Integrand size = 20, antiderivative size = 569 \[ \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x^{5/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (24 a b+\left (7 b^2+20 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {3 \left (7 b^3+36 a b c+\sqrt {b^2-4 a c} \left (7 b^2+20 a c\right )\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (7 b^2+20 a c-\frac {7 b^3+36 a b c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (7 b^3+36 a b c+\sqrt {b^2-4 a c} \left (7 b^2+20 a c\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 \left (7 b^2+20 a c-\frac {7 b^3+36 a b c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} \sqrt [4]{c} \left (b^2-4 a c\right )^2 \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \]
1/4*x^(5/2)*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2-3/64*arctan(2^(1/4) *c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(7*b^2+20*a*c+(-36*a*b*c-7 *b^3)/(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(1/4)/(-4*a*c+b^2)^2/(-b+(-4*a*c+b^2)^ (1/2))^(3/4)-3/64*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^ (1/4))*(7*b^2+20*a*c+(-36*a*b*c-7*b^3)/(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(1/4) /(-4*a*c+b^2)^2/(-b+(-4*a*c+b^2)^(1/2))^(3/4)-3/64*arctan(2^(1/4)*c^(1/4)* x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(7*b^3+36*a*b*c+(20*a*c+7*b^2)*(-4* a*c+b^2)^(1/2))*2^(3/4)/c^(1/4)/(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2)) ^(3/4)-3/64*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)) *(7*b^3+36*a*b*c+(20*a*c+7*b^2)*(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(1/4)/(-4*a* c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)+1/16*(24*a*b+(20*a*c+7*b^2)*x^2 )*x^(1/2)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.59 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.70 \[ \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{64} \left (\frac {4 \sqrt {x} \left (12 a^2 \left (2 b-c x^2\right )+b^2 x^4 \left (11 b+7 c x^2\right )+a \left (39 b^2 x^2+28 b c x^4+20 c^2 x^6\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2}+\frac {24 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )-5 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4+2 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{a c^2 \left (-b^2+4 a c\right )}+\frac {3 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {8 b^5 \log \left (\sqrt {x}-\text {$\#$1}\right )-72 a b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+152 a^2 b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+8 b^4 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-9 a b^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-44 a^2 c^3 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{a c^2 \left (b^2-4 a c\right )^2}\right ) \]
((4*Sqrt[x]*(12*a^2*(2*b - c*x^2) + b^2*x^4*(11*b + 7*c*x^2) + a*(39*b^2*x ^2 + 28*b*c*x^4 + 20*c^2*x^6)))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^2) + (24*RootSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 5*a*b*c*Log[S qrt[x] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 + 2*a*c^2*Log[Sqrt[x] - #1]*#1 ^4)/(b*#1^3 + 2*c*#1^7) & ])/(a*c^2*(-b^2 + 4*a*c)) + (3*RootSum[a + b*#1^ 4 + c*#1^8 & , (8*b^5*Log[Sqrt[x] - #1] - 72*a*b^3*c*Log[Sqrt[x] - #1] + 1 52*a^2*b*c^2*Log[Sqrt[x] - #1] + 8*b^4*c*Log[Sqrt[x] - #1]*#1^4 - 9*a*b^2* c^2*Log[Sqrt[x] - #1]*#1^4 - 44*a^2*c^3*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(a*c^2*(b^2 - 4*a*c)^2))/64
Time = 0.71 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1435, 1701, 1822, 27, 1752, 756, 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.
| \(\displaystyle \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1435 |
| \(\displaystyle 2 \int \frac {x^6}{\left (c x^4+b x^2+a\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 1701 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int \frac {x^2 \left (10 a-7 b x^2\right )}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1822 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {-\frac {\int -\frac {3 \left (8 a b-\left (7 b^2+20 a c\right ) x^2\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {3 \int \frac {8 a b-\left (7 b^2+20 a c\right ) x^2}{c x^4+b x^2+a}d\sqrt {x}}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {3 \left (-\frac {1}{2} \left (-\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}d\sqrt {x}-\frac {1}{2} \left (\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}d\sqrt {x}\right )}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {3 \left (-\frac {1}{2} \left (\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}\right )-\frac {1}{2} \left (-\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}\right )\right )}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {3 \left (-\frac {1}{2} \left (\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )-\frac {1}{2} \left (-\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )\right )}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (\frac {x^{5/2} \left (2 a+b x^2\right )}{8 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\frac {3 \left (-\frac {1}{2} \left (\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )-\frac {1}{2} \left (-\frac {36 a b c+7 b^3}{\sqrt {b^2-4 a c}}+20 a c+7 b^2\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )\right )}{4 \left (b^2-4 a c\right )}-\frac {\sqrt {x} \left (x^2 \left (20 a c+7 b^2\right )+24 a b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 \left (b^2-4 a c\right )}\right )\) |
