Integrand size = 23, antiderivative size = 519 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx=\frac {(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\sqrt {b^2-4 a c} (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 \left (c d^2-b d e+a e^2\right )^4}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}+\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 e \left (c d^2-b d e+a e^2\right )^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4} \]
1/12*(-b*e+2*c*d)*n/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^3+1/8*(2*c^2*d^2+b^2*e^2 -2*c*e*(a*e+b*d))*n/e/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^2+1/4*(-b*e+2*c*d)*(c^ 2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*n/e/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)-1/4*(2*c^ 4*d^4+b^4*e^4-4*b^2*c*e^3*(a*e+b*d)-4*c^3*d^2*e*(3*a*e+b*d)+2*c^2*e^2*(a^2 *e^2+6*a*b*d*e+3*b^2*d^2))*n*ln(e*x+d)/e/(a*e^2-b*d*e+c*d^2)^4+1/8*(2*c^4* d^4+b^4*e^4-4*b^2*c*e^3*(a*e+b*d)-4*c^3*d^2*e*(3*a*e+b*d)+2*c^2*e^2*(a^2*e ^2+6*a*b*d*e+3*b^2*d^2))*n*ln(c*x^2+b*x+a)/e/(a*e^2-b*d*e+c*d^2)^4-1/4*ln( d*(c*x^2+b*x+a)^n)/e/(e*x+d)^4+1/4*(-b*e+2*c*d)*(2*c^2*d^2+b^2*e^2-2*c*e*( a*e+b*d))*n*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))*(-4*a*c+b^2)^(1/2)/(a*e^ 2-b*d*e+c*d^2)^4
Time = 1.13 (sec) , antiderivative size = 469, normalized size of antiderivative = 0.90 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx=\frac {\frac {n (d+e x) \left (2 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3+3 \left (c d^2+e (-b d+a e)\right )^2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) (d+e x)+6 (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) (d+e x)^2+6 \sqrt {b^2-4 a c} e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) (d+e x)^3 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-6 \left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) (d+e x)^3 \log (d+e x)+3 \left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) (d+e x)^3 \log (a+x (b+c x))\right )}{\left (c d^2+e (-b d+a e)\right )^4}-6 \log \left (d (a+x (b+c x))^n\right )}{24 e (d+e x)^4} \]
((n*(d + e*x)*(2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3 + 3*(c*d^2 + e *(-(b*d) + a*e))^2*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*(d + e*x) + 6 *(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(d + e*x)^2 + 6*Sqrt[b^2 - 4*a*c]*e*(2*c*d - b*e)*(2*c^2*d^2 + b^2 *e^2 - 2*c*e*(b*d + a*e))*(d + e*x)^3*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c ]] - 6*(2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*(d + e*x)^3*Log[d + e*x] + 3*(2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b* d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*(d + e*x)^3*Log[ a + x*(b + c*x)]))/(c*d^2 + e*(-(b*d) + a*e))^4 - 6*Log[d*(a + x*(b + c*x) )^n])/(24*e*(d + e*x)^4)
Time = 1.03 (sec) , antiderivative size = 500, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3005, 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.
