Integrand size = 20, antiderivative size = 544 \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=-\frac {b x^{3/2}}{2 c \left (b^2-4 a c\right )}+\frac {x^{7/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^3-20 a b c+\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\left (3 b^3-20 a b c-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\left (3 b^3-20 a b c+\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\left (3 b^3-20 a b c-\left (3 b^2-14 a c\right ) \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{7/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \]
-1/2*b*x^(3/2)/c/(-4*a*c+b^2)+1/2*x^(7/2)*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+ b*x^2+a)-1/8*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)) *(3*b^3-20*a*b*c-(-14*a*c+3*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(7/4)/(-4*a *c+b^2)^(3/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)+1/8*arctanh(2^(1/4)*c^(1/4)*x^ (1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(3*b^3-20*a*b*c-(-14*a*c+3*b^2)*(-4*a *c+b^2)^(1/2))*2^(1/4)/c^(7/4)/(-4*a*c+b^2)^(3/2)/(-b+(-4*a*c+b^2)^(1/2))^ (1/4)+1/8*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(3 *b^3-20*a*b*c+(-14*a*c+3*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(7/4)/(-4*a*c+ b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)-1/8*arctanh(2^(1/4)*c^(1/4)*x^(1/ 2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(3*b^3-20*a*b*c+(-14*a*c+3*b^2)*(-4*a*c+ b^2)^(1/2))*2^(1/4)/c^(7/4)/(-4*a*c+b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/ 4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.48 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.43 \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=-\frac {4 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b \log \left (\sqrt {x}-\text {$\#$1}\right )-c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]+\frac {\frac {4 c x^{3/2} \left (a b+b^2 x^2-2 a c x^2\right )}{a+b x^2+c x^4}+\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-4 b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+13 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-2 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{b^2-4 a c}}{8 c^2} \]
-1/8*(4*RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[Sqrt[x] - #1] - c*Log[Sqrt[ x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ] + ((4*c*x^(3/2)*(a*b + b^2*x^2 - 2*a* c*x^2))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (-4*b^3*Log[ Sqrt[x] - #1] + 13*a*b*c*Log[Sqrt[x] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 - 2*a*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(b^2 - 4*a*c))/c^ 2
Time = 0.68 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.81, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1435, 1701, 1826, 27, 1834, 27, 827, 218, 221}
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.
| \(\displaystyle \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1435 |
| \(\displaystyle 2 \int \frac {x^7}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 1701 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {x^3 \left (3 b x^2+14 a\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1826 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {b x^{3/2}}{c}-\frac {\int \frac {3 x \left (\left (3 b^2-14 a c\right ) x^2+3 a b\right )}{c x^4+b x^2+a}d\sqrt {x}}{3 c}}{4 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {b x^{3/2}}{c}-\frac {\int \frac {x \left (\left (3 b^2-14 a c\right ) x^2+3 a b\right )}{c x^4+b x^2+a}d\sqrt {x}}{c}}{4 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {b x^{3/2}}{c}-\frac {\frac {1}{2} \left (-\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \int \frac {2 x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}+\frac {1}{2} \left (\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \int \frac {2 x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{c}}{4 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {b x^{3/2}}{c}-\frac {\left (-\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \int \frac {x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}+\left (\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \int \frac {x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{c}}{4 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {b x^{3/2}}{c}-\frac {\left (\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )+\left (-\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{c}}{4 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {b x^{3/2}}{c}-\frac {\left (\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )+\left (-\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{c}}{4 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {b x^{3/2}}{c}-\frac {\left (\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )+\left (-\frac {3 b^3-20 a b c}{\sqrt {b^2-4 a c}}-14 a c+3 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{c}}{4 \left (b^2-4 a c\right )}\right )\) |