2*((x^(5/2)*(2*a + b*x^2))/(8*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (-1/4 *(Sqrt[x]*(24*a*b + (7*b^2 + 20*a*c)*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x ^4)) + (3*(-1/2*((7*b^2 + 20*a*c + (7*b^3 + 36*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^(1/4 )*c^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2^(1/4)*c^(1/4)*Sqrt [x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b - Sqrt[b^2 - 4*a *c])^(3/4)))) - ((7*b^2 + 20*a*c - (7*b^3 + 36*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^(1/4 )*c^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2^(1/4)*c^(1/4)*Sqrt [x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b + Sqrt[b^2 - 4*a *c])^(3/4))))/2))/(4*(b^2 - 4*a*c)))/(8*(b^2 - 4*a*c)))
3.11.82.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) 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[((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[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.53 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.42
| method | result | size |
| derivativedivides | \(\frac {\frac {3 a^{2} b \sqrt {x}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 \left (4 a c -13 b^{2}\right ) a \,x^{\frac {5}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 b \left (28 a c +11 b^{2}\right ) x^{\frac {9}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}+\frac {2 c \left (20 a c +7 b^{2}\right ) x^{\frac {13}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (20 a c +7 b^{2}\right ) \textit {\_R}^{4}-8 a b \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(241\) |
| default | \(\frac {\frac {3 a^{2} b \sqrt {x}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 \left (4 a c -13 b^{2}\right ) a \,x^{\frac {5}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 b \left (28 a c +11 b^{2}\right ) x^{\frac {9}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}+\frac {2 c \left (20 a c +7 b^{2}\right ) x^{\frac {13}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (20 a c +7 b^{2}\right ) \textit {\_R}^{4}-8 a b \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(241\) |
2*(3/4*a^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)-3/32*(4*a*c-13*b^2)*a/(16* a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)+1/32*b*(28*a*c+11*b^2)/(16*a^2*c^2-8*a*b^2* c+b^4)*x^(9/2)+1/32*c*(20*a*c+7*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(13/2))/ (c*x^4+b*x^2+a)^2+3/64/(16*a^2*c^2-8*a*b^2*c+b^4)*sum(((20*a*c+7*b^2)*_R^4 -8*a*b)/(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. 12814 vs. \(2 (469) = 938\).
Time = 3.42 (sec) , antiderivative size = 12814, normalized size of antiderivative = 22.52 \[ \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {11}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
-1/16*(24*b*c^2*x^(17/2) + (41*b^2*c - 20*a*c^2)*x^(13/2) + (13*b^3 + 20*a *b*c)*x^(9/2) + 3*(3*a*b^2 + 4*a^2*c)*x^(5/2))/((b^4*c^2 - 8*a*b^2*c^3 + 1 6*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(3/32*(8*b*c*x^(7/2) + 5*(3*b^2 + 4*a*c)*x^(3/2))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^ 2 + 16*a^2*c^3)*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^2), x)
\[ \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{\frac {11}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
Time = 17.52 (sec) , antiderivative size = 45495, normalized size of antiderivative = 79.96 \[ \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
((x^(9/2)*(11*b^3 + 28*a*b*c))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*x^ (5/2)*(13*a*b^2 - 4*a^2*c))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (c*x^(13 /2)*(20*a*c + 7*b^2))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*a^2*b*x^(1/ 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((((((3*((81*(2401*b^4*(-(4*a*c - b^2)^25) ^(1/2) - 2401*b^29 - 704643072000*a^14*b*c^14 + 1323600*a^2*b^25*c^2 - 282 43200*a^3*b^23*c^3 + 271415040*a^4*b^21*c^4 - 1437284352*a^5*b^19*c^5 + 39 89852160*a^6*b^17*c^6 - 2793799680*a^7*b^15*c^7 - 13327073280*a^8*b^13*c^8 + 19977994240*a^9*b^11*c^9 + 66059239424*a^10*b^9*c^10 - 143696855040*a^1 1*b^7*c^11 - 230770606080*a^12*b^5*c^12 + 887850270720*a^13*b^3*c^13 + 100 00*a^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 9400*a*b^27*c + 9400*a*b^2*c*(-(4*a *c - b^2)^25)^(1/2)))/(33554432*(b^40*c + 1099511627776*a^20*c^21 - 80*a*b ^38*c^2 + 3040*a^2*b^36*c^3 - 72960*a^3*b^34*c^4 + 1240320*a^4*b^32*c^5 - 15876096*a^5*b^30*c^6 + 158760960*a^6*b^28*c^7 - 1270087680*a^7*b^26*c^8 + 8255569920*a^8*b^24*c^9 - 44029706240*a^9*b^22*c^10 + 193730707456*a^10*b ^20*c^11 - 704475299840*a^11*b^18*c^12 + 2113425899520*a^12*b^16*c^13 - 52 02279137280*a^13*b^14*c^14 + 10404558274560*a^14*b^12*c^15 - 1664729323929 6*a^15*b^10*c^16 + 20809116549120*a^16*b^8*c^17 - 19585050869760*a^17*b^6* c^18 + 13056700579840*a^18*b^4*c^19 - 5497558138880*a^19*b^2*c^20)))^(1/4) *(351843720888320*a^13*c^15 + 251658240*a^2*b^22*c^4 - 9730785280*a^3*b...