Integrand size = 30, antiderivative size = 210 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx=\frac {a^2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) x}{b^6}-\frac {a \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) x^3}{3 b^5}+\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) x^5}{5 b^4}+\frac {\left (b^2 B-a b C+a^2 D\right ) x^7}{7 b^3}+\frac {(b C-a D) x^9}{9 b^2}+\frac {D x^{11}}{11 b}-\frac {a^{5/2} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{13/2}} \] Output:
a^2*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*x/b^6-1/3*a*(A*b^3-a*(B*b^2-C*a*b+D*a^2) )*x^3/b^5+1/5*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*x^5/b^4+1/7*(B*b^2-C*a*b+D*a^2 )*x^7/b^3+1/9*(C*b-D*a)*x^9/b^2+1/11*D*x^11/b-a^(5/2)*(A*b^3-a*(B*b^2-C*a* b+D*a^2))*arctan(b^(1/2)*x/a^(1/2))/b^(13/2)
Time = 0.16 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx=-\frac {a^2 \left (-A b^3+a b^2 B-a^2 b C+a^3 D\right ) x}{b^6}+\frac {a \left (-A b^3+a b^2 B-a^2 b C+a^3 D\right ) x^3}{3 b^5}+\frac {\left (A b^3-a b^2 B+a^2 b C-a^3 D\right ) x^5}{5 b^4}+\frac {\left (b^2 B-a b C+a^2 D\right ) x^7}{7 b^3}+\frac {(b C-a D) x^9}{9 b^2}+\frac {D x^{11}}{11 b}+\frac {a^{5/2} \left (-A b^3+a b^2 B-a^2 b C+a^3 D\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{13/2}} \] Input:
Integrate[(x^6*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2),x]
Output:
-((a^2*(-(A*b^3) + a*b^2*B - a^2*b*C + a^3*D)*x)/b^6) + (a*(-(A*b^3) + a*b ^2*B - a^2*b*C + a^3*D)*x^3)/(3*b^5) + ((A*b^3 - a*b^2*B + a^2*b*C - a^3*D )*x^5)/(5*b^4) + ((b^2*B - a*b*C + a^2*D)*x^7)/(7*b^3) + ((b*C - a*D)*x^9) /(9*b^2) + (D*x^11)/(11*b) + (a^(5/2)*(-(A*b^3) + a*b^2*B - a^2*b*C + a^3* D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(13/2)
Time = 0.45 (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.
| \(\displaystyle \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \int \left (\frac {a^2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^6}-\frac {a x^2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^5}+\frac {x^4 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^4}+\frac {x^6 \left (a^2 D-a b C+b^2 B\right )}{b^3}+\frac {a^6 D-a^5 b C+a^4 b^2 B-a^3 A b^3}{b^6 \left (a+b x^2\right )}+\frac {x^8 (b C-a D)}{b^2}+\frac {D x^{10}}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 x \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^6}-\frac {a x^3 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^5}+\frac {x^5 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{5 b^4}+\frac {x^7 \left (a^2 D-a b C+b^2 B\right )}{7 b^3}-\frac {a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^{13/2}}+\frac {x^9 (b C-a D)}{9 b^2}+\frac {D x^{11}}{11 b}\) |
Input:
Int[(x^6*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2),x]
Output:
(a^2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*x)/b^6 - (a*(A*b^3 - a*(b^2*B - a *b*C + a^2*D))*x^3)/(3*b^5) + ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*x^5)/(5 *b^4) + ((b^2*B - a*b*C + a^2*D)*x^7)/(7*b^3) + ((b*C - a*D)*x^9)/(9*b^2) + (D*x^11)/(11*b) - (a^(5/2)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*ArcTan[(S qrt[b]*x)/Sqrt[a]])/b^(13/2)
