Integrand size = 25, antiderivative size = 93 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx=-\frac {a (b B-a D)-b (A b-a C) x}{2 a b^2 \left (a+b x^2\right )}+\frac {(A b+a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {D \log \left (a+b x^2\right )}{2 b^2} \] Output:
-1/2*(a*(B*b-D*a)-b*(A*b-C*a)*x)/a/b^2/(b*x^2+a)+1/2*(A*b+C*a)*arctan(b^(1 /2)*x/a^(1/2))/a^(3/2)/b^(3/2)+1/2*D*ln(b*x^2+a)/b^2
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {a^2 D+A b^2 x-a b (B+C x)}{a \left (a+b x^2\right )}+\frac {\sqrt {b} (A b+a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}+D \log \left (a+b x^2\right )}{2 b^2} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^2,x]
Output:
((a^2*D + A*b^2*x - a*b*(B + C*x))/(a*(a + b*x^2)) + (Sqrt[b]*(A*b + a*C)* ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2) + D*Log[a + b*x^2])/(2*b^2)
Time = 0.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2345, 25, 27, 452, 218, 240}
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.
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle -\frac {\int -\frac {A b+a C+2 a D x}{b \left (b x^2+a\right )}dx}{2 a}-\frac {a \left (B-\frac {a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {A b+a C+2 a D x}{b \left (b x^2+a\right )}dx}{2 a}-\frac {a \left (B-\frac {a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {A b+a C+2 a D x}{b x^2+a}dx}{2 a b}-\frac {a \left (B-\frac {a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 452 |
\(\displaystyle \frac {(a C+A b) \int \frac {1}{b x^2+a}dx+2 a D \int \frac {x}{b x^2+a}dx}{2 a b}-\frac {a \left (B-\frac {a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 a D \int \frac {x}{b x^2+a}dx+\frac {(a C+A b) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}}{2 a b}-\frac {a \left (B-\frac {a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {\frac {(a C+A b) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {a D \log \left (a+b x^2\right )}{b}}{2 a b}-\frac {a \left (B-\frac {a D}{b}\right )-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^2,x]
Output:
-1/2*(a*(B - (a*D)/b) - (A*b - a*C)*x)/(a*b*(a + b*x^2)) + (((A*b + a*C)*A rcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + (a*D*Log[a + b*x^2])/b)/(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[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
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 = 0.45 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\frac {\left (A b -C a \right ) x}{2 a b}-\frac {B b -D a}{2 b^{2}}}{b \,x^{2}+a}+\frac {\frac {D a \ln \left (b \,x^{2}+a \right )}{b}+\frac {\left (A b +C a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{2 a b}\) | \(88\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
(1/2*(A*b-C*a)/a/b*x-1/2*(B*b-D*a)/b^2)/(b*x^2+a)+1/2/a/b*(D*a/b*ln(b*x^2+ a)+(A*b+C*a)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.76 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx=\left [\frac {2 \, D a^{3} - 2 \, B a^{2} b - {\left (C a^{2} + A a b + {\left (C a b + A b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, {\left (C a^{2} b - A a b^{2}\right )} x + 2 \, {\left (D a^{2} b x^{2} + D a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, \frac {D a^{3} - B a^{2} b + {\left (C a^{2} + A a b + {\left (C a b + A b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (C a^{2} b - A a b^{2}\right )} x + {\left (D a^{2} b x^{2} + D a^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}\right ] \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/4*(2*D*a^3 - 2*B*a^2*b - (C*a^2 + A*a*b + (C*a*b + A*b^2)*x^2)*sqrt(-a* b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - 2*(C*a^2*b - A*a*b^2)*x + 2*(D*a^2*b*x^2 + D*a^3)*log(b*x^2 + a))/(a^2*b^3*x^2 + a^3*b^2), 1/2*(D *a^3 - B*a^2*b + (C*a^2 + A*a*b + (C*a*b + A*b^2)*x^2)*sqrt(a*b)*arctan(sq rt(a*b)*x/a) - (C*a^2*b - A*a*b^2)*x + (D*a^2*b*x^2 + D*a^3)*log(b*x^2 + a ))/(a^2*b^3*x^2 + a^3*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (82) = 164\).
