^9)*
x^3 + 3*(a*b^10*c^4 - 15*a^2*b^8*c^5 + 80*a^3*b^6*c^6 - 160*a^4*b^4*c^7 + 256*a^6*c^9)*x^2 + 3*(a^2*b^9*c^4 -
16*a^3*b^7*c^5 + 96*a^4*b^5*c^6 - 256*a^5*b^3*c^7 + 256*a^6*b*c^8)*x)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2565 vs. \(2 (284) = 568\).

Time = 11.19 (sec) , antiderivative size = 2565, normalized size of antiderivative = 8.81 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]

[In]

integrate(x**7/(c*x**2+b*x+a)**4,x)

[Out]

(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*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*c**4))*log(x + (-256*a**4*c**7*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3
- 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*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*c**4)) +
128*a**4*c**3 + 256*a**3*b**2*c**6*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4
*c - b**6)/(2*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*c**4)) - 58*a**3*b**2*c**2 - 96*a**2*b**
4*c**5*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*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*c**4)) + 13*a**2*b**4*c + 16*a*b**6*c**4*(-b*sqrt(-(4*a*c - b**2)**7
)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(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*c**4)) - a*b**6 - b**8*c**3*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**
4*c - b**6)/(2*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 + 224
0*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)))/(140*a**3*b*c**3 - 70*a**2*b**3
*c**2 + 14*a*b**5*c - b**7)) + (b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c -
b**6)/(2*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*c**4))*log(x + (-256*a**4*c**7*(b*sqrt(-(4*a*
c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*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*c**4)) + 128*a**4*c**3 + 256*a**3*b**2*c**6*(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 7
0*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*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*c**4)) - 58*
a**3*b**2*c**2 - 96*a**2*b**4*c**5*(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*
c - b**6)/(2*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*c**4)) + 13*a**2*b**4*c + 16*a*b**6*c**4*
(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*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*c**4)) - a*b**6 - b**8*c**3*(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*
a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*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*c**4)))/(140*a
**3*b*c**3 - 70*a**2*b**3*c**2 + 14*a*b**5*c - b**7)) + (352*a**6*c**3 - 438*a**5*b**2*c**2 + 124*a**4*b**4*c
- 11*a**3*b**6 + x**5*(924*a**3*b*c**5 - 798*a**2*b**3*c**4 + 210*a*b**5*c**3 - 18*b**7*c**2) + x**4*(576*a**4
*c**5 + 726*a**3*b**2*c**4 - 1023*a**2*b**4*c**3 + 300*a*b**6*c**2 - 27*b**8*c) + x**3*(2272*a**4*b*c**4 - 169
8*a**3*b**3*c**3 + 117*a**2*b**5*c**2 + 76*a*b**7*c - 11*b**9) + x**2*(864*a**5*c**4 + 456*a**4*b**2*c**3 - 11
43*a**3*b**4*c**2 + 357*a**2*b**6*c - 33*a*b**8) + x*(1284*a**5*b*c**3 - 1380*a**4*b**3*c**2 + 378*a**3*b**5*c
- 33*a**2*b**7))/(384*a**6*c**7 - 288*a**5*b**2*c**6 + 72*a**4*b**4*c**5 - 6*a**3*b**6*c**4 + x**6*(384*a**3*
c**10 - 288*a**2*b**2*c**9 + 72*a*b**4*c**8 - 6*b**6*c**7) + x**5*(1152*a**3*b*c**9 - 864*a**2*b**3*c**8 + 216
*a*b**5*c**7 - 18*b**7*c**6) + x**4*(1152*a**4*c**9 + 288*a**3*b**2*c**8 - 648*a**2*b**4*c**7 + 198*a*b**6*c**
6 - 18*b**8*c**5) + x**3*(2304*a**4*b*c**8 - 1344*a**3*b**3*c**7 + 144*a**2*b**5*c**6 + 36*a*b**7*c**5 - 6*b**
9*c**4) + x**2*(1152*a**5*c**8 + 288*a**4*b**2*c**7 - 648*a**3*b**4*c**6 + 198*a**2*b**6*c**5 - 18*a*b**8*c**4
) + x*(1152*a**5*b*c**7 - 864*a**4*b**3*c**6 + 216*a**3*b**5*c**5 - 18*a**2*b**7*c**4))

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^7/(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 deta

