Integrand size = 18, antiderivative size = 121 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 a x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {6 a^2 \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \] Output:
1/4*x^6*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2-3/2*a*x^2*(b*x^2+2*a)/( -4*a*c+b^2)^2/(c*x^4+b*x^2+a)-6*a^2*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2) )/(-4*a*c+b^2)^(5/2)
Time = 0.12 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.60 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{4} \left (\frac {b^5-8 a b^3 c+22 a^2 b c^2-2 b^4 c x^2+16 a b^2 c^2 x^2-20 a^2 c^3 x^2}{c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b^4 x^2+a b^2 \left (b-4 c x^2\right )+a^2 c \left (-3 b+2 c x^2\right )}{c^3 \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {24 a^2 \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}\right ) \] Input:
Integrate[x^9/(a + b*x^2 + c*x^4)^3,x]
Output:
((b^5 - 8*a*b^3*c + 22*a^2*b*c^2 - 2*b^4*c*x^2 + 16*a*b^2*c^2*x^2 - 20*a^2 *c^3*x^2)/(c^3*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^4*x^2 + a*b^2*(b - 4*c*x^2) + a^2*c*(-3*b + 2*c*x^2))/(c^3*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^ 4)^2) + (24*a^2*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^( 5/2))/4
Time = 0.48 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1434, 1153, 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.
| \(\displaystyle \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {x^8}{\left (c x^4+b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 1153 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 a \int \frac {x^4}{\left (c x^4+b x^2+a\right )^2}dx^2}{b^2-4 a c}\right )\) |
| \(\Big \downarrow \) 1153 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 a \left (\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {2 a \int \frac {1}{c x^4+b x^2+a}dx^2}{b^2-4 a c}\right )}{b^2-4 a c}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 a \left (\frac {4 a \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{b^2-4 a c}+\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {3 a \left (\frac {4 a \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}\right )\) |
Input:
Int[x^9/(a + b*x^2 + c*x^4)^3,x]
Output:
((x^6*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*a*((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)))/(b^2 - 4*a*c))/2
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[(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. \(266\) vs. \(2(113)=226\).
Time = 0.17 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.21
| method | result | size |
| default | \(\frac {-\frac {\left (10 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) x^{6}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \left (2 a^{2} c^{2}+8 a \,b^{2} c -b^{4}\right ) x^{4}}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (6 a^{2} c^{2}-10 a \,b^{2} c +b^{4}\right ) x^{2}}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}+\frac {a^{2} b \left (10 a c -b^{2}\right )}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {6 a^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}\) | \(267\) |
| risch | \(\frac {-\frac {\left (10 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) x^{6}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \left (2 a^{2} c^{2}+8 a \,b^{2} c -b^{4}\right ) x^{4}}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (6 a^{2} c^{2}-10 a \,b^{2} c +b^{4}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}+\frac {a^{2} b \left (10 a c -b^{2}\right )}{4 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 a^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 a^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(356\) |
Input:
int(x^9/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
1/2*(-1/c*(10*a^2*c^2-8*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*b* (2*a^2*c^2+8*a*b^2*c-b^4)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-a*(6*a^2*c^2- 10*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x^2+1/2*a^2*b*(10*a*c-b^2)/ c^2/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+6*a^2/(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. 474 vs. \(2 (113) = 226\).
