Integrand size = 22, antiderivative size = 107 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {a}{2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {c}{2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {(b c+a d) \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {(b c+a d) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
1/2*a/(-a*d+b*c)^2/(b*x^2+a)+1/2*c/(-a*d+b*c)^2/(d*x^2+c)+1/2*(a*d+b*c)*ln (b*x^2+a)/(-a*d+b*c)^3-1/2*(a*d+b*c)*ln(d*x^2+c)/(-a*d+b*c)^3
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {\frac {a (b c-a d)}{a+b x^2}+\frac {c (b c-a d)}{c+d x^2}+(b c+a d) \log \left (a+b x^2\right )-(b c+a d) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
((a*(b*c - a*d))/(a + b*x^2) + (c*(b*c - a*d))/(c + d*x^2) + (b*c + a*d)*L og[a + b*x^2] - (b*c + a*d)*Log[c + d*x^2])/(2*(b*c - a*d)^3)
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 86, 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.
| \(\displaystyle \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^2}dx^2\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {(b c+a d) b}{(b c-a d)^3 \left (b x^2+a\right )}-\frac {a b}{(b c-a d)^2 \left (b x^2+a\right )^2}-\frac {d (b c+a d)}{(b c-a d)^3 \left (d x^2+c\right )}-\frac {c d}{(b c-a d)^2 \left (d x^2+c\right )^2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {a}{\left (a+b x^2\right ) (b c-a d)^2}+\frac {c}{\left (c+d x^2\right ) (b c-a d)^2}+\frac {(a d+b c) \log \left (a+b x^2\right )}{(b c-a d)^3}-\frac {(a d+b c) \log \left (c+d x^2\right )}{(b c-a d)^3}\right )\) |
(a/((b*c - a*d)^2*(a + b*x^2)) + c/((b*c - a*d)^2*(c + d*x^2)) + ((b*c + a *d)*Log[a + b*x^2])/(b*c - a*d)^3 - ((b*c + a*d)*Log[c + d*x^2])/(b*c - a* d)^3)/2
3.4.2.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 2.70 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(-\frac {b \left (\frac {\left (a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 \left (a d -b c \right )^{3}}+\frac {d \left (\frac {\left (a d +b c \right ) \ln \left (d \,x^{2}+c \right )}{d}+\frac {\left (a d -b c \right ) c}{d \left (d \,x^{2}+c \right )}\right )}{2 \left (a d -b c \right )^{3}}\) | \(113\) |
| norman | \(\frac {\frac {a c}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {\left (a d +b c \right ) x^{2}}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}-\frac {\left (a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (a d +b c \right ) \ln \left (d \,x^{2}+c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}\) | \(193\) |
| risch | \(\frac {\frac {a c}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {\left (a d +b c \right ) x^{2}}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+\frac {d \ln \left (d \,x^{2}+c \right ) a}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}+\frac {c \ln \left (d \,x^{2}+c \right ) b}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}-\frac {\ln \left (-b \,x^{2}-a \right ) a d}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\ln \left (-b \,x^{2}-a \right ) b c}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(289\) |
| parallelrisch | \(-\frac {\ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{2} d^{3}+\ln \left (b \,x^{2}+a \right ) x^{4} b^{3} c \,d^{2}-\ln \left (d \,x^{2}+c \right ) x^{4} a \,b^{2} d^{3}-\ln \left (d \,x^{2}+c \right ) x^{4} b^{3} c \,d^{2}+\ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b \,d^{3}+\ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c^{2} d -2 a^{2} b c \,d^{2}+2 a \,b^{2} c^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{2} c \,d^{2}-\ln \left (d \,x^{2}+c \right ) x^{2} a^{2} b \,d^{3}-\ln \left (d \,x^{2}+c \right ) x^{2} b^{3} c^{2} d -x^{2} a^{2} b \,d^{3}+x^{2} b^{3} c^{2} d +\ln \left (b \,x^{2}+a \right ) a^{2} b c \,d^{2}+\ln \left (b \,x^{2}+a \right ) a \,b^{2} c^{2} d -2 \ln \left (d \,x^{2}+c \right ) x^{2} a \,b^{2} c \,d^{2}-\ln \left (d \,x^{2}+c \right ) a^{2} b c \,d^{2}-\ln \left (d \,x^{2}+c \right ) a \,b^{2} c^{2} d}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) b d}\) | \(377\) |
-1/2*b/(a*d-b*c)^3*((a*d+b*c)/b*ln(b*x^2+a)-(a*d-b*c)*a/b/(b*x^2+a))+1/2*d /(a*d-b*c)^3*((a*d+b*c)/d*ln(d*x^2+c)+(a*d-b*c)*c/d/(d*x^2+c))
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (99) = 198\).
