Integrand size = 20, antiderivative size = 569 \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (24 a b+\left (5 b^2+28 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 a c\right )\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^2 \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 a c\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^2 \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \]
1/4*x^(7/2)*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/16*x^(3/2)*(24*a* b+(28*a*c+5*b^2)*x^2)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/64*arctan(2^(1/4)*c ^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^2+28*a*c+(-172*a*b*c-5* b^3)/(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^2/(-b+(-4*a*c+b^2)^( 1/2))^(1/4)-1/64*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^( 1/4))*(5*b^2+28*a*c+(-172*a*b*c-5*b^3)/(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(3/4) /(-4*a*c+b^2)^2/(-b+(-4*a*c+b^2)^(1/2))^(1/4)+1/64*arctan(2^(1/4)*c^(1/4)* x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^3+172*a*b*c+(28*a*c+5*b^2)*(-4 *a*c+b^2)^(1/2))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2) )^(1/4)-1/64*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4) )*(5*b^3+172*a*b*c+(28*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(3/4)/(-4* a*c+b^2)^(5/2)/(-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 = 0.70 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.69 \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{64} \left (\frac {4 x^{3/2} \left (4 a^2 \left (6 b-c x^2\right )+b^2 x^4 \left (9 b+5 c x^2\right )+a \left (37 b^2 x^2+36 b c x^4+28 c^2 x^6\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2}+\frac {8 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )-13 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4+2 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{a c^2 \left (-b^2+4 a c\right )}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {8 b^5 \log \left (\sqrt {x}-\text {$\#$1}\right )-136 a b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+344 a^2 b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+8 b^4 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-11 a b^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-36 a^2 c^3 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{a c^2 \left (b^2-4 a c\right )^2}\right ) \]
((4*x^(3/2)*(4*a^2*(6*b - c*x^2) + b^2*x^4*(9*b + 5*c*x^2) + a*(37*b^2*x^2 + 36*b*c*x^4 + 28*c^2*x^6)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2) + (8 *RootSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 13*a*b*c*Log[Sqr t[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) & ])/(a*c^2*(-b^2 + 4*a*c)) + RootSum[a + b*#1^4 + c*# 1^8 & , (8*b^5*Log[Sqrt[x] - #1] - 136*a*b^3*c*Log[Sqrt[x] - #1] + 344*a^2 *b*c^2*Log[Sqrt[x] - #1] + 8*b^4*c*Log[Sqrt[x] - #1]*#1^4 - 11*a*b^2*c^2*L og[Sqrt[x] - #1]*#1^4 - 36*a^2*c^3*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^ 5) & ]/(a*c^2*(b^2 - 4*a*c)^2))/64
Time = 0.74 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1435, 1701, 1822, 25, 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.
| \(\displaystyle \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1435 |
| \(\displaystyle 2 \int \frac {x^7}{\left (c x^4+b x^2+a\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 1701 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int \frac {x^3 \left (14 a-5 b x^2\right )}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1822 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {\int -\frac {x \left (72 a b-\left (5 b^2+28 a c\right ) x^2\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x^2 \left (28 a c+5 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {\int \frac {x \left (72 a b-\left (5 b^2+28 a c\right ) x^2\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x^2 \left (28 a c+5 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {-\frac {1}{2} \left (-\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \int \frac {2 x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}-\frac {1}{2} \left (\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \int \frac {2 x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{4 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x^2 \left (28 a c+5 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {-\left (\left (-\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \int \frac {x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}\right )-\left (\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \int \frac {x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{4 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x^2 \left (28 a c+5 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {-\left (\left (\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )\right )-\left (-\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{4 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x^2 \left (28 a c+5 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {-\left (\left (\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )\right )-\left (-\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{4 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x^2 \left (28 a c+5 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (\frac {x^{7/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {-\left (\left (\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )\right )-\left (-\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{4 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (x^2 \left (28 a c+5 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