| \(\displaystyle \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 3005 |
| \(\displaystyle \frac {n \int \frac {b+2 c x}{(d+e x)^4 \left (c x^2+b x+a\right )}dx}{4 e}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {n \int \left (\frac {e (b e-2 c d)}{\left (c d^2-b e d+a e^2\right ) (d+e x)^4}+\frac {e \left (-2 c^4 d^4+4 c^3 e (b d+3 a e) d^2-b^4 e^4+4 b^2 c e^3 (b d+a e)-2 c^2 e^2 \left (3 b^2 d^2+6 a b e d+a^2 e^2\right )\right )}{\left (c d^2-b e d+a e^2\right )^4 (d+e x)}+\frac {e^4 b^5-4 c d e^3 b^4+c e^2 \left (6 c d^2-5 a e^2\right ) b^3-4 c^2 d e \left (c d^2-4 a e^2\right ) b^2+c^2 \left (c^2 d^4-18 a c e^2 d^2+5 a^2 e^4\right ) b+8 a c^3 d e \left (c d^2-a e^2\right )+c \left (2 c^4 d^4-4 c^3 e (b d+3 a e) d^2+b^4 e^4-4 b^2 c e^3 (b d+a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b e d+a^2 e^2\right )\right ) x}{\left (c d^2-b e d+a e^2\right )^4 \left (c x^2+b x+a\right )}+\frac {e (2 c d-b e) \left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right )}{\left (c d^2-b e d+a e^2\right )^3 (d+e x)^2}+\frac {e \left (-2 c^2 d^2-b^2 e^2+2 c e (b d+a e)\right )}{\left (c d^2-b e d+a e^2\right )^2 (d+e x)^3}\right )dx}{4 e}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {n \left (\frac {\left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}-\frac {\log (d+e x) \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right )}{\left (a e^2-b d e+c d^2\right )^4}+\frac {e \sqrt {b^2-4 a c} (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^4}+\frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac {-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac {2 c d-b e}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\right )}{4 e}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}\) |
(n*((2*c*d - b*e)/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + (2*c^2*d^2 + b ^2*e^2 - 2*c*e*(b*d + a*e))/(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + (( 2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))/((c*d^2 - b*d*e + a* e^2)^3*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*e*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^ 2 - 2*c*e*(b*d + a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*d^2 - b* d*e + a*e^2)^4 - ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d ^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 + ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a ^2*e^2))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^4)))/(4*e) - Log [d*(a + b*x + c*x^2)^n]/(4*e*(d + e*x)^4)
3.1.91.3.1 Defintions of rubi rules used
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[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_. ), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))) , x] - Simp[b*n*(p/(e*(m + 1))) Int[SimplifyIntegrand[(d + e*x)^(m + 1)*( a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]
Time = 8.09 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.40
| method | result | size |
| parts | \(-\frac {\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )}{4 e \left (e x +d \right )^{4}}+\frac {n \left (-\frac {\left (2 a^{2} c^{2} e^{4}-4 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 a \,c^{3} d^{2} e^{2}+b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{4}}-\frac {2 a c \,e^{2}-e^{2} b^{2}+2 b c d e -2 c^{2} d^{2}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}-\frac {b e -2 c d}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )^{3}}+\frac {3 a b c \,e^{3}-6 a \,c^{2} d \,e^{2}-b^{3} e^{3}+3 b^{2} c d \,e^{2}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )}+\frac {\frac {\left (2 a^{2} c^{3} e^{4}-4 a \,b^{2} c^{2} e^{4}+12 a b \,c^{3} d \,e^{3}-12 a \,c^{4} d^{2} e^{2}+b^{4} c \,e^{4}-4 