Integrand size = 18, antiderivative size = 119 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {x^6 \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 b x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 a b \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
-1/4*x^6*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+3/4*b*x^2*(b*x^2+2*a)/ (-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+3*a*b*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2 ))/(-4*a*c+b^2)^(5/2)
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {8 a^3 c+b^4 x^4+a b x^2 \left (2 b^2+b c x^2+6 c^2 x^4\right )+a^2 \left (b^2+10 b c x^2+16 c^2 x^4\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2}-\frac {3 a b \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}} \]
-1/4*(8*a^3*c + b^4*x^4 + a*b*x^2*(2*b^2 + b*c*x^2 + 6*c^2*x^4) + a^2*(b^2 + 10*b*c*x^2 + 16*c^2*x^4))/(c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2) - ( 3*a*b*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2)
Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1434, 1156, 1153, 1083, 219}
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.
| \(\displaystyle \int \frac {x^7}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {x^6}{\left (c x^4+b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 1156 |
| \(\displaystyle \frac {1}{2} \left (\frac {3 b \int \frac {x^4}{\left (c x^4+b x^2+a\right )^2}dx^2}{2 \left (b^2-4 a c\right )}-\frac {x^6 \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1153 |
| \(\displaystyle \frac {1}{2} \left (\frac {3 b \left (\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 a \int \frac {1}{c x^4+b x^2+a}dx^2}{b^2-4 a c}\right )}{2 \left (b^2-4 a c\right )}-\frac {x^6 \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (\frac {3 b \left (\frac {4 a \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{b^2-4 a c}+\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {x^6 \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {3 b \left (\frac {4 a \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {x^6 \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
(-1/2*(x^6*(b + 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*b*((x ^2*(2*a + b*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (4*a*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(2*(b^2 - 4*a*c)))/2
3.9.75.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 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[2*(2*p + 3)*((c*d^2 - b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^m*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* (b^2 - 4*a*c))), x] + Simp[m*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a*c))) Int[ (d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(111)=222\).
Time = 0.14 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.93
| method | result | size |
| default | \(\frac {-\frac {3 a b c \,x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {\left (16 a^{2} c^{2}+a \,b^{2} c +b^{4}\right ) x^{4}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (5 a c +b^{2}\right ) a b \,x^{2}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a^{2} \left (8 a c +b^{2}\right )}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 a b \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}\) | \(230\) |
| risch | \(\frac {-\frac {3 a b c \,x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (16 a^{2} c^{2}+a \,b^{2} c +b^{4}\right ) x^{4}}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (5 a c +b^{2}\right ) a b \,x^{2}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a^{2} \left (8 a c +b^{2}\right )}{4 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 a b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right )}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 a b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right )}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(322\) |
