\(\int \frac {x^7}{(a+b x+c x^2)^3} \, dx\) [244]

Optimal result

Integrand size = 16, antiderivative size = 315 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {3 b x}{c^4}+\frac {x^2}{2 c^3}+\frac {a \left (b^2-2 a c\right ) \left (b^4-4 a b^2 c+a^2 c^2\right )+b \left (b^6-7 a b^4 c+14 a^2 b^2 c^2-7 a^3 c^3\right ) x}{2 c^6 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {17 a b^6-\frac {b^8}{c}-88 a^2 b^4 c+153 a^3 b^2 c^2-48 a^4 c^3+2 b \left (4 b^6-35 a b^4 c+91 a^2 b^2 c^2-63 a^3 c^3\right ) x}{2 c^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{5/2}}+\frac {3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5} \] Output:

-3*b*x/c^4+1/2*x^2/c^3+1/2*(a*(-2*a*c+b^2)*(a^2*c^2-4*a*b^2*c+b^4)+b*(-7*a 
^3*c^3+14*a^2*b^2*c^2-7*a*b^4*c+b^6)*x)/c^6/(-4*a*c+b^2)/(c*x^2+b*x+a)^2+1 
/2*(17*a*b^6-b^8/c-88*a^2*b^4*c+153*a^3*b^2*c^2-48*a^4*c^3+2*b*(-63*a^3*c^ 
3+91*a^2*b^2*c^2-35*a*b^4*c+4*b^6)*x)/c^5/(-4*a*c+b^2)^2/(c*x^2+b*x+a)+3*b 
*(-70*a^3*c^3+70*a^2*b^2*c^2-21*a*b^4*c+2*b^6)*arctanh((2*c*x+b)/(-4*a*c+b 
^2)^(1/2))/c^5/(-4*a*c+b^2)^(5/2)+3/2*(-a*c+2*b^2)*ln(c*x^2+b*x+a)/c^5
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.95 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx=\frac {-6 b c^2 x+c^3 x^2-\frac {b^8-17 a b^6 c+88 a^2 b^4 c^2-153 a^3 b^2 c^3+48 a^4 c^4-8 b^7 c x+70 a b^5 c^2 x-182 a^2 b^3 c^3 x+126 a^3 b c^4 x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {-2 a^4 c^3+b^7 x+a b^5 (b-7 c x)+a^3 b c^2 (9 b-7 c x)+2 a^2 b^3 c (-3 b+7 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {6 b c \left (-2 b^6+21 a b^4 c-70 a^2 b^2 c^2+70 a^3 c^3\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}-3 c \left (-2 b^2+a c\right ) \log (a+x (b+c x))}{2 c^6} \] Input:

Integrate[x^7/(a + b*x + c*x^2)^3,x]
 

Output:

(-6*b*c^2*x + c^3*x^2 - (b^8 - 17*a*b^6*c + 88*a^2*b^4*c^2 - 153*a^3*b^2*c 
^3 + 48*a^4*c^4 - 8*b^7*c*x + 70*a*b^5*c^2*x - 182*a^2*b^3*c^3*x + 126*a^3 
*b*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-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))^2) + (6*b*c*(-2*b^6 + 21*a*b^4*c - 70*a^2*b^2 
*c^2 + 70*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^ 
(5/2) - 3*c*(-2*b^2 + a*c)*Log[a + x*(b + c*x)])/(2*c^6)
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1164, 27, 1233, 27, 1200, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle \frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {3 x^5 (4 a+b x)}{\left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \int \frac {x^5 (4 a+b x)}{\left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \left (\frac {\int \frac {2 x^3 \left (2 a \left (b^2-8 a c\right )+b \left (2 b^2-11 a c\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}-\frac {x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \left (\frac {2 \int \frac {x^3 \left (2 a \left (b^2-8 a c\right )+b \left (2 b^2-11 a c\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}-\frac {x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \left (\frac {2 \int \left (\frac {b \left (2 b^2-11 a c\right ) x^2}{c}-\frac {\left (2 b^4-13 a c b^2+16 a^2 c^2\right ) x}{c^2}+\frac {b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^3}-\frac {\left (2 b^2-a c\right ) x \left (b^2-4 a c\right )^2+a b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^3 \left (c x^2+b x+a\right )}\right )dx}{c \left (b^2-4 a c\right )}-\frac {x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {3 \left (\frac {2 \left (-\frac {x^2 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 c^2}-\frac {b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \sqrt {b^2-4 a c}}-\frac {\left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {b x \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^3}+\frac {b x^3 \left (2 b^2-11 a c\right )}{3 c}\right )}{c \left (b^2-4 a c\right )}-\frac {x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}\)

