Integrand size = 16, antiderivative size = 349 \[ \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx=\frac {4 \left (b^6-11 a b^4 c+38 a^2 b^2 c^2-35 a^3 c^3\right ) x}{c^4 \left (b^2-4 a c\right )^3}-\frac {2 b \left (b^4-10 a b^2 c+29 a^2 c^2\right ) x^2}{c^3 \left (b^2-4 a c\right )^3}+\frac {x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {x^5 \left (a \left (b^2-14 a c\right )+b \left (b^2-9 a c\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (4 a \left (b^4-9 a b^2 c+35 a^2 c^2\right )+b \left (4 b^4-39 a b^2 c+122 a^2 c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {4 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{7/2}}-\frac {2 b \log \left (a+b x+c x^2\right )}{c^5} \]
4*(-35*a^3*c^3+38*a^2*b^2*c^2-11*a*b^4*c+b^6)*x/c^4/(-4*a*c+b^2)^3-2*b*(29 *a^2*c^2-10*a*b^2*c+b^4)*x^2/c^3/(-4*a*c+b^2)^3+1/3*x^7*(b*x+2*a)/(-4*a*c+ b^2)/(c*x^2+b*x+a)^3+1/3*x^5*(a*(-14*a*c+b^2)+b*(-9*a*c+b^2)*x)/c/(-4*a*c+ b^2)^2/(c*x^2+b*x+a)^2+1/3*x^3*(4*a*(35*a^2*c^2-9*a*b^2*c+b^4)+b*(122*a^2* c^2-39*a*b^2*c+4*b^4)*x)/c^2/(-4*a*c+b^2)^3/(c*x^2+b*x+a)-4*(70*a^4*c^4-14 0*a^3*b^2*c^3+70*a^2*b^4*c^2-14*a*b^6*c+b^8)*arctanh((2*c*x+b)/(-4*a*c+b^2 )^(1/2))/c^5/(-4*a*c+b^2)^(7/2)-2*b*ln(c*x^2+b*x+a)/c^5
Time = 0.37 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.25 \[ \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx=\frac {3 c^3 x+\frac {-b^8 x+a^2 b^4 c (7 b-20 c x)-a b^6 (b-8 c x)-2 a^3 b^2 c^2 (7 b-8 c x)+a^4 c^3 (7 b-2 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^3}+\frac {b^9-17 a b^7 c+95 a^2 b^5 c^2-202 a^3 b^3 c^3+125 a^4 b c^4-7 b^8 c x+68 a b^6 c^2 x-212 a^2 b^4 c^3 x+220 a^3 b^2 c^4 x-38 a^4 c^5 x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {6 c \left (-b^9+15 a b^7 c-83 a^2 b^5 c^2+198 a^3 b^3 c^3-163 a^4 b c^4+3 b^8 c x-36 a b^6 c^2 x+146 a^2 b^4 c^3 x-212 a^3 b^2 c^4 x+58 a^4 c^5 x\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}-\frac {12 c^2 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}-6 b c^2 \log (a+x (b+c x))}{3 c^7} \]
(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)
Time = 0.79 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1164, 27, 1233, 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.
| \(\displaystyle \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle \frac {x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {\int \frac {2 x^6 (7 a+b x)}{\left (c x^2+b x+a\right )^3}dx}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {2 \int \frac {x^6 (7 a+b x)}{\left (c x^2+b x+a\right )^3}dx}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {2 \left (\frac {\int \frac {x^4 \left (5 a \left (b^2-14 a c\right )+2 b \left (2 b^2-13 a c\right ) x\right )}{\left (c x^2+b x+a\right )^2}dx}{2 c \left (b^2-4 a c\right )}-\frac {x^5 \left (b x \left (b^2-9 a c\right )+a \left (b^2-14 a c\right )\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {2 \left (\frac {\frac {\int \frac {12 x^2 \left (a \left (b^4-9 a c b^2+35 a^2 c^2\right )+b \left (b^4-10 a c b^2+29 a^2 c^2\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}-\frac {x^3 \left (b x \left (122 a^2 c^2-39 a b^2 c+4 b^4\right )+4 a \left (35 a^2 c^2-9 a b^2 c+b^4\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^5 \left (b x \left (b^2-9 a c\right )+a \left (b^2-14 a c\right )\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {2 \left (\frac {\frac {12 \int \frac {x^2 \left (a \left (b^4-9 a c b^2+35 a^2 c^2\right )+b \left (b^4-10 a c b^2+29 a^2 c^2\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}-\frac {x^3 \left (b x \left (122 a^2 c^2-39 a b^2 c+4 b^4\right )+4 a \left (35 a^2 c^2-9 a b^2 c+b^4\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^5 \left (b x \left (b^2-9 a c\right )+a \left (b^2-14 a c\right )\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {2 \left (\frac {\frac {12 \int \left (-\frac {b^6}{c^2}+\frac {11 a b^4}{c}-38 a^2 b^2+\frac {\left (b^4-10 a c b^2+29 a^2 c^2\right ) x b}{c}+35 a^3 c+\frac {b x \left (b^2-4 a c\right )^3+a \left (b^6-11 a c b^4+38 a^2 c^2 b^2-35 a^3 c^3\right )}{c^2 \left (c x^2+b x+a\right )}\right )dx}{c \left (b^2-4 a c\right )}-\frac {x^3 \left (b x \left (122 a^2 c^2-39 a b^2 c+4 b^4\right )+4 a \left (35 a^2 c^2-9 a b^2 c+b^4\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^5 \left (b x \left (b^2-9 a c\right )+a \left (b^2-14 a c\right )\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^7 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {2 \left (\frac {\frac {12 \left (\frac {b x^2 \left (29 a^2 c^2-10 a b^2 c+b^4\right )}{2 c}-x \left (-35 a^3 c+38 a^2 b^2-\frac {11 a b^4}{c}+\frac {b^6}{c^2}\right )+\frac {\left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {b \left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )}{2 c^3}\right )}{c \left (b^2-4 a c\right )}-\frac {x^3 \left (b x \left (122 a^2 c^2-39 a b^2 c+4 b^4\right )+4 a \left (35 a^2 c^2-9 a b^2 c+b^4\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 c \left (b^2-4 a c\right )}-\frac {x^5 \left (b x \left (b^2-9 a c\right )+a \left (b^2-14 a c\right )\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 \left (b^2-4 a c\right )}\) |
(x^7*(2*a + b*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (2*(-1/2*(x^5*(a *(b^2 - 14*a*c) + b*(b^2 - 9*a*c)*x))/(c*(b^2 - 4*a*c)*(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))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2))) + (12*(-((38*a^2*b^2 + b^6/c^2 - (11*a*b^4)/c - 35*a^3*c)*x) + (b*(b^4 - 10*a*b^2*c + 29*a^2*c ^2)*x^2)/(2*c) + ((b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 7 0*a^4*c^4)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) + (b*(b^2 - 4*a*c)^3*Log[a + b*x + c*x^2])/(2*c^3)))/(c*(b^2 - 4*a*c)))/( 2*c*(b^2 - 4*a*c))))/(3*(b^2 - 4*a*c))
3.23.10.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(729\) vs. \(2(339)=678\).
Time = 16.96 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.09
| method | result | size |
| default | \(\frac {x}{c^{4}}-\frac {\frac {-\frac {2 c \left (58 a^{4} c^{4}-212 a^{3} b^{2} c^{3}+146 a^{2} b^{4} c^{2}-36 a \,b^{6} c +3 b^{8}\right ) x^{5}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {2 b \left (47 a^{4} c^{4}+226 a^{3} b^{2} c^{3}-209 a^{2} b^{4} c^{2}+57 a \,b^{6} c -5 b^{8}\right ) x^{4}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {\left (544 a^{5} c^{5}-3234 a^{4} b^{2} c^{4}+1788 a^{3} b^{4} c^{3}-68 a^{2} b^{6} c^{2}-96 a \,b^{8} c +13 b^{10}\right ) x^{3}}{3 c \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {a b \left (304 a^{4} c^{4}+387 a^{3} b^{2} c^{3}-486 a^{2} b^{4} c^{2}+143 a \,b^{6} c -13 b^{8}\right ) x^{2}}{c \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {a^{2} \left (76 a^{4} c^{4}-694 a^{3} b^{2} c^{3}+567 a^{2} b^{4} c^{2}-150 a \,b^{6} c +13 b^{8}\right ) x}{c \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {\left (590 c^{3} a^{3}-535 a^{2} b^{2} c^{2}+147 a \,b^{4} c -13 b^{6}\right ) a^{3} b}{3 c \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {\frac {2 \left (64 a^{3} c^{3} b -48 a^{2} c^{2} b^{3}+12 a \,b^{5} c -b^{7}\right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {8 \left (35 a^{4} c^{3}-38 a^{3} b^{2} c^{2}+11 a^{2} c \,b^{4}-a \,b^{6}-\frac {\left (64 a^{3} c^{3} b -48 a^{2} c^{2} b^{3}+12 a \,b^{5} c -b^{7}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}}{c^{4}}\) | \(730\) |
| risch | \(\text {Expression too large to display}\) | \(2979\) |