Time = 0.25 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.77 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {2 \, a b c^{2} - 2 \, a^{2} c d + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2} + {\left ({\left (b^{2} c d + a b d^{2}\right )} x^{4} + a b c^{2} + a^{2} c d + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - {\left ({\left (b^{2} c d + a b d^{2}\right )} x^{4} + a b c^{2} + a^{2} c d + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )}} \]
1/2*(2*a*b*c^2 - 2*a^2*c*d + (b^2*c^2 - a^2*d^2)*x^2 + ((b^2*c*d + a*b*d^2 )*x^4 + a*b*c^2 + a^2*c*d + (b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x^2)*log(b*x^2 + a) - ((b^2*c*d + a*b*d^2)*x^4 + a*b*c^2 + a^2*c*d + (b^2*c^2 + 2*a*b*c* d + a^2*d^2)*x^2)*log(d*x^2 + c))/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c ^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3* b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (90) = 180\).
Time = 2.58 (sec) , antiderivative size = 507, normalized size of antiderivative = 4.74 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {2 a c + x^{2} \left (a d + b c\right )}{2 a^{3} c d^{2} - 4 a^{2} b c^{2} d + 2 a b^{2} c^{3} + x^{4} \cdot \left (2 a^{2} b d^{3} - 4 a b^{2} c d^{2} + 2 b^{3} c^{2} d\right ) + x^{2} \cdot \left (2 a^{3} d^{3} - 2 a^{2} b c d^{2} - 2 a b^{2} c^{2} d + 2 b^{3} c^{3}\right )} + \frac {\left (a d + b c\right ) \log {\left (x^{2} + \frac {- \frac {a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} + \frac {4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d - \frac {b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac {\left (a d + b c\right ) \log {\left (x^{2} + \frac {\frac {a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} - \frac {4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d + \frac {b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{2 \left (a d - b c\right )^{3}} \]
(2*a*c + x**2*(a*d + b*c))/(2*a**3*c*d**2 - 4*a**2*b*c**2*d + 2*a*b**2*c** 3 + x**4*(2*a**2*b*d**3 - 4*a*b**2*c*d**2 + 2*b**3*c**2*d) + x**2*(2*a**3* d**3 - 2*a**2*b*c*d**2 - 2*a*b**2*c**2*d + 2*b**3*c**3)) + (a*d + b*c)*log (x**2 + (-a**4*d**4*(a*d + b*c)/(a*d - b*c)**3 + 4*a**3*b*c*d**3*(a*d + b* c)/(a*d - b*c)**3 - 6*a**2*b**2*c**2*d**2*(a*d + b*c)/(a*d - b*c)**3 + a** 2*d**2 + 4*a*b**3*c**3*d*(a*d + b*c)/(a*d - b*c)**3 + 2*a*b*c*d - b**4*c** 4*(a*d + b*c)/(a*d - b*c)**3 + b**2*c**2)/(2*a*b*d**2 + 2*b**2*c*d))/(2*(a *d - b*c)**3) - (a*d + b*c)*log(x**2 + (a**4*d**4*(a*d + b*c)/(a*d - b*c)* *3 - 4*a**3*b*c*d**3*(a*d + b*c)/(a*d - b*c)**3 + 6*a**2*b**2*c**2*d**2*(a *d + b*c)/(a*d - b*c)**3 + a**2*d**2 - 4*a*b**3*c**3*d*(a*d + b*c)/(a*d - b*c)**3 + 2*a*b*c*d + b**4*c**4*(a*d + b*c)/(a*d - b*c)**3 + b**2*c**2)/(2 *a*b*d**2 + 2*b**2*c*d))/(2*(a*d - b*c)**3)
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (99) = 198\).