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]
Time = 0.57 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(\frac {\frac {1}{11} D x^{11} b^{5}+\frac {1}{9} C \,b^{5} x^{9}-\frac {1}{9} D a \,b^{4} x^{9}+\frac {1}{7} b^{5} B \,x^{7}-\frac {1}{7} C a \,b^{4} x^{7}+\frac {1}{7} D a^{2} b^{3} x^{7}+\frac {1}{5} A \,b^{5} x^{5}-\frac {1}{5} B a \,b^{4} x^{5}+\frac {1}{5} C \,a^{2} b^{3} x^{5}-\frac {1}{5} D a^{3} b^{2} x^{5}-\frac {1}{3} A a \,b^{4} x^{3}+\frac {1}{3} B \,a^{2} b^{3} x^{3}-\frac {1}{3} C \,a^{3} b^{2} x^{3}+\frac {1}{3} D a^{4} b \,x^{3}+A \,a^{2} b^{3} x -B \,a^{3} b^{2} x +C \,a^{4} b x -D a^{5} x}{b^{6}}-\frac {a^{3} \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{6} \sqrt {a b}}\) | \(233\) |
Input:
int(x^6*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/b^6*(1/11*D*x^11*b^5+1/9*C*b^5*x^9-1/9*D*a*b^4*x^9+1/7*b^5*B*x^7-1/7*C*a *b^4*x^7+1/7*D*a^2*b^3*x^7+1/5*A*b^5*x^5-1/5*B*a*b^4*x^5+1/5*C*a^2*b^3*x^5 -1/5*D*a^3*b^2*x^5-1/3*A*a*b^4*x^3+1/3*B*a^2*b^3*x^3-1/3*C*a^3*b^2*x^3+1/3 *D*a^4*b*x^3+A*a^2*b^3*x-B*a^3*b^2*x+C*a^4*b*x-D*a^5*x)-a^3*(A*b^3-B*a*b^2 +C*a^2*b-D*a^3)/b^6/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
Time = 0.08 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.15 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx=\left [\frac {630 \, D b^{5} x^{11} - 770 \, {\left (D a b^{4} - C b^{5}\right )} x^{9} + 990 \, {\left (D a^{2} b^{3} - C a b^{4} + B b^{5}\right )} x^{7} - 1386 \, {\left (D a^{3} b^{2} - C a^{2} b^{3} + B a b^{4} - A b^{5}\right )} x^{5} + 2310 \, {\left (D a^{4} b - C a^{3} b^{2} + B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 3465 \, {\left (D a^{5} - C a^{4} b + B a^{3} b^{2} - A a^{2} b^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 6930 \, {\left (D a^{5} - C a^{4} b + B a^{3} b^{2} - A a^{2} b^{3}\right )} x}{6930 \, b^{6}}, \frac {315 \, D b^{5} x^{11} - 385 \, {\left (D a b^{4} - C b^{5}\right )} x^{9} + 495 \, {\left (D a^{2} b^{3} - C a b^{4} + B b^{5}\right )} x^{7} - 693 \, {\left (D a^{3} b^{2} - C a^{2} b^{3} + B a b^{4} - A b^{5}\right )} x^{5} + 1155 \, {\left (D a^{4} b - C a^{3} b^{2} + B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 3465 \, {\left (D a^{5} - C a^{4} b + B a^{3} b^{2} - A a^{2} b^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 3465 \, {\left (D a^{5} - C a^{4} b + B a^{3} b^{2} - A a^{2} b^{3}\right )} x}{3465 \, b^{6}}\right ] \] Input:
integrate(x^6*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x, algorithm="fricas")
Output:
[1/6930*(630*D*b^5*x^11 - 770*(D*a*b^4 - C*b^5)*x^9 + 990*(D*a^2*b^3 - C*a *b^4 + B*b^5)*x^7 - 1386*(D*a^3*b^2 - C*a^2*b^3 + B*a*b^4 - A*b^5)*x^5 + 2 310*(D*a^4*b - C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*x^3 - 3465*(D*a^5 - C*a^4* b + B*a^3*b^2 - A*a^2*b^3)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/( b*x^2 + a)) - 6930*(D*a^5 - C*a^4*b + B*a^3*b^2 - A*a^2*b^3)*x)/b^6, 1/346 5*(315*D*b^5*x^11 - 385*(D*a*b^4 - C*b^5)*x^9 + 495*(D*a^2*b^3 - C*a*b^4 + B*b^5)*x^7 - 693*(D*a^3*b^2 - C*a^2*b^3 + B*a*b^4 - A*b^5)*x^5 + 1155*(D* a^4*b - C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*x^3 + 3465*(D*a^5 - C*a^4*b + B*a ^3*b^2 - A*a^2*b^3)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 3465*(D*a^5 - C*a^ 4*b + B*a^3*b^2 - A*a^2*b^3)*x)/b^6]