Time = 0.90 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.51 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx=\left (\frac {D}{2 b^{2}} - \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right ) \log {\left (x + \frac {- 2 D a^{2} + 4 a^{2} b^{2} \left (\frac {D}{2 b^{2}} - \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right )}{A b^{2} + C a b} \right )} + \left (\frac {D}{2 b^{2}} + \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right ) \log {\left (x + \frac {- 2 D a^{2} + 4 a^{2} b^{2} \left (\frac {D}{2 b^{2}} + \frac {\sqrt {- a^{3} b^{5}} \left (A b + C a\right )}{4 a^{3} b^{4}}\right )}{A b^{2} + C a b} \right )} + \frac {- B a b + D a^{2} + x \left (A b^{2} - C a b\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)
Output:
(D/(2*b**2) - sqrt(-a**3*b**5)*(A*b + C*a)/(4*a**3*b**4))*log(x + (-2*D*a* *2 + 4*a**2*b**2*(D/(2*b**2) - sqrt(-a**3*b**5)*(A*b + C*a)/(4*a**3*b**4)) )/(A*b**2 + C*a*b)) + (D/(2*b**2) + sqrt(-a**3*b**5)*(A*b + C*a)/(4*a**3*b **4))*log(x + (-2*D*a**2 + 4*a**2*b**2*(D/(2*b**2) + sqrt(-a**3*b**5)*(A*b + C*a)/(4*a**3*b**4)))/(A*b**2 + C*a*b)) + (-B*a*b + D*a**2 + x*(A*b**2 - C*a*b))/(2*a**2*b**2 + 2*a*b**3*x**2)
Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx=\frac {D a^{2} - B a b - {\left (C a b - A b^{2}\right )} x}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {D \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {{\left (C a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
1/2*(D*a^2 - B*a*b - (C*a*b - A*b^2)*x)/(a*b^3*x^2 + a^2*b^2) + 1/2*D*log( b*x^2 + a)/b^2 + 1/2*(C*a + A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b)
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx=\frac {D \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {{\left (C a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} - \frac {{\left (C a - A b\right )} x - \frac {D a^{2} - B a b}{b}}{2 \, {\left (b x^{2} + a\right )} a b} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x, algorithm="giac")
Output:
1/2*D*log(b*x^2 + a)/b^2 + 1/2*(C*a + A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b )*a*b) - 1/2*((C*a - A*b)*x - (D*a^2 - B*a*b)/b)/((b*x^2 + a)*a*b)
Time = 0.57 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx=\frac {\left (\ln \left (b\,x^2+a\right )+\frac {a}{b\,x^2+a}\right )\,D}{2\,b^2}-\frac {B}{2\,b\,\left (b\,x^2+a\right )}+\frac {A\,x}{2\,a\,\left (b\,x^2+a\right )}-\frac {C\,x}{2\,b\,\left (b\,x^2+a\right )}+\frac {A\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}}+\frac {C\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,b^{3/2}} \] Input:
int((A + B*x + C*x^2 + x^3*D)/(a + b*x^2)^2,x)
Output:
((log(a + b*x^2) + a/(a + b*x^2))*D)/(2*b^2) - B/(2*b*(a + b*x^2)) + (A*x) /(2*a*(a + b*x^2)) - (C*x)/(2*b*(a + b*x^2)) + (A*atan((b^(1/2)*x)/a^(1/2) ))/(2*a^(3/2)*b^(1/2)) + (C*atan((b^(1/2)*x)/a^(1/2)))/(2*a^(1/2)*b^(3/2))
Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.68 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a c +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b c \,x^{2}+\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} d +\mathrm {log}\left (b \,x^{2}+a \right ) a b d \,x^{2}+a \,b^{2} x -a b c x -a b d \,x^{2}+b^{3} x^{2}}{2 a \,b^{2} \left (b \,x^{2}+a \right )} \] Input:
int((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x)
Output:
(sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b + sqrt(b)*sqrt(a)*atan( (b*x)/(sqrt(b)*sqrt(a)))*a*c + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a) ))*b**2*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b*c*x**2 + lo g(a + b*x**2)*a**2*d + log(a + b*x**2)*a*b*d*x**2 + a*b**2*x - a*b*c*x - a *b*d*x**2 + b**3*x**2)/(2*a*b**2*(a + b*x**2))