1/2*(-3*a*b*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6-1/2*(16*a^2*c^2+a*b^2*c+b^4)/ c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-(5*a*c+b^2)*a*b/c/(16*a^2*c^2-8*a*b^2*c+b ^4)*x^2-1/2*a^2*(8*a*c+b^2)/c/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^ 2-3*a*b/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4 *a*c-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (111) = 222\).
Time = 0.26 (sec) , antiderivative size = 892, normalized size of antiderivative = 7.50 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^3} \, dx=\left [-\frac {6 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} + 4 \, a^{3} b^{2} c - 32 \, a^{4} c^{2} + {\left (b^{6} - 3 \, a b^{4} c + 12 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} x^{4} + 2 \, {\left (a b^{5} + a^{2} b^{3} c - 20 \, a^{3} b c^{2}\right )} x^{2} - 6 \, {\left (a b c^{3} x^{8} + 2 \, a b^{2} c^{2} x^{6} + 2 \, a^{2} b^{2} c x^{2} + a^{3} b c + {\left (a b^{3} c + 2 \, a^{2} b c^{2}\right )} x^{4}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right )}{4 \, {\left (a^{2} b^{6} c - 12 \, a^{3} b^{4} c^{2} + 48 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{8} + 2 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{6} + {\left (b^{8} c - 10 \, a b^{6} c^{2} + 24 \, a^{2} b^{4} c^{3} + 32 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5}\right )} x^{4} + 2 \, {\left (a b^{7} c - 12 \, a^{2} b^{5} c^{2} + 48 \, a^{3} b^{3} c^{3} - 64 \, a^{4} b c^{4}\right )} x^{2}\right )}}, -\frac {6 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} + 4 \, a^{3} b^{2} c - 32 \, a^{4} c^{2} + {\left (b^{6} - 3 \, a b^{4} c + 12 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} x^{4} + 2 \, {\left (a b^{5} + a^{2} b^{3} c - 20 \, a^{3} b c^{2}\right )} x^{2} - 12 \, {\left (a b c^{3} x^{8} + 2 \, a b^{2} c^{2} x^{6} + 2 \, a^{2} b^{2} c x^{2} + a^{3} b c + {\left (a b^{3} c + 2 \, a^{2} b c^{2}\right )} x^{4}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{4 \, {\left (a^{2} b^{6} c - 12 \, a^{3} b^{4} c^{2} + 48 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{8} + 2 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{6} + {\left (b^{8} c - 10 \, a b^{6} c^{2} + 24 \, a^{2} b^{4} c^{3} + 32 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5}\right )} x^{4} + 2 \, {\left (a b^{7} c - 12 \, a^{2} b^{5} c^{2} + 48 \, a^{3} b^{3} c^{3} - 64 \, a^{4} b c^{4}\right )} x^{2}\right )}}\right ] \]
[-1/4*(6*(a*b^3*c^2 - 4*a^2*b*c^3)*x^6 + a^2*b^4 + 4*a^3*b^2*c - 32*a^4*c^ 2 + (b^6 - 3*a*b^4*c + 12*a^2*b^2*c^2 - 64*a^3*c^3)*x^4 + 2*(a*b^5 + a^2*b ^3*c - 20*a^3*b*c^2)*x^2 - 6*(a*b*c^3*x^8 + 2*a*b^2*c^2*x^6 + 2*a^2*b^2*c* x^2 + a^3*b*c + (a*b^3*c + 2*a^2*b*c^2)*x^4)*sqrt(b^2 - 4*a*c)*log((2*c^2* x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)))/(a^2*b^6*c - 12*a^3*b^4*c^2 + 48*a^4*b^2*c^3 - 64*a^5*c^4 + ( b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^8 + 2*(b^7*c^2 - 1 2*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^6 + (b^8*c - 10*a*b^6*c^2 + 24*a^2*b^4*c^3 + 32*a^3*b^2*c^4 - 128*a^4*c^5)*x^4 + 2*(a*b^7*c - 12*a^2* b^5*c^2 + 48*a^3*b^3*c^3 - 64*a^4*b*c^4)*x^2), -1/4*(6*(a*b^3*c^2 - 4*a^2* b*c^3)*x^6 + a^2*b^4 + 4*a^3*b^2*c - 32*a^4*c^2 + (b^6 - 3*a*b^4*c + 12*a^ 2*b^2*c^2 - 64*a^3*c^3)*x^4 + 2*(a*b^5 + a^2*b^3*c - 20*a^3*b*c^2)*x^2 - 1 2*(a*b*c^3*x^8 + 2*a*b^2*c^2*x^6 + 2*a^2*b^2*c*x^2 + a^3*b*c + (a*b^3*c + 2*a^2*b*c^2)*x^4)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a *c)/(b^2 - 4*a*c)))/(a^2*b^6*c - 12*a^3*b^4*c^2 + 48*a^4*b^2*c^3 - 64*a^5* c^4 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^8 + 2*(b^7* c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^6 + (b^8*c - 10*a*b^ 6*c^2 + 24*a^2*b^4*c^3 + 32*a^3*b^2*c^4 - 128*a^4*c^5)*x^4 + 2*(a*b^7*c - 12*a^2*b^5*c^2 + 48*a^3*b^3*c^3 - 64*a^4*b*c^4)*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (112) = 224\).
