Integrand size = 27, antiderivative size = 118 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx=\frac {(b C-2 a D) x}{b^3}+\frac {D x^3}{3 b^2}+\frac {\left (\frac {A}{a}-\frac {b^2 B-a b C+a^2 D}{b^3}\right ) x}{2 \left (a+b x^2\right )}+\frac {\left (A b^3+a \left (b^2 B-3 a b C+5 a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}} \] Output:
(C*b-2*D*a)*x/b^3+1/3*D*x^3/b^2+(A/a-(B*b^2-C*a*b+D*a^2)/b^3)*x/(2*b*x^2+2 *a)+1/2*(A*b^3+a*(B*b^2-3*C*a*b+5*D*a^2))*arctan(b^(1/2)*x/a^(1/2))/a^(3/2 )/b^(7/2)
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx=\frac {(b C-2 a D) x}{b^3}+\frac {D x^3}{3 b^2}-\frac {\left (-A b^3+a b^2 B-a^2 b C+a^3 D\right ) x}{2 a b^3 \left (a+b x^2\right )}+\frac {\left (A b^3+a b^2 B-3 a^2 b C+5 a^3 D\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^2,x]
Output:
((b*C - 2*a*D)*x)/b^3 + (D*x^3)/(3*b^2) - ((-(A*b^3) + a*b^2*B - a^2*b*C + a^3*D)*x)/(2*a*b^3*(a + b*x^2)) + ((A*b^3 + a*b^2*B - 3*a^2*b*C + 5*a^3*D )*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(7/2))
Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2345, 25, 1467, 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.
\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\int -\frac {\frac {2 a D x^4}{b}+\frac {2 a (b C-a D) x^2}{b^2}+\frac {D a^3-b C a^2+b^2 B a+A b^3}{b^3}}{b x^2+a}dx}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\frac {2 a D x^4}{b}+\frac {2 a (b C-a D) x^2}{b^2}+A+\frac {a \left (D a^2-b C a+b^2 B\right )}{b^3}}{b x^2+a}dx}{2 a}+\frac {x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 1467 |
\(\displaystyle \frac {\int \left (\frac {2 a D x^2}{b^2}+\frac {2 a (b C-2 a D)}{b^3}+\frac {5 D a^3-3 b C a^2+b^2 B a+A b^3}{b^3 \left (b x^2+a\right )}\right )dx}{2 a}+\frac {x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a \left (5 a^2 D-3 a b C+b^2 B\right )+A b^3\right )}{\sqrt {a} b^{7/2}}+\frac {2 a x (b C-2 a D)}{b^3}+\frac {2 a D x^3}{3 b^2}}{2 a}+\frac {x \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}\) |
Input:
Int[(A + B*x^2 + C*x^4 + D*x^6)/(a + b*x^2)^2,x]
Output:
((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x)/(2*a*(a + b*x^2)) + ((2*a*(b*C - 2*a*D)*x)/b^3 + (2*a*D*x^3)/(3*b^2) + ((A*b^3 + a*(b^2*B - 3*a*b*C + 5*a^ 2*D))*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2)))/(2*a)
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.53 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\frac {1}{3} D x^{3} b +C b x -2 D a x}{b^{3}}+\frac {\frac {\left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (b^{3} A +a \,b^{2} B -3 a^{2} b C +5 a^{3} D\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{3}}\) | \(112\) |
Input:
int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
1/b^3*(1/3*D*x^3*b+C*b*x-2*D*a*x)+1/b^3*(1/2*(A*b^3-B*a*b^2+C*a^2*b-D*a^3) /a*x/(b*x^2+a)+1/2*(A*b^3+B*a*b^2-3*C*a^2*b+5*D*a^3)/a/(a*b)^(1/2)*arctan( b*x/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.08 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, D a^{2} b^{3} x^{5} - 4 \, {\left (5 \, D a^{3} b^{2} - 3 \, C a^{2} b^{3}\right )} x^{3} - 3 \, {\left (5 \, D a^{4} - 3 \, C a^{3} b + B a^{2} b^{2} + A a b^{3} + {\left (5 \, D a^{3} b - 3 \, C a^{2} b^{2} + B a b^{3} + A b^{4}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 6 \, {\left (5 \, D a^{4} b - 3 \, C a^{3} b^{2} + B a^{2} b^{3} - A a b^{4}\right )} x}{12 \, {\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}, \frac {2 \, D a^{2} b^{3} x^{5} - 2 \, {\left (5 \, D a^{3} b^{2} - 3 \, C a^{2} b^{3}\right )} x^{3} + 3 \, {\left (5 \, D a^{4} - 3 \, C a^{3} b + B a^{2} b^{2} + A a b^{3} + {\left (5 \, D a^{3} b - 3 \, C a^{2} b^{2} + B a b^{3} + A b^{4}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 3 \, {\left (5 \, D a^{4} b - 3 \, C a^{3} b^{2} + B a^{2} b^{3} - A a b^{4}\right )} x}{6 \, {\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}\right ] \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/12*(4*D*a^2*b^3*x^5 - 4*(5*D*a^3*b^2 - 3*C*a^2*b^3)*x^3 - 3*(5*D*a^4 - 3*C*a^3*b + B*a^2*b^2 + A*a*b^3 + (5*D*a^3*b - 3*C*a^2*b^2 + B*a*b^3 + A*b ^4)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - 6*(5*D *a^4*b - 3*C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*x)/(a^2*b^5*x^2 + a^3*b^4), 1/ 6*(2*D*a^2*b^3*x^5 - 2*(5*D*a^3*b^2 - 3*C*a^2*b^3)*x^3 + 3*(5*D*a^4 - 3*C* a^3*b + B*a^2*b^2 + A*a*b^3 + (5*D*a^3*b - 3*C*a^2*b^2 + B*a*b^3 + A*b^4)* x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - 3*(5*D*a^4*b - 3*C*a^3*b^2 + B*a^2* b^3 - A*a*b^4)*x)/(a^2*b^5*x^2 + a^3*b^4)]
