Integrand size = 22, antiderivative size = 83 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx=\frac {C x}{b^2}+\frac {\left (\frac {A}{a}-\frac {b B-a C}{b^2}\right ) x}{2 \left (a+b x^2\right )}+\frac {\left (A b^2+a (b B-3 a C)\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}} \] Output:
C*x/b^2+(A/a-(B*b-C*a)/b^2)*x/(2*b*x^2+2*a)+1/2*(A*b^2+a*(B*b-3*C*a))*arct an(b^(1/2)*x/a^(1/2))/a^(3/2)/b^(5/2)
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx=\frac {C x}{b^2}+\frac {\left (A b^2-a b B+a^2 C\right ) x}{2 a b^2 \left (a+b x^2\right )}-\frac {\left (-A b^2-a b B+3 a^2 C\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(a + b*x^2)^2,x]
Output:
(C*x)/b^2 + ((A*b^2 - a*b*B + a^2*C)*x)/(2*a*b^2*(a + b*x^2)) - ((-(A*b^2) - a*b*B + 3*a^2*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(5/2))
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1471, 25, 27, 299, 218}
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.
\(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1471 |
\(\displaystyle \frac {x \left (A b^2-a (b B-a C)\right )}{2 a b^2 \left (a+b x^2\right )}-\frac {\int -\frac {2 a C x^2+b \left (A+\frac {a (b B-a C)}{b^2}\right )}{b \left (b x^2+a\right )}dx}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {2 a C x^2+A b+a \left (B-\frac {a C}{b}\right )}{b \left (b x^2+a\right )}dx}{2 a}+\frac {x \left (A b^2-a (b B-a C)\right )}{2 a b^2 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {2 a C x^2+A b+a \left (B-\frac {a C}{b}\right )}{b x^2+a}dx}{2 a b}+\frac {x \left (A b^2-a (b B-a C)\right )}{2 a b^2 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\frac {\left (a (b B-3 a C)+A b^2\right ) \int \frac {1}{b x^2+a}dx}{b}+\frac {2 a C x}{b}}{2 a b}+\frac {x \left (A b^2-a (b B-a C)\right )}{2 a b^2 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a (b B-3 a C)+A b^2\right )}{\sqrt {a} b^{3/2}}+\frac {2 a C x}{b}}{2 a b}+\frac {x \left (A b^2-a (b B-a C)\right )}{2 a b^2 \left (a+b x^2\right )}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(a + b*x^2)^2,x]
Output:
((A*b^2 - a*(b*B - a*C))*x)/(2*a*b^2*(a + b*x^2)) + ((2*a*C*x)/b + ((A*b^2 + a*(b*B - 3*a*C))*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(3/2)))/(2*a*b )
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], 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] && LtQ[q, -1]
Time = 0.50 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {C x}{b^{2}}+\frac {\frac {\left (b^{2} A -a b B +a^{2} C \right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (b^{2} A +a b B -3 a^{2} C \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{2}}\) | \(79\) |
risch | \(\frac {C x}{b^{2}}+\frac {\left (b^{2} A -a b B +a^{2} C \right ) x}{2 a \,b^{2} \left (b \,x^{2}+a \right )}-\frac {A \ln \left (b x +\sqrt {-a b}\right )}{4 \sqrt {-a b}\, a}-\frac {\ln \left (b x +\sqrt {-a b}\right ) B}{4 b \sqrt {-a b}}+\frac {3 a \ln \left (b x +\sqrt {-a b}\right ) C}{4 b^{2} \sqrt {-a b}}+\frac {A \ln \left (-b x +\sqrt {-a b}\right )}{4 \sqrt {-a b}\, a}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) B}{4 b \sqrt {-a b}}-\frac {3 a \ln \left (-b x +\sqrt {-a b}\right ) C}{4 b^{2} \sqrt {-a b}}\) | \(185\) |
Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
C*x/b^2+1/b^2*(1/2*(A*b^2-B*a*b+C*a^2)/a*x/(b*x^2+a)+1/2*(A*b^2+B*a*b-3*C* a^2)/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.23 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, C a^{2} b^{2} x^{3} + {\left (3 \, C a^{3} - B a^{2} b - A a b^{2} + {\left (3 \, C a^{2} b - B a b^{2} - A b^{3}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (3 \, C a^{3} b - B a^{2} b^{2} + A a b^{3}\right )} x}{4 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, \frac {2 \, C a^{2} b^{2} x^{3} - {\left (3 \, C a^{3} - B a^{2} b - A a b^{2} + {\left (3 \, C a^{2} b - B a b^{2} - A b^{3}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (3 \, C a^{3} b - B a^{2} b^{2} + A a b^{3}\right )} x}{2 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/4*(4*C*a^2*b^2*x^3 + (3*C*a^3 - B*a^2*b - A*a*b^2 + (3*C*a^2*b - B*a*b^ 2 - A*b^3)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*(3*C*a^3*b - B*a^2*b^2 + A*a*b^3)*x)/(a^2*b^4*x^2 + a^3*b^3), 1/2*(2*C* a^2*b^2*x^3 - (3*C*a^3 - B*a^2*b - A*a*b^2 + (3*C*a^2*b - B*a*b^2 - A*b^3) *x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + (3*C*a^3*b - B*a^2*b^2 + A*a*b^3)* x)/(a^2*b^4*x^2 + a^3*b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (73) = 146\).
