\(\int \frac {x^{9/2}}{(a+b x^2+c x^4)^2} \, dx\) [932]

Optimal result

Integrand size = 20, antiderivative size = 471 \[ \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x^{3/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (b^2+12 a c+b \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\left (b-\frac {b^2+12 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\left (b^2+12 a c+b \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\left (b-\frac {b^2+12 a c}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \] Output:

1/2*x^(3/2)*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/8*(b^2+12*a*c+b*(-4 
*a*c+b^2)^(1/2))*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1 
/4))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)+1/8* 
(b-(12*a*c+b^2)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4 
*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2 
))^(1/4)-1/8*(b^2+12*a*c+b*(-4*a*c+b^2)^(1/2))*arctanh(2^(1/4)*c^(1/4)*x^( 
1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^(3/2)/(-b 
-(-4*a*c+b^2)^(1/2))^(1/4)-1/8*(b-(12*a*c+b^2)/(-4*a*c+b^2)^(1/2))*arctanh 
(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(1/4)/c^(3/4)/(- 
4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.35 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.40 \[ \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {1}{8} \left (\frac {4 x^{3/2} \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {4 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {\log \left (\sqrt {x}-\text {$\#$1}\right )}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{c}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-4 b^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+10 a c \log \left (\sqrt {x}-\text {$\#$1}\right )+b c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{c \left (b^2-4 a c\right )}\right ) \] Input:

Integrate[x^(9/2)/(a + b*x^2 + c*x^4)^2,x]
 

Output:

((4*x^(3/2)*(2*a + b*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (4*RootSu 
m[a + b*#1^4 + c*#1^8 & , Log[Sqrt[x] - #1]/(b*#1 + 2*c*#1^5) & ])/c + Roo 
tSum[a + b*#1^4 + c*#1^8 & , (-4*b^2*Log[Sqrt[x] - #1] + 10*a*c*Log[Sqrt[x 
] - #1] + b*c*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(c*(b^2 - 4*a* 
c)))/8
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.86, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1435, 1701, 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.

\(\displaystyle \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx\)

\(\Big \downarrow \) 1435

\(\displaystyle 2 \int \frac {x^5}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 1701

\(\displaystyle 2 \left (\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {x \left (6 a-b x^2\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 1834

\(\displaystyle 2 \left (\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\frac {1}{2} \left (b-\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}\right ) \int \frac {2 x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}-\frac {1}{2} \left (\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}+b\right ) \int \frac {2 x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{4 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\left (\left (b-\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}\right ) \int \frac {x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}\right )-\left (\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}+b\right ) \int \frac {x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{4 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 827

\(\displaystyle 2 \left (\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\left (\left (\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )\right )-\left (b-\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{4 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle 2 \left (\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\left (\left (\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )\right )-\left (b-\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{4 \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle 2 \left (\frac {x^{3/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\left (\left (\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )\right )-\left (b-\frac {12 a c+b^2}{\sqrt {b^2-4 a c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{4 \left (b^2-4 a c\right )}\right )\)

Input:

Int[x^(9/2)/(a + b*x^2 + c*x^4)^2,x]
 

Output:

2*((x^(3/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (-((b + 
 (b^2 + 12*a*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)))) - (b - (b^2 + 12*a*c)/S 
qrt[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*(b^2 - 4*a*c)))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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]
 

rule 221

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]
 

rule 827

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]
 

rule 1435

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]
 

rule 1701

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]
 

rule 1834

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.26

method	result	size
derivativedivides	\(\frac {-\frac {b \,x^{\frac {7}{2}}}{2 \left (4 a c -b^{2}\right )}-\frac {a \,x^{\frac {3}{2}}}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{6} b +6 \textit {\_R}^{2} a \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{32 a c -8 b^{2}}\)	\(121\)
default	\(\frac {-\frac {b \,x^{\frac {7}{2}}}{2 \left (4 a c -b^{2}\right )}-\frac {a \,x^{\frac {3}{2}}}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{6} b +6 \textit {\_R}^{2} a \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{32 a c -8 b^{2}}\)	\(121\)

Input:

int(x^(9/2)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

2*(-1/4*b/(4*a*c-b^2)*x^(7/2)-1/2*a/(4*a*c-b^2)*x^(3/2))/(c*x^4+b*x^2+a)+1 
/8/(4*a*c-b^2)*sum((-_R^6*b+6*_R^2*a)/(2*_R^7*c+_R^3*b)*ln(x^(1/2)-_R),_R= 
RootOf(_Z^8*c+_Z^4*b+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11817 vs. \(2 (375) = 750\).