b^{3} c^{2} d \,e^{3}+6 b^{2} c^{3} d^{2} e^{2}-4 b \,c^{4} d^{3} e +2 c^{5} d^{4}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (5 a^{2} b \,c^{2} e^{4}-8 a^{2} c^{3} d \,e^{3}-5 a \,b^{3} e^{4} c +16 a \,b^{2} c^{2} d \,e^{3}-18 a b \,c^{3} d^{2} e^{2}+8 a \,c^{4} d^{3} e +b^{5} e^{4}-4 b^{4} c d \,e^{3}+6 b^{3} c^{2} d^{2} e^{2}-4 b^{2} c^{3} d^{3} e +b \,c^{4} d^{4}-\frac {\left (2 a^{2} c^{3} e^{4}-4 a \,b^{2} c^{2} e^{4}+12 a b \,c^{3} d \,e^{3}-12 a \,c^{4} d^{2} e^{2}+b^{4} c \,e^{4}-4 b^{3} c^{2} d \,e^{3}+6 b^{2} c^{3} d^{2} e^{2}-4 b \,c^{4} d^{3} e +2 c^{5} d^{4}\right ) b}{2 c}\right ) \arctan \left (\frac {2 x c +b}{\sqrt {4 c a -b^{2}}}\right )}{\sqrt {4 c a -b^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{4}}\right )}{4 e}\) | \(725\) |
| risch | \(\text {Expression too large to display}\) | \(232330\) |
-1/4*ln(d*(c*x^2+b*x+a)^n)/e/(e*x+d)^4+1/4/e*n*(-(2*a^2*c^2*e^4-4*a*b^2*c* e^4+12*a*b*c^2*d*e^3-12*a*c^3*d^2*e^2+b^4*e^4-4*b^3*c*d*e^3+6*b^2*c^2*d^2* e^2-4*b*c^3*d^3*e+2*c^4*d^4)/(a*e^2-b*d*e+c*d^2)^4*ln(e*x+d)-1/2*(2*a*c*e^ 2-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^2-1/3*(b*e-2* c*d)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^3+(3*a*b*c*e^3-6*a*c^2*d*e^2-b^3*e^3+3*b^ 2*c*d*e^2-3*b*c^2*d^2*e+2*c^3*d^3)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)+1/(a*e^2- b*d*e+c*d^2)^4*(1/2*(2*a^2*c^3*e^4-4*a*b^2*c^2*e^4+12*a*b*c^3*d*e^3-12*a*c ^4*d^2*e^2+b^4*c*e^4-4*b^3*c^2*d*e^3+6*b^2*c^3*d^2*e^2-4*b*c^4*d^3*e+2*c^5 *d^4)/c*ln(c*x^2+b*x+a)+2*(5*a^2*b*c^2*e^4-8*a^2*c^3*d*e^3-5*a*b^3*e^4*c+1 6*a*b^2*c^2*d*e^3-18*a*b*c^3*d^2*e^2+8*a*c^4*d^3*e+b^5*e^4-4*b^4*c*d*e^3+6 *b^3*c^2*d^2*e^2-4*b^2*c^3*d^3*e+b*c^4*d^4-1/2*(2*a^2*c^3*e^4-4*a*b^2*c^2* e^4+12*a*b*c^3*d*e^3-12*a*c^4*d^2*e^2+b^4*c*e^4-4*b^3*c^2*d*e^3+6*b^2*c^3* d^2*e^2-4*b*c^4*d^3*e+2*c^5*d^4)*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. 2902 vs. \(2 (501) = 1002\).
Time = 48.90 (sec) , antiderivative size = 5824, normalized size of antiderivative = 11.22 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 2191 vs. \(2 (501) = 1002\).
Time = 0.59 (sec) , antiderivative size = 2191, normalized size of antiderivative = 4.22 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx=\text {Too large to display} \]
1/8*(2*c^4*d^4*n - 4*b*c^3*d^3*e*n + 6*b^2*c^2*d^2*e^2*n - 12*a*c^3*d^2*e^ 2*n - 4*b^3*c*d*e^3*n + 12*a*b*c^2*d*e^3*n + b^4*e^4*n - 4*a*b^2*c*e^4*n + 2*a^2*c^2*e^4*n)*log(c*x^2 + b*x + a)/(c^4*d^8*e - 4*b*c^3*d^7*e^2 + 6*b^ 2*c^2*d^6*e^3 + 4*a*c^3*d^6*e^3 - 4*b^3*c*d^5*e^4 - 12*a*b*c^2*d^5*e^4 + b ^4*d^4*e^5 + 12*a*b^2*c*d^4*e^5 + 6*a^2*c^2*d^4*e^5 - 4*a*b^3*d^3*e^6 - 12 *a^2*b*c*d^3*e^6 + 6*a^2*b^2*d^2*e^7 + 4*a^3*c*d^2*e^7 - 4*a^3*b*d*e^8 + a ^4*e^9) - 1/4*n*log(c*x^2 + b*x + a)/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^ 2 + 4*d^3*e^2*x + d^4*e) - 1/4*(2*c^4*d^4*n - 4*b*c^3*d^3*e*n + 6*b^2*c^2* d^2*e^2*n - 12*a*c^3*d^2*e^2*n - 4*b^3*c*d*e^3*n + 12*a*b*c^2*d*e^3*n + b^ 4*e^4*n - 4*a*b^2*c*e^4*n + 2*a^2*c^2*e^4*n)*log(e*x + d)/(c^4*d^8*e - 4*b *c^3*d^7*e^2 + 6*b^2*c^2*d^6*e^3 + 4*a*c^3*d^6*e^3 - 4*b^3*c*d^5*e^4 - 12* a*b*c^2*d^5*e^4 + b^4*d^4*e^5 + 12*a*b^2*c*d^4*e^5 + 6*a^2*c^2*d^4*e^5 - 4 *a*b^3*d^3*e^6 - 12*a^2*b*c*d^3*e^6 + 6*a^2*b^2*d^2*e^7 + 4*a^3*c*d^2*e^7 - 4*a^3*b*d*e^8 + a^4*e^9) - 1/4*(4*b^2*c^3*d^3*n - 16*a*c^4*d^3*n - 6*b^3 *c^2*d^2*e*n + 24*a*b*c^3*d^2*e*n + 4*b^4*c*d*e^2*n - 20*a*b^2*c^2*d*e^2*n + 16*a^2*c^3*d*e^2*n - b^5*e^3*n + 6*a*b^3*c*e^3*n - 8*a^2*b*c^2*e^3*n)*a rctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^ 2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d ^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2 *b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^...
