Integrand size = 22, antiderivative size = 77 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c^3}{a x}+\frac {d^2 (3 b c-a d) x}{b^2}+\frac {d^3 x^3}{3 b}-\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{5/2}} \] Output:
-c^3/a/x+d^2*(-a*d+3*b*c)*x/b^2+1/3*d^3*x^3/b-(-a*d+b*c)^3*arctan(b^(1/2)* x/a^(1/2))/a^(3/2)/b^(5/2)
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c^3}{a x}+\frac {d^2 (3 b c-a d) x}{b^2}+\frac {d^3 x^3}{3 b}+\frac {(-b c+a d)^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{5/2}} \] Input:
Integrate[(c + d*x^2)^3/(x^2*(a + b*x^2)),x]
Output:
-(c^3/(a*x)) + (d^2*(3*b*c - a*d)*x)/b^2 + (d^3*x^3)/(3*b) + ((-(b*c) + a* d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(5/2))
Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {364, 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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 364 |
\(\displaystyle \int \left (\frac {d^2 (3 b c-a d)}{b^2}+\frac {(a d-b c)^3}{a b^2 \left (a+b x^2\right )}+\frac {c^3}{a x^2}+\frac {d^3 x^2}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^3}{a^{3/2} b^{5/2}}+\frac {d^2 x (3 b c-a d)}{b^2}-\frac {c^3}{a x}+\frac {d^3 x^3}{3 b}\) |
Input:
Int[(c + d*x^2)^3/(x^2*(a + b*x^2)),x]
Output:
-(c^3/(a*x)) + (d^2*(3*b*c - a*d)*x)/b^2 + (d^3*x^3)/(3*b) - ((b*c - a*d)^ 3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*b^(5/2))
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^2)^p/(c + d*x^2)), x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && (In tegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Time = 0.46 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23
method | result | size |
default | \(-\frac {d^{2} \left (-\frac {1}{3} b d \,x^{3}+a d x -3 b c x \right )}{b^{2}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \,b^{2} \sqrt {a b}}-\frac {c^{3}}{a x}\) | \(95\) |
risch | \(\frac {d^{3} x^{3}}{3 b}-\frac {d^{3} a x}{b^{2}}+\frac {3 d^{2} c x}{b}-\frac {c^{3}}{a x}-\frac {a^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) d^{3}}{2 b^{2} \sqrt {-a b}}+\frac {3 a \ln \left (-\sqrt {-a b}\, x +a \right ) c \,d^{2}}{2 b \sqrt {-a b}}-\frac {3 \ln \left (-\sqrt {-a b}\, x +a \right ) c^{2} d}{2 \sqrt {-a b}}+\frac {b \ln \left (-\sqrt {-a b}\, x +a \right ) c^{3}}{2 \sqrt {-a b}\, a}+\frac {a^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) d^{3}}{2 b^{2} \sqrt {-a b}}-\frac {3 a \ln \left (-\sqrt {-a b}\, x -a \right ) c \,d^{2}}{2 b \sqrt {-a b}}+\frac {3 \ln \left (-\sqrt {-a b}\, x -a \right ) c^{2} d}{2 \sqrt {-a b}}-\frac {b \ln \left (-\sqrt {-a b}\, x -a \right ) c^{3}}{2 \sqrt {-a b}\, a}\) | \(268\) |
Input:
int((d*x^2+c)^3/x^2/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-d^2/b^2*(-1/3*b*d*x^3+a*d*x-3*b*c*x)+1/a/b^2*(a^3*d^3-3*a^2*b*c*d^2+3*a*b ^2*c^2*d-b^3*c^3)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))-c^3/a/x
Time = 0.08 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.29 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=\left [\frac {2 \, a^{2} b^{2} d^{3} x^{4} - 6 \, a b^{3} c^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a b} x \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{6 \, a^{2} b^{3} x}, \frac {a^{2} b^{2} d^{3} x^{4} - 3 \, a b^{3} c^{3} - 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b} x \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{3 \, a^{2} b^{3} x}\right ] \] Input:
integrate((d*x^2+c)^3/x^2/(b*x^2+a),x, algorithm="fricas")
Output:
[1/6*(2*a^2*b^2*d^3*x^4 - 6*a*b^3*c^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2 *b*c*d^2 - a^3*d^3)*sqrt(-a*b)*x*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 6*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^2)/(a^2*b^3*x), 1/3*(a^2*b^2*d^3* x^4 - 3*a*b^3*c^3 - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)* sqrt(a*b)*x*arctan(sqrt(a*b)*x/a) + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^2)/( a^2*b^3*x)]
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (65) = 130\).