Time = 4.69 (sec) , antiderivative size = 11817, normalized size of antiderivative = 25.09 \[ \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^(9/2)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**(9/2)/(c*x**4+b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {x^{\frac {9}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate(x^(9/2)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")
 

Output:

1/2*(b*x^(7/2) + 2*a*x^(3/2))/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + ( 
b^3 - 4*a*b*c)*x^2) - integrate(-1/4*(b*x^(5/2) - 6*a*sqrt(x))/((b^2*c - 4 
*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2), x)
 

Giac [F]

\[ \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {x^{\frac {9}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate(x^(9/2)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")
 

Output:

integrate(x^(9/2)/(c*x^4 + b*x^2 + a)^2, x)
 

Mupad [B] (verification not implemented)

Time = 19.53 (sec) , antiderivative size = 23808, normalized size of antiderivative = 50.55 \[ \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

int(x^(9/2)/(a + b*x^2 + c*x^4)^2,x)
 

Output:

- ((a*x^(3/2))/(4*a*c - b^2) + (b*x^(7/2))/(2*(4*a*c - b^2)))/(a + b*x^2 + 
 c*x^4) - atan(((((5435817984*a^10*b*c^10 - 4096*a^3*b^15*c^3 + 1425408*a^ 
4*b^13*c^4 - 32833536*a^5*b^11*c^5 + 323747840*a^6*b^9*c^6 - 1714421760*a^ 
7*b^7*c^7 + 5121245184*a^8*b^5*c^8 - 8170504192*a^9*b^3*c^9)/(128*(b^14 - 
16384*a^7*c^7 + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 2 
1504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 28*a*b^12*c)) - (x^(1/2)*((b^4*(-(4 
*a*c - b^2)^15)^(1/2) - b^19 - 12386304*a^9*b*c^9 + 96*a^2*b^15*c^2 - 2752 
*a^3*b^13*c^3 + 55296*a^4*b^11*c^4 - 585216*a^5*b^9*c^5 + 3350528*a^6*b^7* 
c^6 - 10665984*a^7*b^5*c^7 + 17891328*a^8*b^3*c^8 + 324*a^2*c^2*(-(4*a*c - 
 b^2)^15)^(1/2) + 3*a*b^17*c + 27*a*b^2*c*(-(4*a*c - b^2)^15)^(1/2))/(8192 
*(16777216*a^12*c^15 + b^24*c^3 - 48*a*b^22*c^4 + 1056*a^2*b^20*c^5 - 1408 
0*a^3*b^18*c^6 + 126720*a^4*b^16*c^7 - 811008*a^5*b^14*c^8 + 3784704*a^6*b 
^12*c^9 - 12976128*a^7*b^10*c^10 + 32440320*a^8*b^8*c^11 - 57671680*a^9*b^ 
6*c^12 + 69206016*a^10*b^4*c^13 - 50331648*a^11*b^2*c^14)))^(1/4)*(1207959 
552*a^10*c^11 - 204800*a^3*b^14*c^4 + 5210112*a^4*b^12*c^5 - 56229888*a^5* 
b^10*c^6 + 332922880*a^6*b^8*c^7 - 1163919360*a^7*b^6*c^8 + 2390753280*a^8 
*b^4*c^9 - 2650800128*a^9*b^2*c^10))/(16*(b^12 + 4096*a^6*c^6 + 240*a^2*b^ 
8*c^2 - 1280*a^3*b^6*c^3 + 3840*a^4*b^4*c^4 - 6144*a^5*b^2*c^5 - 24*a*b^10 
*c)))*((b^4*(-(4*a*c - b^2)^15)^(1/2) - b^19 - 12386304*a^9*b*c^9 + 96*a^2 
*b^15*c^2 - 2752*a^3*b^13*c^3 + 55296*a^4*b^11*c^4 - 585216*a^5*b^9*c^5...
 

Reduce [F]

\[ \int \frac {x^{9/2}}{\left (a+b x^2+c x^4\right )^2} \, dx=\int \frac {x^{\frac {9}{2}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}d x \] Input:

int(x^(9/2)/(c*x^4+b*x^2+a)^2,x)
 

Output:

int(x^(9/2)/(c*x^4+b*x^2+a)^2,x)