3.2.14 \(\int \frac {x^6 (c+d x^2+e x^4+f x^6)}{a+b x^2} \, dx\) [114]

3.2.14.1 Optimal result

Integrand size = 30, antiderivative size = 210 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx=\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac {(b e-a f) x^9}{9 b^2}+\frac {f x^{11}}{11 b}-\frac {a^{5/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{13/2}} \]

output

a^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/b^6-1/3*a*(-a^3*f+a^2*b*e-a*b^2*d+b^3 
*c)*x^3/b^5+1/5*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^5/b^4+1/7*(a^2*f-a*b*e+b^ 
2*d)*x^7/b^3+1/9*(-a*f+b*e)*x^9/b^2+1/11*f*x^11/b-a^(5/2)*(-a^3*f+a^2*b*e- 
a*b^2*d+b^3*c)*arctan(x*b^(1/2)/a^(1/2))/b^(13/2)
 

3.2.14.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx=-\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{b^6}+\frac {a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x^3}{3 b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac {(b e-a f) x^9}{9 b^2}+\frac {f x^{11}}{11 b}+\frac {a^{5/2} \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{13/2}} \]

input

Integrate[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]
 

output

-((a^2*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/b^6) + (a*(-(b^3*c) + a*b 
^2*d - a^2*b*e + a^3*f)*x^3)/(3*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f 
)*x^5)/(5*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^9) 
/(9*b^2) + (f*x^11)/(11*b) + (a^(5/2)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3* 
f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(13/2)
 

3.2.14.3 Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2333, 2009}

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.

\(\displaystyle \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx\)

\(\Big \downarrow \) 2333

\(\displaystyle \int \left (\frac {x^6 \left (a^2 f-a b e+b^2 d\right )}{b^3}+\frac {a^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}-\frac {a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac {x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac {a^6 f-a^5 b e+a^4 b^2 d-a^3 b^3 c}{b^6 \left (a+b x^2\right )}+\frac {x^8 (b e-a f)}{b^2}+\frac {f x^{10}}{b}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}-\frac {a x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5}+\frac {x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^4}-\frac {a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^{13/2}}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{11}}{11 b}\)

input

Int[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]
 

output

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^6 - (a*(b^3*c - a*b^2*d + a^ 
2*b*e - a^3*f)*x^3)/(3*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5)/(5 
*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^9)/(9*b^2) 
+ (f*x^11)/(11*b) - (a^(5/2)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(S 
qrt[b]*x)/Sqrt[a]])/b^(13/2)
 

3.2.14.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2333

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] 
&& PolyQ[Pq, x] && IGtQ[p, -2]
 

3.2.14.4 Maple [A] (verified)

Time = 3.47 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {-\frac {1}{11} f \,x^{11} b^{5}+\frac {1}{9} a \,b^{4} f \,x^{9}-\frac {1}{9} b^{5} e \,x^{9}-\frac {1}{7} a^{2} b^{3} f \,x^{7}+\frac {1}{7} a \,b^{4} e \,x^{7}-\frac {1}{7} b^{5} d \,x^{7}+\frac {1}{5} a^{3} b^{2} f \,x^{5}-\frac {1}{5} a^{2} b^{3} e \,x^{5}+\frac {1}{5} a \,b^{4} d \,x^{5}-\frac {1}{5} b^{5} c \,x^{5}-\frac {1}{3} a^{4} b f \,x^{3}+\frac {1}{3} a^{3} b^{2} e \,x^{3}-\frac {1}{3} a^{2} b^{3} d \,x^{3}+\frac {1}{3} a \,b^{4} c \,x^{3}+a^{5} f x -a^{4} b e x +a^{3} b^{2} d x -a^{2} b^{3} c x}{b^{6}}+\frac {a^{3} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{6} \sqrt {a b}}\)	\(233\)
risch	\(-\frac {\sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) c}{2 b^{4}}-\frac {\sqrt {-a b}\, a^{5} \ln \left (\sqrt {-a b}\, x +a \right ) f}{2 b^{7}}-\frac {\sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x +a \right ) d}{2 b^{5}}+\frac {e \,x^{9}}{9 b}+\frac {d \,x^{7}}{7 b}+\frac {c \,x^{5}}{5 b}+\frac {\sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x +a \right ) c}{2 b^{4}}+\frac {\sqrt {-a b}\, a^{5} \ln \left (-\sqrt {-a b}\, x +a \right ) f}{2 b^{7}}-\frac {\sqrt {-a b}\, a^{4} \ln \left (-\sqrt {-a b}\, x +a \right ) e}{2 b^{6}}+\frac {\sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x +a \right ) d}{2 b^{5}}+\frac {\sqrt {-a b}\, a^{4} \ln \left (\sqrt {-a b}\, x +a \right ) e}{2 b^{6}}+\frac {f \,x^{11}}{11 b}-\frac {a f \,x^{9}}{9 b^{2}}+\frac {a^{2} f \,x^{7}}{7 b^{3}}-\frac {a e \,x^{7}}{7 b^{2}}-\frac {a^{3} f \,x^{5}}{5 b^{4}}+\frac {a^{2} e \,x^{5}}{5 b^{3}}-\frac {a d \,x^{5}}{5 b^{2}}+\frac {a^{4} f \,x^{3}}{3 b^{5}}-\frac {a^{3} e \,x^{3}}{3 b^{4}}+\frac {a^{2} d \,x^{3}}{3 b^{3}}-\frac {a c \,x^{3}}{3 b^{2}}-\frac {a^{5} f x}{b^{6}}+\frac {a^{4} e x}{b^{5}}-\frac {a^{3} d x}{b^{4}}+\frac {a^{2} c x}{b^{3}}\)	\(400\)