Time = 0.74 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.70 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx=\frac {D x^{3}}{3 b^{2}} + x \left (\frac {C}{b^{2}} - \frac {2 D a}{b^{3}}\right ) + \frac {x \left (A b^{3} - B a b^{2} + C a^{2} b - D a^{3}\right )}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b^{7}}} \left (A b^{3} + B a b^{2} - 3 C a^{2} b + 5 D a^{3}\right ) \log {\left (- a^{2} b^{3} \sqrt {- \frac {1}{a^{3} b^{7}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b^{7}}} \left (A b^{3} + B a b^{2} - 3 C a^{2} b + 5 D a^{3}\right ) \log {\left (a^{2} b^{3} \sqrt {- \frac {1}{a^{3} b^{7}}} + x \right )}}{4} \] Input:
integrate((D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**2,x)
Output:
D*x**3/(3*b**2) + x*(C/b**2 - 2*D*a/b**3) + x*(A*b**3 - B*a*b**2 + C*a**2* b - D*a**3)/(2*a**2*b**3 + 2*a*b**4*x**2) - sqrt(-1/(a**3*b**7))*(A*b**3 + B*a*b**2 - 3*C*a**2*b + 5*D*a**3)*log(-a**2*b**3*sqrt(-1/(a**3*b**7)) + x )/4 + sqrt(-1/(a**3*b**7))*(A*b**3 + B*a*b**2 - 3*C*a**2*b + 5*D*a**3)*log (a**2*b**3*sqrt(-1/(a**3*b**7)) + x)/4
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (D a^{3} - C a^{2} b + B a b^{2} - A b^{3}\right )} x}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {D b x^{3} - 3 \, {\left (2 \, D a - C b\right )} x}{3 \, b^{3}} + \frac {{\left (5 \, D a^{3} - 3 \, C a^{2} b + B a b^{2} + A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{3}} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-1/2*(D*a^3 - C*a^2*b + B*a*b^2 - A*b^3)*x/(a*b^4*x^2 + a^2*b^3) + 1/3*(D* b*x^3 - 3*(2*D*a - C*b)*x)/b^3 + 1/2*(5*D*a^3 - 3*C*a^2*b + B*a*b^2 + A*b^ 3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^3)
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (5 \, D a^{3} - 3 \, C a^{2} b + B a b^{2} + A b^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{3}} - \frac {D a^{3} x - C a^{2} b x + B a b^{2} x - A b^{3} x}{2 \, {\left (b x^{2} + a\right )} a b^{3}} + \frac {D b^{4} x^{3} - 6 \, D a b^{3} x + 3 \, C b^{4} x}{3 \, b^{6}} \] Input:
integrate((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^2,x, algorithm="giac")
Output:
1/2*(5*D*a^3 - 3*C*a^2*b + B*a*b^2 + A*b^3)*arctan(b*x/sqrt(a*b))/(sqrt(a* b)*a*b^3) - 1/2*(D*a^3*x - C*a^2*b*x + B*a*b^2*x - A*b^3*x)/((b*x^2 + a)*a *b^3) + 1/3*(D*b^4*x^3 - 6*D*a*b^3*x + 3*C*b^4*x)/b^6
Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:
int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2)^2,x)
Output:
int((A + B*x^2 + C*x^4 + x^6*D)/(a + b*x^2)^2, x)
Time = 0.16 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.79 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^2} \, dx=\frac {15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} d -9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b c +15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b d \,x^{2}+6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3}-9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c \,x^{2}+6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{4} x^{2}-15 a^{3} b d x +9 a^{2} b^{2} c x -10 a^{2} b^{2} d \,x^{3}+6 a \,b^{3} c \,x^{3}+2 a \,b^{3} d \,x^{5}}{6 a \,b^{4} \left (b \,x^{2}+a \right )} \] Input:
int((D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^2,x)
Output:
(15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*d - 9*sqrt(b)*sqrt( a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*c + 15*sqrt(b)*sqrt(a)*atan((b*x)/ (sqrt(b)*sqrt(a)))*a**2*b*d*x**2 + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s qrt(a)))*a*b**3 - 9*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c *x**2 + 6*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**4*x**2 - 15*a** 3*b*d*x + 9*a**2*b**2*c*x - 10*a**2*b**2*d*x**3 + 6*a*b**3*c*x**3 + 2*a*b* *3*d*x**5)/(6*a*b**4*(a + b*x**2))