Integrand size = 19, antiderivative size = 69 \[ \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx=-\frac {a B-(A b-a C) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \] Output:
-1/2*(a*B-(A*b-C*a)*x)/a/b/(b*x^2+a)+1/2*(A*b+C*a)*arctan(b^(1/2)*x/a^(1/2 ))/a^(3/2)/b^(3/2)
Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx=\frac {-a B+A b x-a C x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a C) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}} \] Input:
Integrate[(A + x*(B + C*x))/(a + b*x^2)^2,x]
Output:
(-(a*B) + A*b*x - a*C*x)/(2*a*b*(a + b*x^2)) + ((A*b + a*C)*ArcTan[(Sqrt[b ]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2))
Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2083, 2345, 25, 27, 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.
\(\displaystyle \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 2083 |
\(\displaystyle \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^2}dx\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle -\frac {\int -\frac {A b+a C}{b \left (b x^2+a\right )}dx}{2 a}-\frac {a B-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {A b+a C}{b \left (b x^2+a\right )}dx}{2 a}-\frac {a B-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(a C+A b) \int \frac {1}{b x^2+a}dx}{2 a b}-\frac {a B-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(a C+A b) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}-\frac {a B-x (A b-a C)}{2 a b \left (a+b x^2\right )}\) |
Input:
Int[(A + x*(B + C*x))/(a + b*x^2)^2,x]
Output:
-1/2*(a*B - (A*b - a*C)*x)/(a*b*(a + b*x^2)) + ((A*b + a*C)*ArcTan[(Sqrt[b ]*x)/Sqrt[a]])/(2*a^(3/2)*b^(3/2))
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[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum [v, x]^q, x] /; FreeQ[{p, q}, x] && QuadraticQ[{u, v}, x] && !QuadraticMat chQ[{u, v}, x]
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 = 0.85 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\frac {\left (A b -C a \right ) x}{2 a b}-\frac {B}{2 b}}{b \,x^{2}+a}+\frac {\left (A b +C a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a b \sqrt {a b}}\) | \(65\) |
risch | \(\frac {\frac {\left (A b -C a \right ) x}{2 a b}-\frac {B}{2 b}}{b \,x^{2}+a}-\frac {\ln \left (b x +\sqrt {-a b}\right ) A}{4 \sqrt {-a b}\, a}-\frac {\ln \left (b x +\sqrt {-a b}\right ) C}{4 \sqrt {-a b}\, b}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) A}{4 \sqrt {-a b}\, a}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) C}{4 \sqrt {-a b}\, b}\) | \(130\) |
Input:
int((A+x*(C*x+B))/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
(1/2*(A*b-C*a)/a/b*x-1/2*B/b)/(b*x^2+a)+1/2/a*(A*b+C*a)/b/(a*b)^(1/2)*arct an(b*x/(a*b)^(1/2))
Time = 0.07 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.83 \[ \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx=\left [-\frac {2 \, B a^{2} b + {\left (C a^{2} + A a b + {\left (C a b + A b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (C a^{2} b - A a b^{2}\right )} x}{4 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, -\frac {B a^{2} b - {\left (C a^{2} + A a b + {\left (C a b + A b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (C a^{2} b - A a b^{2}\right )} x}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}\right ] \] Input:
integrate((A+x*(C*x+B))/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[-1/4*(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)/(a^2*b^ 3*x^2 + a^3*b^2), -1/2*(B*a^2*b - (C*a^2 + A*a*b + (C*a*b + A*b^2)*x^2)*sq rt(a*b)*arctan(sqrt(a*b)*x/a) + (C*a^2*b - A*a*b^2)*x)/(a^2*b^3*x^2 + a^3* b^2)]
Time = 0.39 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.68 \[ \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx=- \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \left (A b + C a\right ) \log {\left (- a^{2} b \sqrt {- \frac {1}{a^{3} b^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \left (A b + C a\right ) \log {\left (a^{2} b \sqrt {- \frac {1}{a^{3} b^{3}}} + x \right )}}{4} + \frac {- B a + x \left (A b - C a\right )}{2 a^{2} b + 2 a b^{2} x^{2}} \] Input:
integrate((A+x*(C*x+B))/(b*x**2+a)**2,x)
Output:
-sqrt(-1/(a**3*b**3))*(A*b + C*a)*log(-a**2*b*sqrt(-1/(a**3*b**3)) + x)/4 + sqrt(-1/(a**3*b**3))*(A*b + C*a)*log(a**2*b*sqrt(-1/(a**3*b**3)) + x)/4 + (-B*a + x*(A*b - C*a))/(2*a**2*b + 2*a*b**2*x**2)
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx=-\frac {B a + {\left (C a - A b\right )} x}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} + \frac {{\left (C a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \] Input:
integrate((A+x*(C*x+B))/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-1/2*(B*a + (C*a - A*b)*x)/(a*b^2*x^2 + a^2*b) + 1/2*(C*a + A*b)*arctan(b* x/sqrt(a*b))/(sqrt(a*b)*a*b)
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (C a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} - \frac {C a x - A b x + B a}{2 \, {\left (b x^{2} + a\right )} a b} \] Input:
integrate((A+x*(C*x+B))/(b*x^2+a)^2,x, algorithm="giac")
Output:
1/2*(C*a + A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b) - 1/2*(C*a*x - A*b*x + B*a)/((b*x^2 + a)*a*b)
Time = 15.71 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b+C\,a\right )}{2\,a^{3/2}\,b^{3/2}}-\frac {\frac {B}{2\,b}-\frac {x\,\left (A\,b-C\,a\right )}{2\,a\,b}}{b\,x^2+a} \] Input:
int((A + x*(B + C*x))/(a + b*x^2)^2,x)
Output:
(atan((b^(1/2)*x)/a^(1/2))*(A*b + C*a))/(2*a^(3/2)*b^(3/2)) - (B/(2*b) - ( x*(A*b - C*a))/(2*a*b))/(a + b*x^2)
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.74 \[ \int \frac {A+x (B+C x)}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a c +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b c \,x^{2}+a \,b^{2} x -a b c x +b^{3} x^{2}}{2 a \,b^{2} \left (b \,x^{2}+a \right )} \] Input:
int((A+x*(C*x+B))/(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 + a* b**2*x - a*b*c*x + b**3*x**2)/(2*a*b**2*(a + b*x**2))