Time = 0.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.13 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {{\left (b c + a d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {{\left (b c + a d\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {{\left (b c + a d\right )} x^{2} + 2 \, a c}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \]
1/2*(b*c + a*d)*log(b*x^2 + a)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/2*(b*c + a*d)*log(d*x^2 + c)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2 *b*c*d^2 - a^3*d^3) + 1/2*((b*c + a*d)*x^2 + 2*a*c)/(a*b^2*c^3 - 2*a^2*b*c ^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)
Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.66 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {\frac {a b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x^{2} + a\right )}} - \frac {{\left (b^{3} c + a b^{2} d\right )} \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {b^{2} c d}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d\right )}}}{2 \, b} \]
1/2*(a*b^3/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*(b*x^2 + a)) - (b^3*c + a*b^2*d)*log(abs(b*c/(b*x^2 + a) - a*d/(b*x^2 + a) + d))/(b^4*c^3 - 3*a*b^ 3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - b^2*c*d/((b*c - a*d)^3*(b*c/(b*x^ 2 + a) - a*d/(b*x^2 + a) + d)))/b
Time = 5.10 (sec) , antiderivative size = 522, normalized size of antiderivative = 4.88 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {b^2\,c^2\,x^2-a^2\,d^2\,x^2+2\,a\,b\,c^2-2\,a^2\,c\,d+a^2\,d^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+b^2\,c^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+a\,b\,c^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+a^2\,c\,d\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+a\,b\,d^2\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+b^2\,c\,d\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+a\,b\,c\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}}{-2\,a^4\,c\,d^3-2\,a^4\,d^4\,x^2+6\,a^3\,b\,c^2\,d^2+4\,a^3\,b\,c\,d^3\,x^2-2\,a^3\,b\,d^4\,x^4-6\,a^2\,b^2\,c^3\,d+6\,a^2\,b^2\,c\,d^3\,x^4+2\,a\,b^3\,c^4-4\,a\,b^3\,c^3\,d\,x^2-6\,a\,b^3\,c^2\,d^2\,x^4+2\,b^4\,c^4\,x^2+2\,b^4\,c^3\,d\,x^4} \]
(b^2*c^2*x^2 - a^2*d^2*x^2 + 2*a*b*c^2 - 2*a^2*c*d + a^2*d^2*x^2*atan((a*d *x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*2i + b^2*c^2*x^2*atan(( a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*2i + a*b*c^2*atan((a *d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*2i + a^2*c*d*atan((a* d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*2i + a*b*d^2*x^4*atan( (a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*2i + b^2*c*d*x^4*at an((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*2i + a*b*c*d*x^2 *atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*4i)/(2*a*b^3* c^4 - 2*a^4*c*d^3 - 2*a^4*d^4*x^2 + 2*b^4*c^4*x^2 - 6*a^2*b^2*c^3*d + 6*a^ 3*b*c^2*d^2 - 2*a^3*b*d^4*x^4 + 2*b^4*c^3*d*x^4 - 4*a*b^3*c^3*d*x^2 + 4*a^ 3*b*c*d^3*x^2 - 6*a*b^3*c^2*d^2*x^4 + 6*a^2*b^2*c*d^3*x^4)