Time = 0.10 (sec) , antiderivative size = 973, normalized size of antiderivative = 8.04 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(x^9/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
Output:
[-1/4*(a^2*b^5 - 14*a^3*b^3*c + 40*a^4*b*c^2 + 2*(b^6*c - 12*a*b^4*c^2 + 4 2*a^2*b^2*c^3 - 40*a^3*c^4)*x^6 + (b^7 - 12*a*b^5*c + 30*a^2*b^3*c^2 + 8*a ^3*b*c^3)*x^4 + 2*(a*b^6 - 14*a^2*b^4*c + 46*a^3*b^2*c^2 - 24*a^4*c^3)*x^2 - 12*(a^2*c^4*x^8 + 2*a^2*b*c^3*x^6 + 2*a^3*b*c^2*x^2 + a^4*c^2 + (a^2*b^ 2*c^2 + 2*a^3*c^3)*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^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4* c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^8 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^ 2*b^3*c^5 - 64*a^3*b*c^6)*x^6 + (b^8*c^2 - 10*a*b^6*c^3 + 24*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4*c^6)*x^4 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^ 3*b^3*c^4 - 64*a^4*b*c^5)*x^2), -1/4*(a^2*b^5 - 14*a^3*b^3*c + 40*a^4*b*c^ 2 + 2*(b^6*c - 12*a*b^4*c^2 + 42*a^2*b^2*c^3 - 40*a^3*c^4)*x^6 + (b^7 - 12 *a*b^5*c + 30*a^2*b^3*c^2 + 8*a^3*b*c^3)*x^4 + 2*(a*b^6 - 14*a^2*b^4*c + 4 6*a^3*b^2*c^2 - 24*a^4*c^3)*x^2 + 24*(a^2*c^4*x^8 + 2*a^2*b*c^3*x^6 + 2*a^ 3*b*c^2*x^2 + a^4*c^2 + (a^2*b^2*c^2 + 2*a^3*c^3)*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^2 - 12 *a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48* a^2*b^2*c^6 - 64*a^3*c^7)*x^8 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^6 + (b^8*c^2 - 10*a*b^6*c^3 + 24*a^2*b^4*c^4 + 32*a^3*b ^2*c^5 - 128*a^4*c^6)*x^4 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*...
Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (112) = 224\).
Time = 1.99 (sec) , antiderivative size = 554, normalized size of antiderivative = 4.58 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx=- 3 a^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x^{2} + \frac {- 192 a^{5} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{4} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a^{3} b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b}{6 a^{2} c} \right )} + 3 a^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x^{2} + \frac {192 a^{5} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{4} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a^{3} b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 3 a^{2} b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b}{6 a^{2} c} \right )} + \frac {10 a^{3} b c - a^{2} b^{3} + x^{6} \left (- 20 a^{2} c^{3} + 16 a b^{2} c^{2} - 2 b^{4} c\right ) + x^{4} \cdot \left (2 a^{2} b c^{2} + 8 a b^{3} c - b^{5}\right ) + x^{2} \left (- 12 a^{3} c^{2} + 20 a^{2} b^{2} c - 2 a b^{4}\right )}{64 a^{4} c^{4} - 32 a^{3} b^{2} c^{3} + 4 a^{2} b^{4} c^{2} + x^{8} \cdot \left (64 a^{2} c^{6} - 32 a b^{2} c^{5} + 4 b^{4} c^{4}\right ) + x^{6} \cdot \left (128 a^{2} b c^{5} - 64 a b^{3} c^{4} + 8 b^{5} c^{3}\right ) + x^{4} \cdot \left (128 a^{3} c^{5} - 24 a b^{4} c^{3} + 4 b^{6} c^{2}\right ) + x^{2} \cdot \left (128 a^{3} b c^{4} - 64 a^{2} b^{3} c^{3} + 8 a b^{5} c^{2}\right )} \] Input:
integrate(x**9/(c*x**4+b*x**2+a)**3,x)
Output:
-3*a**2*sqrt(-1/(4*a*c - b**2)**5)*log(x**2 + (-192*a**5*c**3*sqrt(-1/(4*a *c - b**2)**5) + 144*a**4*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) - 36*a**3*b **4*c*sqrt(-1/(4*a*c - b**2)**5) + 3*a**2*b**6*sqrt(-1/(4*a*c - b**2)**5) + 3*a**2*b)/(6*a**2*c)) + 3*a**2*sqrt(-1/(4*a*c - b**2)**5)*log(x**2 + (19 2*a**5*c**3*sqrt(-1/(4*a*c - b**2)**5) - 144*a**4*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) + 36*a**3*b**4*c*sqrt(-1/(4*a*c - b**2)**5) - 3*a**2*b**6*sqr t(-1/(4*a*c - b**2)**5) + 3*a**2*b)/(6*a**2*c)) + (10*a**3*b*c - a**2*b**3 + x**6*(-20*a**2*c**3 + 16*a*b**2*c**2 - 2*b**4*c) + x**4*(2*a**2*b*c**2 + 8*a*b**3*c - b**5) + x**2*(-12*a**3*c**2 + 20*a**2*b**2*c - 2*a*b**4))/( 64*a**4*c**4 - 32*a**3*b**2*c**3 + 4*a**2*b**4*c**2 + x**8*(64*a**2*c**6 - 32*a*b**2*c**5 + 4*b**4*c**4) + x**6*(128*a**2*b*c**5 - 64*a*b**3*c**4 + 8*b**5*c**3) + x**4*(128*a**3*c**5 - 24*a*b**4*c**3 + 4*b**6*c**2) + x**2* (128*a**3*b*c**4 - 64*a**2*b**3*c**3 + 8*a*b**5*c**2))