Time = 0.54 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.83 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx=\frac {D x^{11}}{11 b} + x^{9} \left (\frac {C}{9 b} - \frac {D a}{9 b^{2}}\right ) + x^{7} \left (\frac {B}{7 b} - \frac {C a}{7 b^{2}} + \frac {D a^{2}}{7 b^{3}}\right ) + x^{5} \left (\frac {A}{5 b} - \frac {B a}{5 b^{2}} + \frac {C a^{2}}{5 b^{3}} - \frac {D a^{3}}{5 b^{4}}\right ) + x^{3} \left (- \frac {A a}{3 b^{2}} + \frac {B a^{2}}{3 b^{3}} - \frac {C a^{3}}{3 b^{4}} + \frac {D a^{4}}{3 b^{5}}\right ) + x \left (\frac {A a^{2}}{b^{3}} - \frac {B a^{3}}{b^{4}} + \frac {C a^{4}}{b^{5}} - \frac {D a^{5}}{b^{6}}\right ) - \frac {\sqrt {- \frac {a^{5}}{b^{13}}} \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \log {\left (- \frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right )}{- A a^{2} b^{3} + B a^{3} b^{2} - C a^{4} b + D a^{5}} + x \right )}}{2} + \frac {\sqrt {- \frac {a^{5}}{b^{13}}} \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \log {\left (\frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right )}{- A a^{2} b^{3} + B a^{3} b^{2} - C a^{4} b + D a^{5}} + x \right )}}{2} \] Input:
integrate(x**6*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a),x)
Output:
D*x**11/(11*b) + x**9*(C/(9*b) - D*a/(9*b**2)) + x**7*(B/(7*b) - C*a/(7*b* *2) + D*a**2/(7*b**3)) + x**5*(A/(5*b) - B*a/(5*b**2) + C*a**2/(5*b**3) - D*a**3/(5*b**4)) + x**3*(-A*a/(3*b**2) + B*a**2/(3*b**3) - C*a**3/(3*b**4) + D*a**4/(3*b**5)) + x*(A*a**2/b**3 - B*a**3/b**4 + C*a**4/b**5 - D*a**5/ b**6) - sqrt(-a**5/b**13)*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)*log(-b* *6*sqrt(-a**5/b**13)*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)/(-A*a**2*b** 3 + B*a**3*b**2 - C*a**4*b + D*a**5) + x)/2 + sqrt(-a**5/b**13)*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)*log(b**6*sqrt(-a**5/b**13)*(-A*b**3 + B*a*b **2 - C*a**2*b + D*a**3)/(-A*a**2*b**3 + B*a**3*b**2 - C*a**4*b + D*a**5) + x)/2
Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.01 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx=\frac {{\left (D a^{6} - C a^{5} b + B a^{4} b^{2} - A a^{3} b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{6}} + \frac {315 \, D b^{5} x^{11} - 385 \, {\left (D a b^{4} - C b^{5}\right )} x^{9} + 495 \, {\left (D a^{2} b^{3} - C a b^{4} + B b^{5}\right )} x^{7} - 693 \, {\left (D a^{3} b^{2} - C a^{2} b^{3} + B a b^{4} - A b^{5}\right )} x^{5} + 1155 \, {\left (D a^{4} b - C a^{3} b^{2} + B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 3465 \, {\left (D a^{5} - C a^{4} b + B a^{3} b^{2} - A a^{2} b^{3}\right )} x}{3465 \, b^{6}} \] Input:
integrate(x^6*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x, algorithm="maxima")