input

int(x^6*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

output

-1/b^6*(-1/11*f*x^11*b^5+1/9*a*b^4*f*x^9-1/9*b^5*e*x^9-1/7*a^2*b^3*f*x^7+1 
/7*a*b^4*e*x^7-1/7*b^5*d*x^7+1/5*a^3*b^2*f*x^5-1/5*a^2*b^3*e*x^5+1/5*a*b^4 
*d*x^5-1/5*b^5*c*x^5-1/3*a^4*b*f*x^3+1/3*a^3*b^2*e*x^3-1/3*a^2*b^3*d*x^3+1 
/3*a*b^4*c*x^3+a^5*f*x-a^4*b*e*x+a^3*b^2*d*x-a^2*b^3*c*x)+a^3*(a^3*f-a^2*b 
*e+a*b^2*d-b^3*c)/b^6/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
 

3.2.14.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.15 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx=\left [\frac {630 \, b^{5} f x^{11} + 770 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 990 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 1386 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 2310 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} - 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6930 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{6930 \, b^{6}}, \frac {315 \, b^{5} f x^{11} + 385 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 495 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 693 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} - 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3465 \, b^{6}}\right ] \]

input

integrate(x^6*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="fricas")
 

output

[1/6930*(630*b^5*f*x^11 + 770*(b^5*e - a*b^4*f)*x^9 + 990*(b^5*d - a*b^4*e 
 + a^2*b^3*f)*x^7 + 1386*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^5 - 2 
310*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^3 - 3465*(a^2*b^3*c - a^ 
3*b^2*d + a^4*b*e - a^5*f)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/( 
b*x^2 + a)) + 6930*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*x)/b^6, 1/346 
5*(315*b^5*f*x^11 + 385*(b^5*e - a*b^4*f)*x^9 + 495*(b^5*d - a*b^4*e + a^2 
*b^3*f)*x^7 + 693*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^5 - 1155*(a* 
b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^3 - 3465*(a^2*b^3*c - a^3*b^2*d 
 + a^4*b*e - a^5*f)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + 3465*(a^2*b^3*c - 
a^3*b^2*d + a^4*b*e - a^5*f)*x)/b^6]
 

3.2.14.6 Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.83 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx=x^{9} \left (- \frac {a f}{9 b^{2}} + \frac {e}{9 b}\right ) + x^{7} \left (\frac {a^{2} f}{7 b^{3}} - \frac {a e}{7 b^{2}} + \frac {d}{7 b}\right ) + x^{5} \left (- \frac {a^{3} f}{5 b^{4}} + \frac {a^{2} e}{5 b^{3}} - \frac {a d}{5 b^{2}} + \frac {c}{5 b}\right ) + x^{3} \left (\frac {a^{4} f}{3 b^{5}} - \frac {a^{3} e}{3 b^{4}} + \frac {a^{2} d}{3 b^{3}} - \frac {a c}{3 b^{2}}\right ) + x \left (- \frac {a^{5} f}{b^{6}} + \frac {a^{4} e}{b^{5}} - \frac {a^{3} d}{b^{4}} + \frac {a^{2} c}{b^{3}}\right ) - \frac {\sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )}}{2} + \frac {\sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )}}{2} + \frac {f x^{11}}{11 b} \]

input

integrate(x**6*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a),x)
 

