Integrand size = 22, antiderivative size = 114 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {b x}{c \left (b^2-4 a c\right )}+\frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {\log \left (a+b x+c x^2\right )}{2 c^2} \]
-b*x/c/(-4*a*c+b^2)+x^2*(b*x+2*a)/(-4*a*c+b^2)/(c*x^2+b*x+a)+b*(-6*a*c+b^2 )*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2)^(3/2)+1/2*ln(c*x^ 2+b*x+a)/c^2
Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {\frac {2 \left (-2 a^2 c+b^3 x+a b (b-3 c x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 b \left (b^2-6 a c\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+\log (a+x (b+c x))}{2 c^2} \]
((2*(-2*a^2*c + b^3*x + a*b*(b - 3*c*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x)) ) + (2*b*(b^2 - 6*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a *c)^(3/2) + Log[a + x*(b + c*x)])/(2*c^2)
Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {9, 1164, 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.
| \(\displaystyle \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {x^3}{\left (a+b x+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle \frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {x (4 a+b x)}{c x^2+b x+a}dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \left (\frac {b}{c}-\frac {a b+\left (b^2-4 a c\right ) x}{c \left (c x^2+b x+a\right )}\right )dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {-\frac {b \left (b^2-6 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {\left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {b x}{c}}{b^2-4 a c}\) |
(x^2*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) - ((b*x)/c - (b*(b^2 - 6*a*c)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*Log[a + b*x + c*x^2])/(2*c^2))/(b^2 - 4*a*c)
3.1.20.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, 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]
Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.48
| method | result | size |
| default | \(\frac {\frac {b \left (3 a c -b^{2}\right ) x}{c^{2} \left (4 a c -b^{2}\right )}+\frac {a \left (2 a c -b^{2}\right )}{\left (4 a c -b^{2}\right ) c^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a c -b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a b -\frac {\left (4 a c -b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c \left (4 a c -b^{2}\right )}\) | \(169\) |
| risch | \(\frac {\frac {b \left (3 a c -b^{2}\right ) x}{c^{2} \left (4 a c -b^{2}\right )}+\frac {a \left (2 a c -b^{2}\right )}{\left (4 a c -b^{2}\right ) c^{2}}}{c \,x^{2}+b x +a}+\frac {8 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{\left (4 a c -b^{2}\right )^{2}}-\frac {4 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{\left (4 a c -b^{2}\right )^{2} c}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}+\frac {8 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{\left (4 a c -b^{2}\right )^{2}}-\frac {4 \ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{\left (4 a c -b^{2}\right )^{2} c}+\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}-\frac {\ln \left (-24 a^{2} b \,c^{2}+10 a \,b^{3} c -b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (6 a c -b^{2}\right )^{2}}}{2 \left (4 a c -b^{2}\right )^{2} c^{2}}\) | \(979\) |
(b/c^2*(3*a*c-b^2)/(4*a*c-b^2)*x+a*(2*a*c-b^2)/(4*a*c-b^2)/c^2)/(c*x^2+b*x +a)+1/c/(4*a*c-b^2)*(1/2*(4*a*c-b^2)/c*ln(c*x^2+b*x+a)+2*(-a*b-1/2*(4*a*c- b^2)*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. 308 vs. \(2 (108) = 216\).
Time = 0.27 (sec) , antiderivative size = 635, normalized size of antiderivative = 5.57 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\left [\frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (a b^{3} - 6 \, a^{2} b c + {\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} + {\left (b^{4} - 6 \, a b^{2} c\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x\right )}}, \frac {2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \, {\left (a b^{3} - 6 \, a^{2} b c + {\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} + {\left (b^{4} - 6 \, a b^{2} c\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x\right )}}\right ] \]
[1/2*(2*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2 + (a*b^3 - 6*a^2*b*c + (b^3*c - 6*a*b*c^2)*x^2 + (b^4 - 6*a*b^2*c)*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*(b^5 - 7*a*b^3*c + 12*a^2*b*c^2)*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2 )*x)*log(c*x^2 + b*x + a))/(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4 + (b^4* c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^2 + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^ 4)*x), 1/2*(2*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2 + 2*(a*b^3 - 6*a^2*b*c + ( b^3*c - 6*a*b*c^2)*x^2 + (b^4 - 6*a*b^2*c)*x)*sqrt(-b^2 + 4*a*c)*arctan(-s qrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(b^5 - 7*a*b^3*c + 12*a^2 *b*c^2)*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 16* a^2*c^3)*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x)*log(c*x^2 + b*x + a))/( a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c ^5)*x^2 + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x)]
Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (104) = 208\).