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.44 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {{\left (b^{7} - 14 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {\log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {11 \, a^{3} b^{6} - 124 \, a^{4} b^{4} c + 438 \, a^{5} b^{2} c^{2} - 352 \, a^{6} c^{3} + 6 \, {\left (3 \, b^{7} c^{2} - 35 \, a b^{5} c^{3} + 133 \, a^{2} b^{3} c^{4} - 154 \, a^{3} b c^{5}\right )} x^{5} + 3 \, {\left (9 \, b^{8} c - 100 \, a b^{6} c^{2} + 341 \, a^{2} b^{4} c^{3} - 242 \, a^{3} b^{2} c^{4} - 192 \, a^{4} c^{5}\right )} x^{4} + {\left (11 \, b^{9} - 76 \, a b^{7} c - 117 \, a^{2} b^{5} c^{2} + 1698 \, a^{3} b^{3} c^{3} - 2272 \, a^{4} b c^{4}\right )} x^{3} + 3 \, {\left (11 \, a b^{8} - 119 \, a^{2} b^{6} c + 381 \, a^{3} b^{4} c^{2} - 152 \, a^{4} b^{2} c^{3} - 288 \, a^{5} c^{4}\right )} x^{2} + 3 \, {\left (11 \, a^{2} b^{7} - 126 \, a^{3} b^{5} c + 460 \, a^{4} b^{3} c^{2} - 428 \, a^{5} b c^{3}\right )} x}{6 \, {\left (c x^{2} + b x + a\right )}^{3} {\left (b^{2} - 4 \, a c\right )}^{3} c^{4}} \]

[In]

integrate(x^7/(c*x^2+b*x+a)^4,x, algorithm="giac")

[Out]

-(b^7 - 14*a*b^5*c + 70*a^2*b^3*c^2 - 140*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6*c^4 - 12*a*b
^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*sqrt(-b^2 + 4*a*c)) + 1/2*log(c*x^2 + b*x + a)/c^4 + 1/6*(11*a^3*b^6 - 1
24*a^4*b^4*c + 438*a^5*b^2*c^2 - 352*a^6*c^3 + 6*(3*b^7*c^2 - 35*a*b^5*c^3 + 133*a^2*b^3*c^4 - 154*a^3*b*c^5)*
x^5 + 3*(9*b^8*c - 100*a*b^6*c^2 + 341*a^2*b^4*c^3 - 242*a^3*b^2*c^4 - 192*a^4*c^5)*x^4 + (11*b^9 - 76*a*b^7*c
- 117*a^2*b^5*c^2 + 1698*a^3*b^3*c^3 - 2272*a^4*b*c^4)*x^3 + 3*(11*a*b^8 - 119*a^2*b^6*c + 381*a^3*b^4*c^2 -
152*a^4*b^2*c^3 - 288*a^5*c^4)*x^2 + 3*(11*a^2*b^7 - 126*a^3*b^5*c + 460*a^4*b^3*c^2 - 428*a^5*b*c^3)*x)/((c*x
^2 + b*x + a)^3*(b^2 - 4*a*c)^3*c^4)

Mupad [B] (verification not implemented)

Time = 11.19 (sec) , antiderivative size = 1055, normalized size of antiderivative = 3.63 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^4} \, dx=\frac {b\,\mathrm {atan}\left (\frac {\left (\frac {b\,x\,\left (-140\,a^3\,c^3+70\,a^2\,b^2\,c^2-14\,a\,b^4\,c+b^6\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^7}-\frac {b^2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )\,\left (-140\,a^3\,c^3+70\,a^2\,b^2\,c^2-14\,a\,b^4\,c+b^6\right )}{2\,c^7\,{\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 (128\,a^3\,c^7\,{\left (4\,a\,c-b^2\right )}^{7/2}-2\,b^6\,c^4\,{\left (4\,a\,c-b^2\right )}^{7/2}+24\,a\,b^4\,c^5\,{\left (4\,a\,c-b^2\right )}^{7/2}-96\,a^2\,b^2\,c^6\,{\left (4\,a\,c-b^2\right )}^{7/2}\right )}{-140\,a^3\,b\,c^3+70\,a^2\,b^3\,c^2-14\,a\,b^5\,c+b^7}\right )\,\left (-140\,a^3\,c^3+70\,a^2\,b^2\,c^2-14\,a\,b^4\,c+b^6\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^{7/2}}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-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}\right )}{2\,\left (16384\,a^7\,c^{11}-28672\,a^6\,b^2\,c^{10}+21504\,a^5\,b^4\,c^9-8960\,a^4\,b^6\,c^8+2240\,a^3\,b^8\,c^7-336\,a^2\,b^{10}\,c^6+28\,a\,b^{12}\,c^5-b^{14}\,c^4\right )}-\frac {\frac {x^4\,\left (192\,a^4\,c^4+242\,a^3\,b^2\,c^3-341\,a^2\,b^4\,c^2+100\,a\,b^6\,c-9\,b^8\right )}{2\,c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {a^3\,\left (-352\,a^3\,c^3+438\,a^2\,b^2\,c^2-124\,a\,b^4\,c+11\,b^6\right )}{6\,c^4\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {b\,x^5\,\left (-154\,a^3\,c^3+133\,a^2\,b^2\,c^2-35\,a\,b^4\,c+3\,b^6\right )}{c^2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a\,x^2\,\left (288\,a^4\,c^4+152\,a^3\,b^2\,c^3-381\,a^2\,b^4\,c^2+119\,a\,b^6\,c-11\,b^8\right )}{2\,c^4\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {b\,x^3\,\left (2272\,a^4\,c^4-1698\,a^3\,b^2\,c^3+117\,a^2\,b^4\,c^2+76\,a\,b^6\,c-11\,b^8\right )}{6\,c^4\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {a^2\,b\,x\,\left (-428\,a^3\,c^3+460\,a^2\,b^2\,c^2-126\,a\,b^4\,c+11\,b^6\right )}{2\,c^4\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x} \]