Time = 20.48 (sec) , antiderivative size = 4334, normalized size of antiderivative = 8.35 \[ \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx=\text {Too large to display} \]
(log(10*d*e^5*(b^2 - 4*a*c)^(7/2) + 3*e^6*x*(b^2 - 4*a*c)^(7/2) - 6*a*e^6* (4*a*c - b^2)^3 + 96*c^5*d^6*(4*a*c - b^2) - 10*b*e^6*x*(4*a*c - b^2)^3 - 10*b^5*e^6*x*(4*a*c - b^2) + 29*b^2*e^6*x*(b^2 - 4*a*c)^(5/2) + 29*b^4*e^6 *x*(b^2 - 4*a*c)^(3/2) + 3*b^6*e^6*x*(b^2 - 4*a*c)^(1/2) + 192*c^6*d^6*x*( b^2 - 4*a*c)^(1/2) + 44*a*b^2*e^6*(4*a*c - b^2)^2 - 16*b^3*d*e^5*(4*a*c - b^2)^2 + 58*c*d^2*e^4*(4*a*c - b^2)^3 + 176*c^2*d^3*e^3*(b^2 - 4*a*c)^(5/2 ) + 44*b^3*e^6*x*(4*a*c - b^2)^2 + 14*a*b*e^6*(b^2 - 4*a*c)^(5/2) - 232*c^ 3*d^4*e^2*(4*a*c - b^2)^2 - 14*a*b^4*e^6*(4*a*c - b^2) + 44*a*b^3*e^6*(b^2 - 4*a*c)^(3/2) + 6*a*b^5*e^6*(b^2 - 4*a*c)^(1/2) + 96*b*c^5*d^6*(b^2 - 4* a*c)^(1/2) - 48*b*d*e^5*(4*a*c - b^2)^3 + 32*b^5*d*e^5*(4*a*c - b^2) + 74* b^2*d*e^5*(b^2 - 4*a*c)^(5/2) - 66*b^4*d*e^5*(b^2 - 4*a*c)^(3/2) - 18*b^6* d*e^5*(b^2 - 4*a*c)^(1/2) + 160*c^4*d^5*e*(b^2 - 4*a*c)^(3/2) + 288*b*c^2* d^3*e^3*(4*a*c - b^2)^2 - 84*b^2*c*d^2*e^4*(4*a*c - b^2)^2 - 40*b^2*c^3*d^ 4*e^2*(4*a*c - b^2) + 160*b^3*c^2*d^3*e^3*(4*a*c - b^2) - 64*b^2*c^2*d^3*e ^3*(b^2 - 4*a*c)^(3/2) + 360*b^3*c^3*d^4*e^2*(b^2 - 4*a*c)^(1/2) - 240*b^4 *c^2*d^3*e^3*(b^2 - 4*a*c)^(1/2) - 352*c^3*d^3*e^3*x*(4*a*c - b^2)^2 - 128 *b*c^4*d^5*e*(4*a*c - b^2) - 206*b*c*d^2*e^4*(b^2 - 4*a*c)^(5/2) + 20*c*d* e^5*x*(4*a*c - b^2)^3 + 320*c^5*d^5*e*x*(4*a*c - b^2) - 110*b^4*c*d^2*e^4* (4*a*c - b^2) - 168*b*c^3*d^4*e^2*(b^2 - 4*a*c)^(3/2) - 288*b^2*c^4*d^5*e* (b^2 - 4*a*c)^(1/2) + 148*b^3*c*d^2*e^4*(b^2 - 4*a*c)^(3/2) + 90*b^5*c*...