Time = 0.73 (sec) , antiderivative size = 729, normalized size of antiderivative = 6.39 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) \log {\left (x + \frac {- 16 a^{2} c^{3} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a c - b^{2}\right )}{2 c^{2} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac {1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac {2 a^{2} c - a b^{2} + x \left (3 a b c - b^{3}\right )}{4 a^{2} c^{3} - a b^{2} c^{2} + x^{2} \cdot \left (4 a c^{4} - b^{2} c^{3}\right ) + x \left (4 a b c^{3} - b^{3} c^{2}\right )} \]
(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a** 2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2))*log(x + (-16*a**2*c**3*(- b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2* b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)) + 8*a**2*c + 8*a*b**2*c**2* (-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a** 2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)) - a*b**2 - b**4*c*(-b*sqr t(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2* c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)))/(6*a*b*c - b**3)) + (b*sqrt(-(4 *a*c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2))*log(x + (-16*a**2*c**3*(b*sqrt(-(4*a* c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 1 2*a*b**4*c - b**6)) + 1/(2*c**2)) + 8*a**2*c + 8*a*b**2*c**2*(b*sqrt(-(4*a *c - b**2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(2*c**2)) - a*b**2 - b**4*c*(b*sqrt(-(4*a*c - b** 2)**3)*(6*a*c - b**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b** 4*c - b**6)) + 1/(2*c**2)))/(6*a*b*c - b**3)) + (2*a**2*c - a*b**2 + x*(3* a*b*c - b**3))/(4*a**2*c**3 - a*b**2*c**2 + x**2*(4*a*c**4 - b**2*c**3) + x*(4*a*b*c**3 - b**3*c**2))
Exception generated. \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.10 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=-\frac {{\left (b^{3} - 6 \, a b c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {\log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac {a b^{2} - 2 \, a^{2} c + {\left (b^{3} - 3 \, a b c\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \]
-(b^3 - 6*a*b*c)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^2 - 4*a*c^ 3)*sqrt(-b^2 + 4*a*c)) + 1/2*log(c*x^2 + b*x + a)/c^2 + (a*b^2 - 2*a^2*c + (b^3 - 3*a*b*c)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^2)
Time = 8.81 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.45 \[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {\frac {a\,\left (2\,a\,c-b^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}+\frac {b\,x\,\left (3\,a\,c-b^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{2\,\left (64\,a^3\,c^5-48\,a^2\,b^2\,c^4+12\,a\,b^4\,c^3-b^6\,c^2\right )}+\frac {b\,\mathrm {atan}\left (\frac {c^2\,{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (\frac {2\,b\,x\,\left (6\,a\,c-b^2\right )}{c\,{\left (4\,a\,c-b^2\right )}^3}+\frac {b^2\,\left (4\,a\,c^2-b^2\,c\right )\,\left (6\,a\,c-b^2\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^4}\right )}{b^3-6\,a\,b\,c}\right )\,\left (6\,a\,c-b^2\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]
((a*(2*a*c - b^2))/(c^2*(4*a*c - b^2)) + (b*x*(3*a*c - b^2))/(c^2*(4*a*c - b^2)))/(a + b*x + c*x^2) - (log(a + b*x + c*x^2)*(b^6 - 64*a^3*c^3 + 48*a ^2*b^2*c^2 - 12*a*b^4*c))/(2*(64*a^3*c^5 - b^6*c^2 + 12*a*b^4*c^3 - 48*a^2 *b^2*c^4)) + (b*atan((c^2*(4*a*c - b^2)^(5/2)*((2*b*x*(6*a*c - b^2))/(c*(4 *a*c - b^2)^3) + (b^2*(4*a*c^2 - b^2*c)*(6*a*c - b^2))/(c^3*(4*a*c - b^2)^ 4)))/(b^3 - 6*a*b*c))*(6*a*c - b^2))/(c^2*(4*a*c - b^2)^(3/2))