Integrand size = 27, antiderivative size = 100 \[ \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx=\frac {\left (b^2 B-a b C+a^2 D\right ) x}{b^3}+\frac {(b C-a D) x^3}{3 b^2}+\frac {D x^5}{5 b}+\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}} \] Output:
(B*b^2-C*a*b+D*a^2)*x/b^3+1/3*(C*b-D*a)*x^3/b^2+1/5*D*x^5/b+(A*b^3-a*(B*b^ 2-C*a*b+D*a^2))*arctan(b^(1/2)*x/a^(1/2))/a^(1/2)/b^(7/2)
Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx=\frac {x \left (15 a^2 D-5 a b \left (3 C+D x^2\right )+b^2 \left (15 B+5 C x^2+3 D x^4\right )\right )}{15 b^3}+\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2),x]
Output:
(x*(15*a^2*D - 5*a*b*(3*C + D*x^2) + b^2*(15*B + 5*C*x^2 + 3*D*x^4)))/(15* b^3) + ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/( Sqrt[a]*b^(7/2))
Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2341, 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.
\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx\) |
\(\Big \downarrow \) 2341 |
\(\displaystyle \int \left (\frac {a^2 D-a b C+b^2 B}{b^3}+\frac {a^3 (-D)+a^2 b C-a b^2 B+A b^3}{b^3 \left (a+b x^2\right )}+\frac {x^2 (b C-a D)}{b^2}+\frac {D x^4}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{\sqrt {a} b^{7/2}}+\frac {x \left (a^2 D-a b C+b^2 B\right )}{b^3}+\frac {x^3 (b C-a D)}{3 b^2}+\frac {D x^5}{5 b}\) |
Input:
Int[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2),x]
Output:
((b^2*B - a*b*C + a^2*D)*x)/b^3 + ((b*C - a*D)*x^3)/(3*b^2) + (D*x^5)/(5*b ) + ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqr t[a]*b^(7/2))
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.48 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\frac {1}{5} D x^{5} b^{2}+\frac {1}{3} C \,b^{2} x^{3}-\frac {1}{3} D a b \,x^{3}+b^{2} B x -C a b x +D a^{2} x}{b^{3}}+\frac {\left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(94\) |
Input:
int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/b^3*(1/5*D*x^5*b^2+1/3*C*b^2*x^3-1/3*D*a*b*x^3+b^2*B*x-C*a*b*x+D*a^2*x)+ (A*b^3-B*a*b^2+C*a^2*b-D*a^3)/b^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
Time = 0.08 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.36 \[ \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx=\left [\frac {6 \, D a b^{3} x^{5} - 10 \, {\left (D a^{2} b^{2} - C a b^{3}\right )} x^{3} + 15 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 30 \, {\left (D a^{3} b - C a^{2} b^{2} + B a b^{3}\right )} x}{30 \, a b^{4}}, \frac {3 \, D a b^{3} x^{5} - 5 \, {\left (D a^{2} b^{2} - C a b^{3}\right )} x^{3} - 15 \, {\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (D a^{3} b - C a^{2} b^{2} + B a b^{3}\right )} x}{15 \, a b^{4}}\right ] \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x, algorithm="fricas")
Output:
[1/30*(6*D*a*b^3*x^5 - 10*(D*a^2*b^2 - C*a*b^3)*x^3 + 15*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a) ) + 30*(D*a^3*b - C*a^2*b^2 + B*a*b^3)*x)/(a*b^4), 1/15*(3*D*a*b^3*x^5 - 5 *(D*a^2*b^2 - C*a*b^3)*x^3 - 15*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*sqrt(a *b)*arctan(sqrt(a*b)*x/a) + 15*(D*a^3*b - C*a^2*b^2 + B*a*b^3)*x)/(a*b^4)]
Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.60 \[ \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx=\frac {D x^{5}}{5 b} + x^{3} \left (\frac {C}{3 b} - \frac {D a}{3 b^{2}}\right ) + x \left (\frac {B}{b} - \frac {C a}{b^{2}} + \frac {D a^{2}}{b^{3}}\right ) + \frac {\sqrt {- \frac {1}{a b^{7}}} \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \log {\left (- a b^{3} \sqrt {- \frac {1}{a b^{7}}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a b^{7}}} \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \log {\left (a b^{3} \sqrt {- \frac {1}{a b^{7}}} + x \right )}}{2} \] Input:
integrate((D*x**6+C*x**4+B*x**2+A)/(b*x**2+a),x)
Output:
D*x**5/(5*b) + x**3*(C/(3*b) - D*a/(3*b**2)) + x*(B/b - C*a/b**2 + D*a**2/ b**3) + sqrt(-1/(a*b**7))*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)*log(-a* b**3*sqrt(-1/(a*b**7)) + x)/2 - sqrt(-1/(a*b**7))*(-A*b**3 + B*a*b**2 - C* a**2*b + D*a**3)*log(a*b**3*sqrt(-1/(a*b**7)) + x)/2
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx=-\frac {{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, D b^{2} x^{5} - 5 \, {\left (D a b - C b^{2}\right )} x^{3} + 15 \, {\left (D a^{2} - C a b + B b^{2}\right )} x}{15 \, b^{3}} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x, algorithm="maxima")
Output:
-(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*D*b^2*x^5 - 5*(D*a*b - C*b^2)*x^3 + 15*(D*a^2 - C*a*b + B*b^2)* x)/b^3
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx=-\frac {{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, D b^{4} x^{5} - 5 \, D a b^{3} x^{3} + 5 \, C b^{4} x^{3} + 15 \, D a^{2} b^{2} x - 15 \, C a b^{3} x + 15 \, B b^{4} x}{15 \, b^{5}} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x, algorithm="giac")
Output:
-(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*D*b^4*x^5 - 5*D*a*b^3*x^3 + 5*C*b^4*x^3 + 15*D*a^2*b^2*x - 15*C *a*b^3*x + 15*B*b^4*x)/b^5
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{b\,x^2+a} \,d x \] Input:
int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2),x)
Output:
int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2), x)
Time = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x^2+C x^4+D x^6}{a+b x^2} \, dx=\frac {-15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} d +15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c +15 a^{2} b d x -15 a \,b^{2} c x -5 a \,b^{2} d \,x^{3}+15 b^{4} x +5 b^{3} c \,x^{3}+3 b^{3} d \,x^{5}}{15 b^{4}} \] Input:
int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a),x)
Output:
( - 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*d + 15*sqrt(b)*s qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*c + 15*a**2*b*d*x - 15*a*b**2*c*x - 5*a*b**2*d*x**3 + 15*b**4*x + 5*b**3*c*x**3 + 3*b**3*d*x**5)/(15*b**4)