2*((x^(7/2)*(2*a + b*x^2))/(8*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (-1/4 *(x^(3/2)*(24*a*b + (5*b^2 + 28*a*c)*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x ^4)) + (-((5*b^2 + 28*a*c + (5*b^3 + 172*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)))) - (5*b^2 + 28*a*c - (5*b^3 + 172*a*b*c)/Sqrt[b^2 - 4*a*c])*(ArcTa n[(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))))/(4*(b^2 - 4*a*c)))/(8*(b^2 - 4*a*c)))
3.11.81.3.1 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[f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*((b*d - 2*a*e - (b*e - 2*c*d)*x^n)/(n*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[f^n/(n*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^(m - n)* (a + b*x^n + c*x^(2*n))^(p + 1)*Simp[(n - m - 1)*(b*d - 2*a*e) + (2*n*p + 2 *n + m + 1)*(b*e - 2*c*d)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] & & EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 1] && IntegerQ[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 = 2.55 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.43
| method | result | size |
| derivativedivides | \(\frac {\frac {3 a^{2} b \,x^{\frac {3}{2}}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (4 a c -37 b^{2}\right ) x^{\frac {7}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {9 b \left (4 a c +b^{2}\right ) x^{\frac {11}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 c \left (28 a c +5 b^{2}\right ) x^{\frac {15}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-28 a c -5 b^{2}\right ) \textit {\_R}^{6}+72 b \,\textit {\_R}^{2} a \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(242\) |
| default | \(\frac {\frac {3 a^{2} b \,x^{\frac {3}{2}}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (4 a c -37 b^{2}\right ) x^{\frac {7}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {9 b \left (4 a c +b^{2}\right ) x^{\frac {11}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 c \left (28 a c +5 b^{2}\right ) x^{\frac {15}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-28 a c -5 b^{2}\right ) \textit {\_R}^{6}+72 b \,\textit {\_R}^{2} a \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(242\) |
2*(3/4*a^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)-1/32*a*(4*a*c-37*b^2)/(16* a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)+9/32*b*(4*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^ 4)*x^(11/2)+1/32*c*(28*a*c+5*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(15/2))/(c* x^4+b*x^2+a)^2-1/64/(16*a^2*c^2-8*a*b^2*c+b^4)*sum(((-28*a*c-5*b^2)*_R^6+7 2*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. 19805 vs. \(2 (469) = 938\).
Time = 44.23 (sec) , antiderivative size = 19805, normalized size of antiderivative = 34.81 \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {13}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
1/16*((5*b^2*c + 28*a*c^2)*x^(15/2) + 9*(b^3 + 4*a*b*c)*x^(11/2) + 24*a^2* b*x^(3/2) + (37*a*b^2 - 4*a^2*c)*x^(7/2))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2 *c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b ^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2* b^3*c + 16*a^3*b*c^2)*x^2) + integrate(1/32*((5*b^2 + 28*a*c)*x^(5/2) - 72 *a*b*sqrt(x))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2 + 1 6*a^2*c^3)*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^2), x)
\[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {13}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
Time = 16.98 (sec) , antiderivative size = 39697, normalized size of antiderivative = 69.77 \[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
((9*x^(11/2)*(b^3 + 4*a*b*c))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^(7/ 2)*(37*a*b^2 - 4*a^2*c))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (c*x^(15/2) *(28*a*c + 5*b^2))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*a^2*b*x^(3/2)) /(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) - atan(((((386183668047020032*a^16*c^16 + 209715200 0*a^3*b^26*c^3 - 7615312560128*a^4*b^24*c^4 + 295658569334784*a^5*b^22*c^5 - 5154027327193088*a^6*b^20*c^6 + 52821290217635840*a^7*b^18*c^7 - 350572 668266741760*a^8*b^16*c^8 + 1560295235622273024*a^9*b^14*c^9 - 46282369669 60300032*a^10*b^12*c^10 + 8604139182719238144*a^11*b^10*c^11 - 79240263697 53743360*a^12*b^8*c^12 - 1942353261163970560*a^13*b^6*c^13 + 1182321565924 2749952*a^14*b^4*c^14 - 8419198028392431616*a^15*b^2*c^15)/(268435456*(b^2 8 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 + 256256* a^4*b^20*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7 *b^14*c^7 + 196804608*a^8*b^12*c^8 - 524812288*a^9*b^10*c^9 + 1049624576*a ^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^12 - 93952 4096*a^13*b^2*c^13 - 56*a*b^26*c)) - (x^(1/2)*(-(625*b^31 + 625*b^6*(-(4*a *c - b^2)^25)^(1/2) - 15192104632320*a^15*b*c^15 - 89000*a^2*b^27*c^2 + 27 186416*a^3*b^25*c^3 - 1342297600*a^4*b^23*c^4 + 25492409600*a^5*b^21*c^5 - 265188833280*a^6*b^19*c^6 + 1688816578560*a^7*b^17*c^7 - 6664504147968*a^ 8*b^15*c^8 + 14462970429440*a^9*b^13*c^9 - 4163326443520*a^10*b^11*c^10...