Input:

Int[x^7/(a + b*x + c*x^2)^3,x]
 

Output:

(x^6*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (3*(-((x^4*(a*(b 
^2 - 8*a*c) + b*(b^2 - 6*a*c)*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2))) + ( 
2*((b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c)*x)/c^3 - ((2*b^4 - 13*a*b^2*c + 16*a^2 
*c^2)*x^2)/(2*c^2) + (b*(2*b^2 - 11*a*c)*x^3)/(3*c) - (b*(2*b^6 - 21*a*b^4 
*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/ 
(c^4*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)^2*(2*b^2 - a*c)*Log[a + b*x + c*x 
^2])/(2*c^4)))/(c*(b^2 - 4*a*c))))/(2*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> 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] + Simp[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] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[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] && 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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.46

method	result	size
default	\(-\frac {-\frac {1}{2} c \,x^{2}+3 b x}{c^{4}}+\frac {\frac {-\frac {b \left (63 a^{3} c^{3}-91 a^{2} b^{2} c^{2}+35 a \,b^{4} c -4 b^{6}\right ) x^{3}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}-\frac {\left (48 a^{4} c^{4}-27 a^{3} b^{2} c^{3}-94 a^{2} b^{4} c^{2}+53 a \,b^{6} c -7 b^{8}\right ) x^{2}}{2 c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {a b \left (73 a^{3} c^{3}-136 a^{2} b^{2} c^{2}+58 a \,b^{4} c -7 b^{6}\right ) x}{c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}-\frac {a^{2} \left (40 a^{3} c^{3}-115 a^{2} b^{2} c^{2}+55 a \,b^{4} c -7 b^{6}\right )}{2 c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {3 \left (-16 a^{3} c^{3}+40 a^{2} b^{2} c^{2}-17 a \,b^{4} c +2 b^{6}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {6 \left (27 a^{3} b \,c^{2}-15 a^{2} b^{3} c +2 a \,b^{5}-\frac {\left (-16 a^{3} c^{3}+40 a^{2} b^{2} c^{2}-17 a \,b^{4} c +2 b^{6}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}}{c^{4}}\)	\(459\)
risch	\(\text {Expression too large to display}\)	\(2479\)

Input:

int(x^7/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/c^4*(-1/2*c*x^2+3*b*x)+1/c^4*((-b*(63*a^3*c^3-91*a^2*b^2*c^2+35*a*b^4*c 
-4*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*(48*a^4*c^4-27*a^3*b^2*c^3-94*a 
^2*b^4*c^2+53*a*b^6*c-7*b^8)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-a/c*b*(73*a^ 
3*c^3-136*a^2*b^2*c^2+58*a*b^4*c-7*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x-1/2*a 
^2*(40*a^3*c^3-115*a^2*b^2*c^2+55*a*b^4*c-7*b^6)/c/(16*a^2*c^2-8*a*b^2*c+b 
^4))/(c*x^2+b*x+a)^2+3/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(-16*a^3*c^3+40*a^2 
*b^2*c^2-17*a*b^4*c+2*b^6)/c*ln(c*x^2+b*x+a)+2*(27*a^3*b*c^2-15*a^2*b^3*c+ 
2*a*b^5-1/2*(-16*a^3*c^3+40*a^2*b^2*c^2-17*a*b^4*c+2*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. 1093 vs. \(2 (303) = 606\).