Time = 0.38 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.87 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=x \left (- \frac {a d^{3}}{b^{2}} + \frac {3 c d^{2}}{b}\right ) - \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log {\left (- \frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log {\left (\frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x^{3}}{3 b} - \frac {c^{3}}{a x} \] Input:
integrate((d*x**2+c)**3/x**2/(b*x**2+a),x)
Output:
x*(-a*d**3/b**2 + 3*c*d**2/b) - sqrt(-1/(a**3*b**5))*(a*d - b*c)**3*log(-a **2*b**2*sqrt(-1/(a**3*b**5))*(a*d - b*c)**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3) + x)/2 + sqrt(-1/(a**3*b**5))*(a*d - b*c)** 3*log(a**2*b**2*sqrt(-1/(a**3*b**5))*(a*d - b*c)**3/(a**3*d**3 - 3*a**2*b* c*d**2 + 3*a*b**2*c**2*d - b**3*c**3) + x)/2 + d**3*x**3/(3*b) - c**3/(a*x )
Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.31 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c^{3}}{a x} + \frac {b d^{3} x^{3} + 3 \, {\left (3 \, b c d^{2} - a d^{3}\right )} x}{3 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b^{2}} \] Input:
integrate((d*x^2+c)^3/x^2/(b*x^2+a),x, algorithm="maxima")
Output:
-c^3/(a*x) + 1/3*(b*d^3*x^3 + 3*(3*b*c*d^2 - a*d^3)*x)/b^2 - (b^3*c^3 - 3* a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a* b^2)
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c^{3}}{a x} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b^{2}} + \frac {b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x - 3 \, a b d^{3} x}{3 \, b^{3}} \] Input:
integrate((d*x^2+c)^3/x^2/(b*x^2+a),x, algorithm="giac")
Output:
-c^3/(a*x) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(b* x/sqrt(a*b))/(sqrt(a*b)*a*b^2) + 1/3*(b^2*d^3*x^3 + 9*b^2*c*d^2*x - 3*a*b* d^3*x)/b^3
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.53 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=\frac {d^3\,x^3}{3\,b}-\frac {c^3}{a\,x}-x\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{3/2}\,b^{5/2}} \] Input:
int((c + d*x^2)^3/(x^2*(a + b*x^2)),x)
Output:
(d^3*x^3)/(3*b) - c^3/(a*x) - x*((a*d^3)/b^2 - (3*c*d^2)/b) + (atan((b^(1/ 2)*x*(a*d - b*c)^3)/(a^(1/2)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b* c*d^2)))*(a*d - b*c)^3)/(a^(3/2)*b^(5/2))
Time = 0.24 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.14 \[ \int \frac {\left (c+d x^2\right )^3}{x^2 \left (a+b x^2\right )} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} d^{3} x -9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b c \,d^{2} x +9 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c^{2} d x -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} c^{3} x -3 a^{3} b \,d^{3} x^{2}+9 a^{2} b^{2} c \,d^{2} x^{2}+a^{2} b^{2} d^{3} x^{4}-3 a \,b^{3} c^{3}}{3 a^{2} b^{3} x} \] Input:
int((d*x^2+c)^3/x^2/(b*x^2+a),x)
Output:
(3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*d**3*x - 9*sqrt(b)*s qrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*c*d**2*x + 9*sqrt(b)*sqrt(a)*a tan((b*x)/(sqrt(b)*sqrt(a)))*a*b**2*c**2*d*x - 3*sqrt(b)*sqrt(a)*atan((b*x )/(sqrt(b)*sqrt(a)))*b**3*c**3*x - 3*a**3*b*d**3*x**2 + 9*a**2*b**2*c*d**2 *x**2 + a**2*b**2*d**3*x**4 - 3*a*b**3*c**3)/(3*a**2*b**3*x)