Time = 1.65 (sec) , antiderivative size = 524, normalized size of antiderivative = 4.40 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {3 a b \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x^{2} + \frac {- 192 a^{4} b c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{3} b^{3} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a^{2} b^{5} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{2}}{6 a b c} \right )}}{2} - \frac {3 a b \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x^{2} + \frac {192 a^{4} b c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{3} b^{3} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a^{2} b^{5} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 3 a b^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{2}}{6 a b c} \right )}}{2} + \frac {- 8 a^{3} c - a^{2} b^{2} - 6 a b c^{2} x^{6} + x^{4} \left (- 16 a^{2} c^{2} - a b^{2} c - b^{4}\right ) + x^{2} \left (- 10 a^{2} b c - 2 a b^{3}\right )}{64 a^{4} c^{3} - 32 a^{3} b^{2} c^{2} + 4 a^{2} b^{4} c + x^{8} \cdot \left (64 a^{2} c^{5} - 32 a b^{2} c^{4} + 4 b^{4} c^{3}\right ) + x^{6} \cdot \left (128 a^{2} b c^{4} - 64 a b^{3} c^{3} + 8 b^{5} c^{2}\right ) + x^{4} \cdot \left (128 a^{3} c^{4} - 24 a b^{4} c^{2} + 4 b^{6} c\right ) + x^{2} \cdot \left (128 a^{3} b c^{3} - 64 a^{2} b^{3} c^{2} + 8 a b^{5} c\right )} \]
3*a*b*sqrt(-1/(4*a*c - b**2)**5)*log(x**2 + (-192*a**4*b*c**3*sqrt(-1/(4*a *c - b**2)**5) + 144*a**3*b**3*c**2*sqrt(-1/(4*a*c - b**2)**5) - 36*a**2*b **5*c*sqrt(-1/(4*a*c - b**2)**5) + 3*a*b**7*sqrt(-1/(4*a*c - b**2)**5) + 3 *a*b**2)/(6*a*b*c))/2 - 3*a*b*sqrt(-1/(4*a*c - b**2)**5)*log(x**2 + (192*a **4*b*c**3*sqrt(-1/(4*a*c - b**2)**5) - 144*a**3*b**3*c**2*sqrt(-1/(4*a*c - b**2)**5) + 36*a**2*b**5*c*sqrt(-1/(4*a*c - b**2)**5) - 3*a*b**7*sqrt(-1 /(4*a*c - b**2)**5) + 3*a*b**2)/(6*a*b*c))/2 + (-8*a**3*c - a**2*b**2 - 6* a*b*c**2*x**6 + x**4*(-16*a**2*c**2 - a*b**2*c - b**4) + x**2*(-10*a**2*b* c - 2*a*b**3))/(64*a**4*c**3 - 32*a**3*b**2*c**2 + 4*a**2*b**4*c + x**8*(6 4*a**2*c**5 - 32*a*b**2*c**4 + 4*b**4*c**3) + x**6*(128*a**2*b*c**4 - 64*a *b**3*c**3 + 8*b**5*c**2) + x**4*(128*a**3*c**4 - 24*a*b**4*c**2 + 4*b**6* c) + x**2*(128*a**3*b*c**3 - 64*a**2*b**3*c**2 + 8*a*b**5*c))
Exception generated. \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Exception raised: ValueError} \]
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 = 1.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.44 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {3 \, a b \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {6 \, a b c^{2} x^{6} + b^{4} x^{4} + a b^{2} c x^{4} + 16 \, a^{2} c^{2} x^{4} + 2 \, a b^{3} x^{2} + 10 \, a^{2} b c x^{2} + a^{2} b^{2} + 8 \, a^{3} c}{4 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} \]
-3*a*b*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2 *c^2)*sqrt(-b^2 + 4*a*c)) - 1/4*(6*a*b*c^2*x^6 + b^4*x^4 + a*b^2*c*x^4 + 1 6*a^2*c^2*x^4 + 2*a*b^3*x^2 + 10*a^2*b*c*x^2 + a^2*b^2 + 8*a^3*c)/((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*(c*x^4 + b*x^2 + a)^2)
Time = 13.13 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.55 \[ \int \frac {x^7}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {\frac {x^2\,\left (5\,c\,a^2\,b+a\,b^3\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^4\,\left (16\,a^2\,c^2+a\,b^2\,c+b^4\right )}{4\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,\left (8\,c\,a^2+a\,b^2\right )}{4\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,a\,b\,c\,x^6}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^8+2\,a\,b\,x^2+2\,b\,c\,x^6}-\frac {3\,a\,b\,\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {9\,a\,b^2\,c^2}{{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {9\,a\,b^3\,\left (32\,a^2\,b\,c^4-16\,a\,b^3\,c^3+2\,b^5\,c^2\right )}{2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {18\,a^2\,b^3\,c^2}{{\left (4\,a\,c-b^2\right )}^{15/2}}\right )\,\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )}{18\,a^2\,b^2\,c^2}\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \]
- ((x^2*(a*b^3 + 5*a^2*b*c))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^4*( b^4 + 16*a^2*c^2 + a*b^2*c))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a*(a* b^2 + 8*a^2*c))/(4*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*a*b*c*x^6)/(2*(b ^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b* x^2 + 2*b*c*x^6) - (3*a*b*atan(((x^2*((9*a*b^2*c^2)/((4*a*c - b^2)^(9/2)*( b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (9*a*b^3*(2*b^5*c^2 - 16*a*b^3*c^3 + 32*a ^2*b*c^4))/(2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))) + (18* a^2*b^3*c^2)/(4*a*c - b^2)^(15/2))*(b^4*(4*a*c - b^2)^5 + 16*a^2*c^2*(4*a* c - b^2)^5 - 8*a*b^2*c*(4*a*c - b^2)^5))/(18*a^2*b^2*c^2)))/(4*a*c - b^2)^ (5/2)