Integrand size = 19, antiderivative size = 82 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{2 a b^2 \left (a+b x^2\right )}+\frac {(b c-a d) (b c+3 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}} \] Output:
d^2*x/b^2+1/2*(-a*d+b*c)^2*x/a/b^2/(b*x^2+a)+1/2*(-a*d+b*c)*(3*a*d+b*c)*ar ctan(b^(1/2)*x/a^(1/2))/a^(3/2)/b^(5/2)
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{2 a b^2 \left (a+b x^2\right )}+\frac {\left (b^2 c^2+2 a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{5/2}} \] Input:
Integrate[(c + d*x^2)^2/(a + b*x^2)^2,x]
Output:
(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(2*a*b^2*(a + b*x^2)) + ((b^2*c^2 + 2*a*b* c*d - 3*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(5/2))
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {300, 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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \int \left (\frac {-a^2 d^2+2 b d x^2 (b c-a d)+b^2 c^2}{b^2 \left (a+b x^2\right )^2}+\frac {d^2}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d) (3 a d+b c)}{2 a^{3/2} b^{5/2}}+\frac {x (b c-a d)^2}{2 a b^2 \left (a+b x^2\right )}+\frac {d^2 x}{b^2}\) |
Input:
Int[(c + d*x^2)^2/(a + b*x^2)^2,x]
Output:
(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(2*a*b^2*(a + b*x^2)) + ((b*c - a*d)*(b*c + 3*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(5/2))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Time = 0.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {d^{2} x}{b^{2}}-\frac {-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{2}}\) | \(94\) |
risch | \(\frac {d^{2} x}{b^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{2 a \,b^{2} \left (b \,x^{2}+a \right )}-\frac {3 a \ln \left (b x -\sqrt {-a b}\right ) d^{2}}{4 b^{2} \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c d}{2 b \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c^{2}}{4 \sqrt {-a b}\, a}+\frac {3 a \ln \left (-b x -\sqrt {-a b}\right ) d^{2}}{4 b^{2} \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c d}{2 b \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c^{2}}{4 \sqrt {-a b}\, a}\) | \(214\) |
Input:
int((d*x^2+c)^2/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
d^2*x/b^2-1/b^2*(-1/2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a*x/(b*x^2+a)+1/2*(3*a^2 *d^2-2*a*b*c*d-b^2*c^2)/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.62 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, a^{2} b^{2} d^{2} x^{3} + {\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x}{4 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, \frac {2 \, a^{2} b^{2} d^{2} x^{3} + {\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x}{2 \, {\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \] Input:
integrate((d*x^2+c)^2/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/4*(4*a^2*b^2*d^2*x^3 + (a*b^2*c^2 + 2*a^2*b*c*d - 3*a^3*d^2 + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*x^2)*sqrt(-a*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + 3*a^3*b*d^2)*x)/(a^2*b^4 *x^2 + a^3*b^3), 1/2*(2*a^2*b^2*d^2*x^3 + (a*b^2*c^2 + 2*a^2*b*c*d - 3*a^3 *d^2 + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*x^2)*sqrt(a*b)*arctan(sqrt(a* b)*x/a) + (a*b^3*c^2 - 2*a^2*b^2*c*d + 3*a^3*b*d^2)*x)/(a^2*b^4*x^2 + a^3* b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (73) = 146\).
Time = 0.35 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.88 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log {\left (- \frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right ) \log {\left (\frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right ) \left (3 a d + b c\right )}{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2}} + x \right )}}{4} + \frac {d^{2} x}{b^{2}} \] Input:
integrate((d*x**2+c)**2/(b*x**2+a)**2,x)
Output:
x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(2*a**2*b**2 + 2*a*b**3*x**2) + sqrt (-1/(a**3*b**5))*(a*d - b*c)*(3*a*d + b*c)*log(-a**2*b**2*sqrt(-1/(a**3*b* *5))*(a*d - b*c)*(3*a*d + b*c)/(3*a**2*d**2 - 2*a*b*c*d - b**2*c**2) + x)/ 4 - sqrt(-1/(a**3*b**5))*(a*d - b*c)*(3*a*d + b*c)*log(a**2*b**2*sqrt(-1/( a**3*b**5))*(a*d - b*c)*(3*a*d + b*c)/(3*a**2*d**2 - 2*a*b*c*d - b**2*c**2 ) + x)/4 + d**2*x/b**2
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {d^{2} x}{b^{2}} + \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} \] Input:
integrate((d*x^2+c)^2/(b*x^2+a)^2,x, algorithm="maxima")
Output:
1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/(a*b^3*x^2 + a^2*b^2) + d^2*x/b^2 + 1/2*(b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b ^2)
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^{2} x}{b^{2}} + \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{2}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \, {\left (b x^{2} + a\right )} a b^{2}} \] Input:
integrate((d*x^2+c)^2/(b*x^2+a)^2,x, algorithm="giac")
Output:
d^2*x/b^2 + 1/2*(b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*arctan(b*x/sqrt(a*b))/(s qrt(a*b)*a*b^2) + 1/2*(b^2*c^2*x - 2*a*b*c*d*x + a^2*d^2*x)/((b*x^2 + a)*a *b^2)
Time = 0.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.51 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d^2\,x}{b^2}+\frac {x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a\,\left (b^3\,x^2+a\,b^2\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+b\,c\right )}{\sqrt {a}\,\left (-3\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+b\,c\right )}{2\,a^{3/2}\,b^{5/2}} \] Input:
int((c + d*x^2)^2/(a + b*x^2)^2,x)
Output:
(d^2*x)/b^2 + (x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(2*a*(a*b^2 + b^3*x^2)) + (atan((b^(1/2)*x*(a*d - b*c)*(3*a*d + b*c))/(a^(1/2)*(b^2*c^2 - 3*a^2*d^ 2 + 2*a*b*c*d)))*(a*d - b*c)*(3*a*d + b*c))/(2*a^(3/2)*b^(5/2))
Time = 0.21 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.62 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} d^{2}+2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b c d -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,d^{2} x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c^{2}+2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c d \,x^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} c^{2} x^{2}+3 a^{3} b \,d^{2} x -2 a^{2} b^{2} c d x +2 a^{2} b^{2} d^{2} x^{3}+a \,b^{3} c^{2} x}{2 a^{2} b^{3} \left (b \,x^{2}+a \right )} \] Input:
int((d*x^2+c)^2/(b*x^2+a)^2,x)
Output:
( - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*d**2 + 2*sqrt(b)* sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*c*d - 3*sqrt(b)*sqrt(a)*atan( (b*x)/(sqrt(b)*sqrt(a)))*a**2*b*d**2*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sq rt(b)*sqrt(a)))*a*b**2*c**2 + 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a )))*a*b**2*c*d*x**2 + sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b**3*c **2*x**2 + 3*a**3*b*d**2*x - 2*a**2*b**2*c*d*x + 2*a**2*b**2*d**2*x**3 + a *b**3*c**2*x)/(2*a**2*b**3*(a + b*x**2))