Integrand size = 20, antiderivative size = 594 \[ \int \frac {x^{5/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {x^{3/2} \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 x^{3/2} \left (b \left (b^2+4 a c\right )+c \left (b^2+12 a c\right ) x^2\right )}{16 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt [4]{c} \left (b^2+12 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 a b c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a \left (b^2-4 a c\right )^2 \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt [4]{c} \left (b^3-68 a b c+\sqrt {b^2-4 a c} \left (b^2+12 a c\right )\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {3 \sqrt [4]{c} \left (b^2+12 a c-\frac {b^3}{\sqrt {b^2-4 a c}}+\frac {68 a b c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a \left (b^2-4 a c\right )^2 \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt [4]{c} \left (b^3-68 a b c+\sqrt {b^2-4 a c} \left (b^2+12 a c\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \]
-1/4*x^(3/2)*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+3/16*x^(3/2)*(b*(4 *a*c+b^2)+c*(12*a*c+b^2)*x^2)/a/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+3/64*c^(1/4 )*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(b^2+12*a* c-b^3/(-4*a*c+b^2)^(1/2)+68*a*b*c/(-4*a*c+b^2)^(1/2))*2^(1/4)/a/(-4*a*c+b^ 2)^2/(-b-(-4*a*c+b^2)^(1/2))^(1/4)-3/64*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x^ (1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(b^2+12*a*c-b^3/(-4*a*c+b^2)^(1/2)+68 *a*b*c/(-4*a*c+b^2)^(1/2))*2^(1/4)/a/(-4*a*c+b^2)^2/(-b-(-4*a*c+b^2)^(1/2) )^(1/4)+3/64*c^(1/4)*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2) )^(1/4))*(b^3-68*a*b*c+(12*a*c+b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a/(-4*a*c+ b^2)^(5/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)-3/64*c^(1/4)*arctanh(2^(1/4)*c^(1 /4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(b^3-68*a*b*c+(12*a*c+b^2)*(-4* a*c+b^2)^(1/2))*2^(1/4)/a/(-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 = 0.36 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.36 \[ \int \frac {x^{5/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 x^{3/2} \left (3 b^2 x^2 \left (b+c x^2\right )^2+4 a^2 c \left (7 b+17 c x^2\right )+a \left (-b^3+7 b^2 c x^2+48 b c^2 x^4+36 c^3 x^6\right )\right )}{\left (a+b x^2+c x^4\right )^2}+3 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )-28 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4+12 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{64 a \left (b^2-4 a c\right )^2} \]
((4*x^(3/2)*(3*b^2*x^2*(b + c*x^2)^2 + 4*a^2*c*(7*b + 17*c*x^2) + a*(-b^3 + 7*b^2*c*x^2 + 48*b*c^2*x^4 + 36*c^3*x^6)))/(a + b*x^2 + c*x^4)^2 + 3*Roo tSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 28*a*b*c*Log[Sqrt[x] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 + 12*a*c^2*Log[Sqrt[x] - #1]*#1^4)/( b*#1 + 2*c*#1^5) & ])/(64*a*(b^2 - 4*a*c)^2)
Time = 0.77 (sec) , antiderivative size = 518, normalized size of antiderivative = 0.87, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1435, 1700, 27, 1824, 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.
| \(\displaystyle \int \frac {x^{5/2}}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1435 |
| \(\displaystyle 2 \int \frac {x^3}{\left (c x^4+b x^2+a\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 1700 |
| \(\displaystyle 2 \left (\frac {\int \frac {3 x \left (b-6 c x^2\right )}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {3 \int \frac {x \left (b-6 c x^2\right )}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1824 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {x^{3/2} \left (c x^2 \left (12 a c+b^2\right )+b \left (4 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {x \left (c \left (b^2+12 a c\right ) x^2+b \left (b^2-28 a c\right )\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}\right )}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {\int \frac {x \left (c \left (b^2+12 a c\right ) x^2+b \left (b^2-28 a c\right )\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (c x^2 \left (12 a c+b^2\right )+b \left (4 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {\frac {1}{2} c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \int \frac {2 x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}+\frac {1}{2} c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \int \frac {2 x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (c x^2 \left (12 a c+b^2\right )+b \left (4 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \int \frac {x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}+c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \int \frac {x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (c x^2 \left (12 a c+b^2\right )+b \left (4 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )+c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (c x^2 \left (12 a c+b^2\right )+b \left (4 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )+c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (c x^2 \left (12 a c+b^2\right )+b \left (4 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (\frac {3 \left (\frac {c \left (\frac {68 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )+c \left (-\frac {68 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+12 a c+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (c x^2 \left (12 a c+b^2\right )+b \left (4 a c+b^2\right )\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{8 \left (b^2-4 a c\right )}-\frac {x^{3/2} \left (b+2 c x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