Time = 0.41 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx=\frac {C x}{b^{2}} + \frac {x \left (A b^{2} - B a b + C a^{2}\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (- A b^{2} - B a b + 3 C a^{2}\right ) \log {\left (- a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (- A b^{2} - B a b + 3 C a^{2}\right ) \log {\left (a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} + x \right )}}{4} \] Input:
integrate((C*x**4+B*x**2+A)/(b*x**2+a)**2,x)
Output:
C*x/b**2 + x*(A*b**2 - B*a*b + C*a**2)/(2*a**2*b**2 + 2*a*b**3*x**2) + sqr t(-1/(a**3*b**5))*(-A*b**2 - B*a*b + 3*C*a**2)*log(-a**2*b**2*sqrt(-1/(a** 3*b**5)) + x)/4 - sqrt(-1/(a**3*b**5))*(-A*b**2 - B*a*b + 3*C*a**2)*log(a* *2*b**2*sqrt(-1/(a**3*b**5)) + x)/4
Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (C a^{2} - B a b + A b^{2}\right )} x}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {C x}{b^{2}} - \frac {{\left (3 \, C a^{2} - B a b - A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
1/2*(C*a^2 - B*a*b + A*b^2)*x/(a*b^3*x^2 + a^2*b^2) + C*x/b^2 - 1/2*(3*C*a ^2 - B*a*b - A*b^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^2)
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx=\frac {C x}{b^{2}} - \frac {{\left (3 \, C a^{2} - B a b - A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} + \frac {C a^{2} x - B a b x + A b^{2} x}{2 \, {\left (b x^{2} + a\right )} a b^{2}} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^2,x, algorithm="giac")
Output:
C*x/b^2 - 1/2*(3*C*a^2 - B*a*b - A*b^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a *b^2) + 1/2*(C*a^2*x - B*a*b*x + A*b^2*x)/((b*x^2 + a)*a*b^2)
Time = 0.41 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx=\frac {C\,x}{b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-3\,C\,a^2+B\,a\,b+A\,b^2\right )}{2\,a^{3/2}\,b^{5/2}}+\frac {x\,\left (C\,a^2-B\,a\,b+A\,b^2\right )}{2\,a\,\left (b^3\,x^2+a\,b^2\right )} \] Input:
int((A + B*x^2 + C*x^4)/(a + b*x^2)^2,x)
Output:
(C*x)/b^2 + (atan((b^(1/2)*x)/a^(1/2))*(A*b^2 - 3*C*a^2 + B*a*b))/(2*a^(3/ 2)*b^(5/2)) + (x*(A*b^2 + C*a^2 - B*a*b))/(2*a*(a*b^2 + b^3*x^2))
Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.54 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^2} \, dx=\frac {-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} c +2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b c \,x^{2}+2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} x^{2}+3 a^{2} b c x +2 a \,b^{2} c \,x^{3}}{2 a \,b^{3} \left (b \,x^{2}+a \right )} \] Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^2,x)
Output:
( - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*c + 2*sqrt(b)*sqr t(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2 - 3*sqrt(b)*sqrt(a)*atan((b*x)/( sqrt(b)*sqrt(a)))*a*b*c*x**2 + 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt( a)))*b**3*x**2 + 3*a**2*b*c*x + 2*a*b**2*c*x**3)/(2*a*b**3*(a + b*x**2))