1/c^4*x-1/c^4*((-2*c*(58*a^4*c^4-212*a^3*b^2*c^3+146*a^2*b^4*c^2-36*a*b^6* c+3*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5+2*b*(47*a^4*c^4+22 6*a^3*b^2*c^3-209*a^2*b^4*c^2+57*a*b^6*c-5*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2 +12*a*b^4*c-b^6)*x^4-1/3*(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)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b ^6)*x^3+a*b*(304*a^4*c^4+387*a^3*b^2*c^3-486*a^2*b^4*c^2+143*a*b^6*c-13*b^ 8)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-a^2*(76*a^4*c^4-694*a^ 3*b^2*c^3+567*a^2*b^4*c^2-150*a*b^6*c+13*b^8)/c/(64*a^3*c^3-48*a^2*b^2*c^2 +12*a*b^4*c-b^6)*x+1/3*(590*a^3*c^3-535*a^2*b^2*c^2+147*a*b^4*c-13*b^6)*a^ 3*b/c/(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+4/(64*a^ 3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*(1/2*(64*a^3*b*c^3-48*a^2*b^3*c^2+12* a*b^5*c-b^7)/c*ln(c*x^2+b*x+a)+2*(35*a^4*c^3-38*a^3*b^2*c^2+11*a^2*c*b^4-a *b^6-1/2*(64*a^3*b*c^3-48*a^2*b^3*c^2+12*a*b^5*c-b^7)*b/c)/(4*a*c-b^2)^(1/ 2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1647 vs. \(2 (339) = 678\).
Time = 0.47 (sec) , antiderivative size = 3314, normalized size of antiderivative = 9.50 \[ \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]
[-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 + 13 40*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 + 201 2*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 + 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)*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^...
Leaf count of result is larger than twice the leaf count of optimal. 2769 vs. \(2 (350) = 700\).
Time = 19.68 (sec) , antiderivative size = 2769, normalized size of antiderivative = 7.93 \[ \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]
(-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**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 - 286 72*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a...
Exception generated. \[ \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.29 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.34 \[ \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx=\frac {4 \, {\left (b^{8} - 14 \, a b^{6} c + 70 \, a^{2} b^{4} c^{2} - 140 \, a^{3} b^{2} c^{3} + 70 \, a^{4} c^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} c^{5} - 12 \, a b^{4} c^{6} + 48 \, a^{2} b^{2} c^{7} - 64 \, a^{3} c^{8}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x}{c^{4}} - \frac {2 \, b \log \left (c x^{2} + b x + a\right )}{c^{5}} - \frac {13 \, a^{3} b^{7} - 147 \, a^{4} b^{5} c + 535 \, a^{5} b^{3} c^{2} - 590 \, a^{6} b c^{3} + 6 \, {\left (3 \, b^{8} c^{2} - 36 \, a b^{6} c^{3} + 146 \, a^{2} b^{4} c^{4} - 212 \, a^{3} b^{2} c^{5} + 58 \, a^{4} c^{6}\right )} x^{5} + 6 \, {\left (5 \, b^{9} c - 57 \, a b^{7} c^{2} + 209 \, a^{2} b^{5} c^{3} - 226 \, a^{3} b^{3} c^{4} - 47 \, a^{4} b c^{5}\right )} x^{4} + {\left (13 \, b^{10} - 96 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 1788 \, a^{3} b^{4} c^{3} - 3234 \, a^{4} b^{2} c^{4} + 544 \, a^{5} c^{5}\right )} x^{3} + 3 \, {\left (13 \, a b^{9} - 143 \, a^{2} b^{7} c + 486 \, a^{3} b^{5} c^{2} - 387 \, a^{4} b^{3} c^{3} - 304 \, a^{5} b c^{4}\right )} x^{2} + 3 \, {\left (13 \, a^{2} b^{8} - 150 \, a^{3} b^{6} c + 567 \, a^{4} b^{4} c^{2} - 694 \, a^{5} b^{2} c^{3} + 76 \, a^{6} c^{4}\right )} x}{3 \, {\left (c x^{2} + b x + a\right )}^{3} {\left (b^{2} - 4 \, a c\right )}^{3} c^{5}} \]
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)*arcta n((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)
Time = 10.94 (sec) , antiderivative size = 1159, normalized size of antiderivative = 3.32 \[ \int \frac {x^8}{\left (a+b x+c x^2\right )^4} \, dx=\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}} \]
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 - 2 12*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 + 56 7*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 + 114688*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...