2*(-1/8*(x^(3/2)*(b + 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3 *((x^(3/2)*(b*(b^2 + 4*a*c) + c*(b^2 + 12*a*c)*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (c*(b^2 + 12*a*c - b^3/Sqrt[b^2 - 4*a*c] + (68*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*(b^2 + 12*a*c + b^3/Sqrt[b^2 - 4*a* c] - (68*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))))/(4*a*(b^2 - 4*a*c))))/( 8*(b^2 - 4*a*c)))
3.11.85.3.1 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^(n - 1)*(d*x)^(m - n + 1)*(b + 2*c*x^n)*((a + b*x^n + c* x^(2*n))^(p + 1)/(n*(p + 1)*(b^2 - 4*a*c))), x] - Simp[d^n/(n*(p + 1)*(b^2 - 4*a*c)) Int[(d*x)^(m - n)*(b*(m - n + 1) + 2*c*(m + 2*n*(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, n - 1] && LeQ[m, 2*n - 1]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(a + b*x^n + c*x^ (2*n))^(p + 1)*((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n)/(a*f*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(a*n*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^m* (a + b*x^n + c*x^(2*n))^(p + 1)*Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + n*(2*p + 3) + 1)*(b*d - 2*a*e) *x^n, x], 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] && LtQ[p, -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 = 2.56 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.47
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b \left (28 a c -b^{2}\right ) x^{\frac {3}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}+\frac {\left (68 a^{2} c^{2}+7 a \,b^{2} c +3 b^{4}\right ) x^{\frac {7}{2}}}{16 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 c b \left (8 a c +b^{2}\right ) x^{\frac {11}{2}}}{8 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 c^{2} \left (12 a c +b^{2}\right ) x^{\frac {15}{2}}}{16 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (-12 a c -b^{2}\right ) \textit {\_R}^{6}+b \left (28 a c -b^{2}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(281\) |
| default | \(\frac {\frac {2 b \left (28 a c -b^{2}\right ) x^{\frac {3}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}+\frac {\left (68 a^{2} c^{2}+7 a \,b^{2} c +3 b^{4}\right ) x^{\frac {7}{2}}}{16 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 c b \left (8 a c +b^{2}\right ) x^{\frac {11}{2}}}{8 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 c^{2} \left (12 a c +b^{2}\right ) x^{\frac {15}{2}}}{16 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (-12 a c -b^{2}\right ) \textit {\_R}^{6}+b \left (28 a c -b^{2}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(281\) |
2*(1/32*b*(28*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)+1/32*(68*a^2*c^2 +7*a*b^2*c+3*b^4)/a/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)+3/16/a*c*b*(8*a*c+b ^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(11/2)+3/32*c^2*(12*a*c+b^2)/a/(16*a^2*c^ 2-8*a*b^2*c+b^4)*x^(15/2))/(c*x^4+b*x^2+a)^2-3/64/a/(16*a^2*c^2-8*a*b^2*c+ b^4)*sum((c*(-12*a*c-b^2)*_R^6+b*(28*a*c-b^2)*_R^2)/(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. 21124 vs. \(2 (492) = 984\).
Time = 62.28 (sec) , antiderivative size = 21124, normalized size of antiderivative = 35.56 \[ \int \frac {x^{5/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^{5/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^{5/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {5}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
1/16*(3*(b^2*c^2 + 12*a*c^3)*x^(15/2) + 6*(b^3*c + 8*a*b*c^2)*x^(11/2) + ( 3*b^4 + 7*a*b^2*c + 68*a^2*c^2)*x^(7/2) - (a*b^3 - 28*a^2*b*c)*x^(3/2))/(( a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*x^8 + a^3*b^4 - 8*a^4*b^2*c + 16*a ^5*c^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x^6 + (a*b^6 - 6*a^2*b ^4*c + 32*a^4*c^3)*x^4 + 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2) + i ntegrate(3/32*((b^2*c + 12*a*c^2)*x^(5/2) + (b^3 - 28*a*b*c)*sqrt(x))/(a^2 *b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*x ^4 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2), x)
\[ \int \frac {x^{5/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {5}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
Time = 17.26 (sec) , antiderivative size = 42197, normalized size of antiderivative = 71.04 \[ \int \frac {x^{5/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
((3*x^(11/2)*(b^3*c + 8*a*b*c^2))/(8*(a*b^4 + 16*a^3*c^2 - 8*a^2*b^2*c)) - (x^(3/2)*(b^3 - 28*a*b*c))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^(7/2) *(3*b^4 + 68*a^2*c^2 + 7*a*b^2*c))/(16*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c^2*x^(15/2)*(12*a*c + b^2))/(16*(a*b^4 + 16*a^3*c^2 - 8*a^2*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(((((27* (3799912185593856*a^15*c^19 + 2097152*b^30*c^4 - 266338304*a*b^28*c^5 + 14 019461120*a^2*b^26*c^6 - 402594463744*a^3*b^24*c^7 + 7074549334016*a^4*b^2 2*c^8 - 81637933056000*a^5*b^20*c^9 + 645335479222272*a^6*b^18*c^10 - 3564 382621532160*a^7*b^16*c^11 + 13728399105196032*a^8*b^14*c^12 - 35694820362 027008*a^9*b^12*c^13 + 56529603635707904*a^10*b^10*c^14 - 3376765135644262 4*a^11*b^8*c^15 - 51215251621806080*a^12*b^6*c^16 + 114542723335192576*a^1 3*b^4*c^17 - 70615034782285824*a^14*b^2*c^18))/(33554432*(a^2*b^28 + 26843 5456*a^16*c^14 - 56*a^3*b^26*c + 1456*a^4*b^24*c^2 - 23296*a^5*b^22*c^3 + 256256*a^6*b^20*c^4 - 2050048*a^7*b^18*c^5 + 12300288*a^8*b^16*c^6 - 56229 888*a^9*b^14*c^7 + 196804608*a^10*b^12*c^8 - 524812288*a^11*b^10*c^9 + 104 9624576*a^12*b^8*c^10 - 1526726656*a^13*b^6*c^11 + 1526726656*a^14*b^4*c^1 2 - 939524096*a^15*b^2*c^13)) - (9*x^(1/2)*(-(81*(b^33 + b^8*(-(4*a*c - b^ 2)^25)^(1/2) - 471104225280*a^16*b*c^16 + 10509*a^2*b^29*c^2 - 394248*a^3* b^27*c^3 + 9219696*a^4*b^25*c^4 - 140233728*a^5*b^23*c^5 + 1424368896*a^6* b^21*c^6 - 9732052992*a^7*b^19*c^7 + 43376799744*a^8*b^17*c^8 - 1084930...