Exception generated. \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^9/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
Output:
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 = 1.20 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.75 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {6 \, a^{2} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{4} c x^{6} - 16 \, a b^{2} c^{2} x^{6} + 20 \, a^{2} c^{3} x^{6} + b^{5} x^{4} - 8 \, a b^{3} c x^{4} - 2 \, a^{2} b c^{2} x^{4} + 2 \, a b^{4} x^{2} - 20 \, a^{2} b^{2} c x^{2} + 12 \, a^{3} c^{2} x^{2} + a^{2} b^{3} - 10 \, a^{3} b c}{4 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} \] Input:
integrate(x^9/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
Output:
6*a^2*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*(2*b^4*c*x^6 - 16*a*b^2*c^2*x^6 + 20*a^2*c^ 3*x^6 + b^5*x^4 - 8*a*b^3*c*x^4 - 2*a^2*b*c^2*x^4 + 2*a*b^4*x^2 - 20*a^2*b ^2*c*x^2 + 12*a^3*c^2*x^2 + a^2*b^3 - 10*a^3*b*c)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^4 + b*x^2 + a)^2)
Time = 0.28 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.67 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {6\,a^2\,\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {36\,a^3\,c^2}{{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {36\,a^3\,b\,\left (16\,a^2\,b\,c^4-8\,a\,b^3\,c^3+b^5\,c^2\right )}{{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {72\,a^4\,b\,c^2}{{\left (4\,a\,c-b^2\right )}^{15/2}}\right )\,\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )}{72\,a^4\,c^2}\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {x^6\,\left (10\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a^2\,\left (b^3-10\,a\,b\,c\right )}{4\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {x^4\,\left (2\,a^2\,b\,c^2+8\,a\,b^3\,c-b^5\right )}{4\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,x^2\,\left (6\,a^2\,c^2-10\,a\,b^2\,c+b^4\right )}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^8+2\,a\,b\,x^2+2\,b\,c\,x^6} \] Input:
int(x^9/(a + b*x^2 + c*x^4)^3,x)
Output:
(6*a^2*atan(((x^2*((36*a^3*c^2)/((4*a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 - 8 *a*b^2*c)) + (36*a^3*b*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4))/((4*a*c - b ^2)^(15/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))) + (72*a^4*b*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))/(72*a^4*c^2)))/(4*a*c - b^2)^(5/2) - ((x^6*(b^4 + 10*a^2*c^ 2 - 8*a*b^2*c))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a^2*(b^3 - 10*a*b* c))/(4*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^4*(2*a^2*b*c^2 - b^5 + 8*a *b^3*c))/(4*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a*x^2*(b^4 + 6*a^2*c^2 - 10*a*b^2*c))/(2*c^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)
Time = 0.25 (sec) , antiderivative size = 1137, normalized size of antiderivative = 9.40 \[ \int \frac {x^9}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^9/(c*x^4+b*x^2+a)^3,x)
Output:
( - 24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt( 2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b* c - 48*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt( 2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b* *2*c*x**2 - 48*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*ata n((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b)) *a**3*b*c**2*x**4 - 24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt( a) + b))*a**2*b**3*c*x**4 - 48*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)* sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt( c)*sqrt(a) + b))*a**2*b**2*c**2*x**6 - 24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt (2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/s qrt(2*sqrt(c)*sqrt(a) + b))*a**2*b*c**3*x**8 - 24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt (c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b*c - 48*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt (c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c*x**2 - 48*sqrt(2*sqrt(c)*s qrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b*c**2*x**4 - 24*sqrt(2*s qrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sq...