Output:
(D*a^6 - C*a^5*b + B*a^4*b^2 - A*a^3*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b) *b^6) + 1/3465*(315*D*b^5*x^11 - 385*(D*a*b^4 - C*b^5)*x^9 + 495*(D*a^2*b^ 3 - C*a*b^4 + B*b^5)*x^7 - 693*(D*a^3*b^2 - C*a^2*b^3 + B*a*b^4 - A*b^5)*x ^5 + 1155*(D*a^4*b - C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*x^3 - 3465*(D*a^5 - C*a^4*b + B*a^3*b^2 - A*a^2*b^3)*x)/b^6
Time = 0.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.16 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx=\frac {{\left (D a^{6} - C a^{5} b + B a^{4} b^{2} - A a^{3} b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{6}} + \frac {315 \, D b^{10} x^{11} - 385 \, D a b^{9} x^{9} + 385 \, C b^{10} x^{9} + 495 \, D a^{2} b^{8} x^{7} - 495 \, C a b^{9} x^{7} + 495 \, B b^{10} x^{7} - 693 \, D a^{3} b^{7} x^{5} + 693 \, C a^{2} b^{8} x^{5} - 693 \, B a b^{9} x^{5} + 693 \, A b^{10} x^{5} + 1155 \, D a^{4} b^{6} x^{3} - 1155 \, C a^{3} b^{7} x^{3} + 1155 \, B a^{2} b^{8} x^{3} - 1155 \, A a b^{9} x^{3} - 3465 \, D a^{5} b^{5} x + 3465 \, C a^{4} b^{6} x - 3465 \, B a^{3} b^{7} x + 3465 \, A a^{2} b^{8} x}{3465 \, b^{11}} \] Input:
integrate(x^6*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x, algorithm="giac")
Output:
(D*a^6 - C*a^5*b + B*a^4*b^2 - A*a^3*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b) *b^6) + 1/3465*(315*D*b^10*x^11 - 385*D*a*b^9*x^9 + 385*C*b^10*x^9 + 495*D *a^2*b^8*x^7 - 495*C*a*b^9*x^7 + 495*B*b^10*x^7 - 693*D*a^3*b^7*x^5 + 693* C*a^2*b^8*x^5 - 693*B*a*b^9*x^5 + 693*A*b^10*x^5 + 1155*D*a^4*b^6*x^3 - 11 55*C*a^3*b^7*x^3 + 1155*B*a^2*b^8*x^3 - 1155*A*a*b^9*x^3 - 3465*D*a^5*b^5* x + 3465*C*a^4*b^6*x - 3465*B*a^3*b^7*x + 3465*A*a^2*b^8*x)/b^11
Timed out. \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx=\int \frac {x^6\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{b\,x^2+a} \,d x \] Input:
int((x^6*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2),x)
Output:
int((x^6*(A + B*x^2 + C*x^4 + x^6*D))/(a + b*x^2), x)
Time = 0.16 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.83 \[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{a+b x^2} \, dx=\frac {3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{5} d -3465 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b c -3465 a^{5} b d x +3465 a^{4} b^{2} c x +1155 a^{4} b^{2} d \,x^{3}-1155 a^{3} b^{3} c \,x^{3}-693 a^{3} b^{3} d \,x^{5}+693 a^{2} b^{4} c \,x^{5}+495 a^{2} b^{4} d \,x^{7}-495 a \,b^{5} c \,x^{7}-385 a \,b^{5} d \,x^{9}+495 b^{7} x^{7}+385 b^{6} c \,x^{9}+315 b^{6} d \,x^{11}}{3465 b^{7}} \] Input:
int(x^6*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x)
Output:
(3465*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**5*d - 3465*sqrt(b)* sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**4*b*c - 3465*a**5*b*d*x + 3465*a* *4*b**2*c*x + 1155*a**4*b**2*d*x**3 - 1155*a**3*b**3*c*x**3 - 693*a**3*b** 3*d*x**5 + 693*a**2*b**4*c*x**5 + 495*a**2*b**4*d*x**7 - 495*a*b**5*c*x**7 - 385*a*b**5*d*x**9 + 495*b**7*x**7 + 385*b**6*c*x**9 + 315*b**6*d*x**11) /(3465*b**7)