Time = 0.12 (sec) , antiderivative size = 2207, normalized size of antiderivative = 7.01 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^7/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

[1/2*(7*a^2*b^8 - 83*a^3*b^6*c + 335*a^4*b^4*c^2 - 500*a^5*b^2*c^3 + 160*a 
^6*c^4 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^6 - 4*(b 
^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^5 - (11*b^8*c^2 - 
 134*a*b^6*c^3 + 552*a^2*b^4*c^4 - 800*a^3*b^2*c^5 + 128*a^4*c^6)*x^4 + 2* 
(b^9*c - 20*a*b^7*c^2 + 147*a^2*b^5*c^3 - 475*a^3*b^3*c^4 + 572*a^4*b*c^5) 
*x^3 + (7*b^10 - 93*a*b^8*c + 451*a^2*b^6*c^2 - 937*a^3*b^4*c^3 + 660*a^4* 
b^2*c^4 + 128*a^5*c^5)*x^2 - 3*(2*a^2*b^7 - 21*a^3*b^5*c + 70*a^4*b^3*c^2 
- 70*a^5*b*c^3 + (2*b^7*c^2 - 21*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70*a^3*b*c^5 
)*x^4 + 2*(2*b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3 - 70*a^3*b^2*c^4)*x^3 + 
 (2*b^9 - 17*a*b^7*c + 28*a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c^4)*x^ 
2 + 2*(2*a*b^8 - 21*a^2*b^6*c + 70*a^3*b^4*c^2 - 70*a^4*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)) + 2*(7*a*b^9 - 89*a^2*b^7*c + 404*a^3*b^5*c^2 
 - 761*a^4*b^3*c^3 + 484*a^5*b*c^4)*x + 3*(2*a^2*b^8 - 25*a^3*b^6*c + 108* 
a^4*b^4*c^2 - 176*a^5*b^2*c^3 + 64*a^6*c^4 + (2*b^8*c^2 - 25*a*b^6*c^3 + 1 
08*a^2*b^4*c^4 - 176*a^3*b^2*c^5 + 64*a^4*c^6)*x^4 + 2*(2*b^9*c - 25*a*b^7 
*c^2 + 108*a^2*b^5*c^3 - 176*a^3*b^3*c^4 + 64*a^4*b*c^5)*x^3 + (2*b^10 - 2 
1*a*b^8*c + 58*a^2*b^6*c^2 + 40*a^3*b^4*c^3 - 288*a^4*b^2*c^4 + 128*a^5*c^ 
5)*x^2 + 2*(2*a*b^9 - 25*a^2*b^7*c + 108*a^3*b^5*c^2 - 176*a^4*b^3*c^3 + 6 
4*a^5*b*c^4)*x)*log(c*x^2 + b*x + a))/(a^2*b^6*c^5 - 12*a^3*b^4*c^6 + 4...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (316) = 632\).

Time = 2.97 (sec) , antiderivative size = 1875, normalized size of antiderivative = 5.95 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x**7/(c*x**2+b*x+a)**3,x)
 

Output:

-3*b*x/c**4 + (-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2* 
c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 
 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a* 
c - 2*b**2)/(2*c**5))*log(x + (96*a**4*c**3 - 159*a**3*b**2*c**2 + 64*a**3 
*c**7*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 2 
1*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a 
**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b* 
*2)/(2*c**5)) + 57*a**2*b**4*c - 48*a**2*b**2*c**6*(-3*b*sqrt(-(4*a*c - b* 
*2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5* 
(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6 
*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c**5)) - 6*a*b**6 + 12 
*a*b**4*c**5*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c 
**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c 
 - 2*b**2)/(2*c**5)) - b**6*c**4*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c 
**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 
1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8* 
c - b**10)) - 3*(a*c - 2*b**2)/(2*c**5)))/(210*a**3*b*c**3 - 210*a**2*b**3 
*c**2 + 63*a*b**5*c - 6*b**7)) + (3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c* 
*3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 ...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^7/(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.05 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {3 \, {\left (2 \, b^{7} - 21 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, {\left (2 \, b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac {c^{3} x^{2} - 6 \, b c^{2} x}{2 \, c^{6}} + \frac {7 \, a^{2} b^{6} - 55 \, a^{3} b^{4} c + 115 \, a^{4} b^{2} c^{2} - 40 \, a^{5} c^{3} + 2 \, {\left (4 \, b^{7} c - 35 \, a b^{5} c^{2} + 91 \, a^{2} b^{3} c^{3} - 63 \, a^{3} b c^{4}\right )} x^{3} + {\left (7 \, b^{8} - 53 \, a b^{6} c + 94 \, a^{2} b^{4} c^{2} + 27 \, a^{3} b^{2} c^{3} - 48 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (7 \, a b^{7} - 58 \, a^{2} b^{5} c + 136 \, a^{3} b^{3} c^{2} - 73 \, a^{4} b c^{3}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{5}} \] Input:

integrate(x^7/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

-3*(2*b^7 - 21*a*b^5*c + 70*a^2*b^3*c^2 - 70*a^3*b*c^3)*arctan((2*c*x + b) 
/sqrt(-b^2 + 4*a*c))/((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt(-b^2 + 4*a 
*c)) + 3/2*(2*b^2 - a*c)*log(c*x^2 + b*x + a)/c^5 + 1/2*(c^3*x^2 - 6*b*c^2 
*x)/c^6 + 1/2*(7*a^2*b^6 - 55*a^3*b^4*c + 115*a^4*b^2*c^2 - 40*a^5*c^3 + 2 
*(4*b^7*c - 35*a*b^5*c^2 + 91*a^2*b^3*c^3 - 63*a^3*b*c^4)*x^3 + (7*b^8 - 5 
3*a*b^6*c + 94*a^2*b^4*c^2 + 27*a^3*b^2*c^3 - 48*a^4*c^4)*x^2 + 2*(7*a*b^7 
 - 58*a^2*b^5*c + 136*a^3*b^3*c^2 - 73*a^4*b*c^3)*x)/((c*x^2 + b*x + a)^2* 
(b^2 - 4*a*c)^2*c^5)
 

Mupad [B] (verification not implemented)

Time = 9.70 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.42 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {a\,\left (-40\,a^4\,c^3+115\,a^3\,b^2\,c^2-55\,a^2\,b^4\,c+7\,a\,b^6\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,x^3\,\left (-63\,a^3\,c^3+91\,a^2\,b^2\,c^2-35\,a\,b^4\,c+4\,b^6\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {x^2\,\left (-48\,a^4\,c^4+27\,a^3\,b^2\,c^3+94\,a^2\,b^4\,c^2-53\,a\,b^6\,c+7\,b^8\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {b\,x\,\left (-73\,a^4\,c^3+136\,a^3\,b^2\,c^2-58\,a^2\,b^4\,c+7\,a\,b^6\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{a^2\,c^4+c^6\,x^4+x^2\,\left (b^2\,c^4+2\,a\,c^5\right )+2\,b\,c^5\,x^3+2\,a\,b\,c^4\,x}+\frac {x^2}{2\,c^3}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (3072\,a^6\,c^6-9984\,a^5\,b^2\,c^5+9600\,a^4\,b^4\,c^4-4320\,a^3\,b^6\,c^3+1020\,a^2\,b^8\,c^2-123\,a\,b^{10}\,c+6\,b^{12}\right )}{2\,\left (1024\,a^5\,c^{10}-1280\,a^4\,b^2\,c^9+640\,a^3\,b^4\,c^8-160\,a^2\,b^6\,c^7+20\,a\,b^8\,c^6-b^{10}\,c^5\right )}-\frac {3\,b\,x}{c^4}-\frac {3\,b\,\mathrm {atan}\left (\frac {\left (\frac {3\,b\,x\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^5}+\frac {3\,b^2\,\left (16\,a^2\,c^6-8\,a\,b^2\,c^5+b^4\,c^4\right )\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{2\,c^9\,{\left (4\,a\,c-b^2\right )}^5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (32\,a^2\,c^7\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^6\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{-210\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2-63\,a\,b^5\,c+6\,b^7}\right )\,\left (-70\,a^3\,c^3+70\,a^2\,b^2\,c^2-21\,a\,b^4\,c+2\,b^6\right )}{c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}} \] Input:

int(x^7/(a + b*x + c*x^2)^3,x)
 

Output:

((a*(7*a*b^6 - 40*a^4*c^3 - 55*a^2*b^4*c + 115*a^3*b^2*c^2))/(2*c*(b^4 + 1 
6*a^2*c^2 - 8*a*b^2*c)) + (b*x^3*(4*b^6 - 63*a^3*c^3 + 91*a^2*b^2*c^2 - 35 
*a*b^4*c))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (x^2*(7*b^8 - 48*a^4*c^4 + 94* 
a^2*b^4*c^2 + 27*a^3*b^2*c^3 - 53*a*b^6*c))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b 
^2*c)) + (b*x*(7*a*b^6 - 73*a^4*c^3 - 58*a^2*b^4*c + 136*a^3*b^2*c^2))/(c* 
(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(a^2*c^4 + c^6*x^4 + x^2*(2*a*c^5 + b^2*c 
^4) + 2*b*c^5*x^3 + 2*a*b*c^4*x) + x^2/(2*c^3) - (log(a + b*x + c*x^2)*(6* 
b^12 + 3072*a^6*c^6 + 1020*a^2*b^8*c^2 - 4320*a^3*b^6*c^3 + 9600*a^4*b^4*c 
^4 - 9984*a^5*b^2*c^5 - 123*a*b^10*c))/(2*(1024*a^5*c^10 - b^10*c^5 + 20*a 
*b^8*c^6 - 160*a^2*b^6*c^7 + 640*a^3*b^4*c^8 - 1280*a^4*b^2*c^9)) - (3*b*x 
)/c^4 - (3*b*atan((((3*b*x*(2*b^6 - 70*a^3*c^3 + 70*a^2*b^2*c^2 - 21*a*b^4 
*c))/(c^4*(4*a*c - b^2)^5) + (3*b^2*(16*a^2*c^6 + b^4*c^4 - 8*a*b^2*c^5)*( 
2*b^6 - 70*a^3*c^3 + 70*a^2*b^2*c^2 - 21*a*b^4*c))/(2*c^9*(4*a*c - b^2)^5* 
(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(32*a^2*c^7*(4*a*c - b^2)^(5/2) + 2*b^4*c 
^5*(4*a*c - b^2)^(5/2) - 16*a*b^2*c^6*(4*a*c - b^2)^(5/2)))/(6*b^7 - 210*a 
^3*b*c^3 + 210*a^2*b^3*c^2 - 63*a*b^5*c))*(2*b^6 - 70*a^3*c^3 + 70*a^2*b^2 
*c^2 - 21*a*b^4*c))/(c^5*(4*a*c - b^2)^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 2048, normalized size of antiderivative = 6.50 \[ \int \frac {x^7}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^7/(c*x^2+b*x+a)^3,x)
 

Output:

(420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b*c**3 - 
 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**3*c** 
2 + 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**2* 
c**3*x + 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4* 
b*c**4*x**2 + 126*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
a**3*b**5*c - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* 
a**3*b**4*c**2*x - 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* 
*2))*a**3*b**3*c**3*x**2 + 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* 
a*c - b**2))*a**3*b**2*c**4*x**3 + 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x) 
/sqrt(4*a*c - b**2))*a**3*b*c**5*x**4 - 12*sqrt(4*a*c - b**2)*atan((b + 2* 
c*x)/sqrt(4*a*c - b**2))*a**2*b**7 + 252*sqrt(4*a*c - b**2)*atan((b + 2*c* 
x)/sqrt(4*a*c - b**2))*a**2*b**6*c*x - 168*sqrt(4*a*c - b**2)*atan((b + 2* 
c*x)/sqrt(4*a*c - b**2))*a**2*b**5*c**2*x**2 - 840*sqrt(4*a*c - b**2)*atan 
((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**4*c**3*x**3 - 420*sqrt(4*a*c - b* 
*2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**4*x**4 - 24*sqrt(4*a 
*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**8*x + 102*sqrt(4*a*c 
- b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**7*c*x**2 + 252*sqrt(4*a* 
c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**6*c**2*x**3 + 126*sqrt 
(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**5*c**3*x**4 - 12* 
sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**9*x**2 - 24*...