2*((x^(7/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((b*x^( 3/2))/c - ((3*b^2 - 14*a*c + (3*b^3 - 20*a*b*c)/Sqrt[b^2 - 4*a*c])*(ArcTan [(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3 /4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(- b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a*c])^ (1/4))) + (3*b^2 - 14*a*c - (3*b^3 - 20*a*b*c)/Sqrt[b^2 - 4*a*c])*(ArcTan[ (2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/ 4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^( 1/4))))/c)/(4*(b^2 - 4*a*c)))
3.11.71.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/d Subst[Int[x^(k*(m + 1) - 1)*(a + b *(x^(2*k)/d^2) + c*(x^(4*k)/d^4))^p, x], x, (d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[(-d^(2*n - 1))*(d*x)^(m - 2*n + 1)*(2*a + b*x^n)*((a + b*x ^n + c*x^(2*n))^(p + 1)/(n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[d^(2*n)/(n*(p + 1)*(b^2 - 4*a*c)) Int[(d*x)^(m - 2*n)*(2*a*(m - 2*n + 1) + b*(m + n*(2 *p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c , d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1 ] && GtQ[m, 2*n - 1]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*( m + n*(2*p + 1) + 1)) Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a* e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x] , x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && Intege rQ[p]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ [{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n , 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.27
| method | result | size |
| derivativedivides | \(\frac {-\frac {\left (2 a c -b^{2}\right ) x^{\frac {7}{2}}}{2 c \left (4 a c -b^{2}\right )}+\frac {a b \,x^{\frac {3}{2}}}{2 c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-14 a c +3 b^{2}\right ) \textit {\_R}^{6}+3 b \,\textit {\_R}^{2} a \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{8 c \left (4 a c -b^{2}\right )}\) | \(149\) |
| default | \(\frac {-\frac {\left (2 a c -b^{2}\right ) x^{\frac {7}{2}}}{2 c \left (4 a c -b^{2}\right )}+\frac {a b \,x^{\frac {3}{2}}}{2 c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-14 a c +3 b^{2}\right ) \textit {\_R}^{6}+3 b \,\textit {\_R}^{2} a \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{8 c \left (4 a c -b^{2}\right )}\) | \(149\) |
2*(-1/4*(2*a*c-b^2)/c/(4*a*c-b^2)*x^(7/2)+1/4*a*b/c/(4*a*c-b^2)*x^(3/2))/( c*x^4+b*x^2+a)-1/8/c/(4*a*c-b^2)*sum(((-14*a*c+3*b^2)*_R^6+3*b*_R^2*a)/(2* _R^7*c+_R^3*b)*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 14601 vs. \(2 (440) = 880\).
Time = 21.50 (sec) , antiderivative size = 14601, normalized size of antiderivative = 26.84 \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {x^{\frac {13}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
-1/2*((b^2 - 2*a*c)*x^(7/2) + a*b*x^(3/2))/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^ 2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^2) + integrate(1/4*((3*b^2 - 14*a* c)*x^(5/2) + 3*a*b*sqrt(x))/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^2), x)
\[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {x^{\frac {13}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
Time = 15.95 (sec) , antiderivative size = 28774, normalized size of antiderivative = 52.89 \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
- ((x^(7/2)*(2*a*c - b^2))/(2*c*(4*a*c - b^2)) - (a*b*x^(3/2))/(2*c*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) - atan(((((46036680704*a^12*c^12 - 110592*a^ 3*b^18*c^3 + 4423680*a^4*b^16*c^4 - 77783040*a^5*b^14*c^5 + 788037632*a^6* b^12*c^6 - 5065015296*a^7*b^10*c^7 + 21401960448*a^8*b^8*c^8 - 59401830400 *a^9*b^6*c^9 + 104312340480*a^10*b^4*c^10 - 104991817728*a^11*b^2*c^11)/(1 28*(16384*a^7*c^10 - b^14*c^3 + 28*a*b^12*c^4 - 336*a^2*b^10*c^5 + 2240*a^ 3*b^8*c^6 - 8960*a^4*b^6*c^7 + 21504*a^5*b^4*c^8 - 28672*a^6*b^2*c^9)) - ( x^(1/2)*((81*b^8*(-(4*a*c - b^2)^15)^(1/2) - 81*b^23 + 741801984*a^11*b*c^ 11 - 90126*a^2*b^19*c^2 + 1201623*a^3*b^17*c^3 - 10588384*a^4*b^15*c^4 + 6 4704576*a^5*b^13*c^5 - 279571968*a^6*b^11*c^6 + 853174784*a^7*b^9*c^7 - 17 99626752*a^8*b^7*c^8 + 2494119936*a^9*b^5*c^9 - 2038693888*a^10*b^3*c^10 + 9604*a^4*c^4*(-(4*a*c - b^2)^15)^(1/2) + 4023*a*b^21*c + 10746*a^2*b^4*c^ 2*(-(4*a*c - b^2)^15)^(1/2) - 26313*a^3*b^2*c^3*(-(4*a*c - b^2)^15)^(1/2) - 1593*a*b^6*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^19 + b^24 *c^7 - 48*a*b^22*c^8 + 1056*a^2*b^20*c^9 - 14080*a^3*b^18*c^10 + 126720*a^ 4*b^16*c^11 - 811008*a^5*b^14*c^12 + 3784704*a^6*b^12*c^13 - 12976128*a^7* b^10*c^14 + 32440320*a^8*b^8*c^15 - 57671680*a^9*b^6*c^16 + 69206016*a^10* b^4*c^17 - 50331648*a^11*b^2*c^18)))^(1/4)*(6576668672*a^11*c^13 + 36864*a ^3*b^16*c^5 - 1302528*a^4*b^14*c^6 + 20480000*a^5*b^12*c^7 - 185991168*a^6 *b^10*c^8 + 1061683200*a^7*b^8*c^9 - 3886022656*a^8*b^6*c^10 + 88835358...