[In]

int(x^7/(a + b*x + c*x^2)^4,x)

[Out]

(b*atan((((b*x*(b^6 - 140*a^3*c^3 + 70*a^2*b^2*c^2 - 14*a*b^4*c))/(c^3*(4*a*c - b^2)^7) - (b^2*(64*a^3*c^6 - b
^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)*(b^6 - 140*a^3*c^3 + 70*a^2*b^2*c^2 - 14*a*b^4*c))/(2*c^7*(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)))*(128*a^3*c^7*(4*a*c - b^2)^(7/2) - 2*b^6*c^4*(4*a*c - b
^2)^(7/2) + 24*a*b^4*c^5*(4*a*c - b^2)^(7/2) - 96*a^2*b^2*c^6*(4*a*c - b^2)^(7/2)))/(b^7 - 140*a^3*b*c^3 + 70*
a^2*b^3*c^2 - 14*a*b^5*c))*(b^6 - 140*a^3*c^3 + 70*a^2*b^2*c^2 - 14*a*b^4*c))/(c^4*(4*a*c - b^2)^(7/2)) - (log
(a + b*x + c*x^2)*(b^14 - 16384*a^7*c^7 + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b
^4*c^5 + 28672*a^6*b^2*c^6 - 28*a*b^12*c))/(2*(16384*a^7*c^11 - b^14*c^4 + 28*a*b^12*c^5 - 336*a^2*b^10*c^6 +
2240*a^3*b^8*c^7 - 8960*a^4*b^6*c^8 + 21504*a^5*b^4*c^9 - 28672*a^6*b^2*c^10)) - ((x^4*(192*a^4*c^4 - 9*b^8 -
341*a^2*b^4*c^2 + 242*a^3*b^2*c^3 + 100*a*b^6*c))/(2*c^3*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (
a^3*(11*b^6 - 352*a^3*c^3 + 438*a^2*b^2*c^2 - 124*a*b^4*c))/(6*c^4*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b
^4*c)) - (b*x^5*(3*b^6 - 154*a^3*c^3 + 133*a^2*b^2*c^2 - 35*a*b^4*c))/(c^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2
- 12*a*b^4*c)) + (a*x^2*(288*a^4*c^4 - 11*b^8 - 381*a^2*b^4*c^2 + 152*a^3*b^2*c^3 + 119*a*b^6*c))/(2*c^4*(b^6
- 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (b*x^3*(2272*a^4*c^4 - 11*b^8 + 117*a^2*b^4*c^2 - 1698*a^3*b^2*
c^3 + 76*a*b^6*c))/(6*c^4*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - (a^2*b*x*(11*b^6 - 428*a^3*c^3 +
460*a^2*b^2*c^2 - 126*a*b^4*c))/(2*c^4*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)))/(x^2*(3*a*b^2 + 3*a
^2*c) + x^4*(3*a*c^2 + 3*b^2*c) + a^3 + x^3*(b^3 + 6*a*b*c) + c^3*x^6 + 3*b*c^2*x^5 + 3*a^2*b*x)