output

x**9*(-a*f/(9*b**2) + e/(9*b)) + x**7*(a**2*f/(7*b**3) - a*e/(7*b**2) + d/ 
(7*b)) + x**5*(-a**3*f/(5*b**4) + a**2*e/(5*b**3) - a*d/(5*b**2) + c/(5*b) 
) + x**3*(a**4*f/(3*b**5) - a**3*e/(3*b**4) + a**2*d/(3*b**3) - a*c/(3*b** 
2)) + x*(-a**5*f/b**6 + a**4*e/b**5 - a**3*d/b**4 + a**2*c/b**3) - sqrt(-a 
**5/b**13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-b**6*sqrt(-a**5/b* 
*13)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**5*f - a**4*b*e + a**3*b** 
2*d - a**2*b**3*c) + x)/2 + sqrt(-a**5/b**13)*(a**3*f - a**2*b*e + a*b**2* 
d - b**3*c)*log(b**6*sqrt(-a**5/b**13)*(a**3*f - a**2*b*e + a*b**2*d - b** 
3*c)/(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c) + x)/2 + f*x**11/(11* 
b)
 

3.2.14.7 Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.01 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx=-\frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{6}} + \frac {315 \, b^{5} f x^{11} + 385 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 495 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 693 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} + 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3465 \, b^{6}} \]

input

integrate(x^6*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="maxima")
 

output

-(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b 
)*b^6) + 1/3465*(315*b^5*f*x^11 + 385*(b^5*e - a*b^4*f)*x^9 + 495*(b^5*d - 
 a*b^4*e + a^2*b^3*f)*x^7 + 693*(b^5*c - a*b^4*d + a^2*b^3*e - a^3*b^2*f)* 
x^5 - 1155*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^3 + 3465*(a^2*b^3 
*c - a^3*b^2*d + a^4*b*e - a^5*f)*x)/b^6
 

3.2.14.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx=-\frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{6}} + \frac {315 \, b^{10} f x^{11} + 385 \, b^{10} e x^{9} - 385 \, a b^{9} f x^{9} + 495 \, b^{10} d x^{7} - 495 \, a b^{9} e x^{7} + 495 \, a^{2} b^{8} f x^{7} + 693 \, b^{10} c x^{5} - 693 \, a b^{9} d x^{5} + 693 \, a^{2} b^{8} e x^{5} - 693 \, a^{3} b^{7} f x^{5} - 1155 \, a b^{9} c x^{3} + 1155 \, a^{2} b^{8} d x^{3} - 1155 \, a^{3} b^{7} e x^{3} + 1155 \, a^{4} b^{6} f x^{3} + 3465 \, a^{2} b^{8} c x - 3465 \, a^{3} b^{7} d x + 3465 \, a^{4} b^{6} e x - 3465 \, a^{5} b^{5} f x}{3465 \, b^{11}} \]

input

integrate(x^6*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="giac")
 

output

-(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b 
)*b^6) + 1/3465*(315*b^10*f*x^11 + 385*b^10*e*x^9 - 385*a*b^9*f*x^9 + 495* 
b^10*d*x^7 - 495*a*b^9*e*x^7 + 495*a^2*b^8*f*x^7 + 693*b^10*c*x^5 - 693*a* 
b^9*d*x^5 + 693*a^2*b^8*e*x^5 - 693*a^3*b^7*f*x^5 - 1155*a*b^9*c*x^3 + 115 
5*a^2*b^8*d*x^3 - 1155*a^3*b^7*e*x^3 + 1155*a^4*b^6*f*x^3 + 3465*a^2*b^8*c 
*x - 3465*a^3*b^7*d*x + 3465*a^4*b^6*e*x - 3465*a^5*b^5*f*x)/b^11
 

3.2.14.9 Mupad [B] (verification not implemented)

Time = 5.91 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.38 \[ \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx=x^9\,\left (\frac {e}{9\,b}-\frac {a\,f}{9\,b^2}\right )+x^7\,\left (\frac {d}{7\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{7\,b}\right )+x^5\,\left (\frac {c}{5\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{5\,b}\right )+\frac {f\,x^{11}}{11\,b}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{f\,a^6-e\,a^5\,b+d\,a^4\,b^2-c\,a^3\,b^3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{b^{13/2}}-\frac {a\,x^3\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{3\,b}+\frac {a^2\,x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{b^2} \]

input

int((x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x)
 

output

x^9*(e/(9*b) - (a*f)/(9*b^2)) + x^7*(d/(7*b) - (a*(e/b - (a*f)/b^2))/(7*b) 
) + x^5*(c/(5*b) - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/(5*b)) + (f*x^11)/( 
11*b) + (a^(5/2)*atan((a^(5/2)*b^(1/2)*x*(b^3*c - a^3*f - a*b^2*d + a^2*b* 
e))/(a^6*f - a^3*b^3*c + a^4*b^2*d - a^5*b*e))*(b^3*c - a^3*f - a*b^2*d + 
a^2*b*e))/b^(13/2) - (a*x^3*(c/b - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/b)) 
/(3*b) + (a^2*x*(c/b - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/b))/b^2