Integrand size = 22, antiderivative size = 83 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {(b c-a d)^2 x}{d^3}-\frac {b (b c-2 a d) x^3}{3 d^2}+\frac {b^2 x^5}{5 d}-\frac {\sqrt {c} (b c-a d)^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{7/2}} \] Output:
(-a*d+b*c)^2*x/d^3-1/3*b*(-2*a*d+b*c)*x^3/d^2+1/5*b^2*x^5/d-c^(1/2)*(-a*d+ b*c)^2*arctan(d^(1/2)*x/c^(1/2))/d^(7/2)
Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {(-b c+a d)^2 x}{d^3}-\frac {b (b c-2 a d) x^3}{3 d^2}+\frac {b^2 x^5}{5 d}-\frac {\sqrt {c} (b c-a d)^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{7/2}} \] Input:
Integrate[(x^2*(a + b*x^2)^2)/(c + d*x^2),x]
Output:
((-(b*c) + a*d)^2*x)/d^3 - (b*(b*c - 2*a*d)*x^3)/(3*d^2) + (b^2*x^5)/(5*d) - (Sqrt[c]*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(7/2)
Time = 0.23 (sec) , antiderivative size = 83, 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.
\(\displaystyle \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx\) |
\(\Big \downarrow \) 364 |
\(\displaystyle \int \left (\frac {-a^2 c d^2+2 a b c^2 d-b^2 c^3}{d^3 \left (c+d x^2\right )}+\frac {(b c-a d)^2}{d^3}-\frac {b x^2 (b c-2 a d)}{d^2}+\frac {b^2 x^4}{d}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {c} (b c-a d)^2 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{7/2}}+\frac {x (b c-a d)^2}{d^3}-\frac {b x^3 (b c-2 a d)}{3 d^2}+\frac {b^2 x^5}{5 d}\) |
Input:
Int[(x^2*(a + b*x^2)^2)/(c + d*x^2),x]
Output:
((b*c - a*d)^2*x)/d^3 - (b*(b*c - 2*a*d)*x^3)/(3*d^2) + (b^2*x^5)/(5*d) - (Sqrt[c]*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(7/2)
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 = 0.45 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {\frac {1}{5} b^{2} d^{2} x^{5}+\frac {2}{3} x^{3} a b \,d^{2}-\frac {1}{3} x^{3} b^{2} c d +a^{2} d^{2} x -2 a b c d x +b^{2} c^{2} x}{d^{3}}-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{d^{3} \sqrt {c d}}\) | \(102\) |
risch | \(\frac {b^{2} x^{5}}{5 d}+\frac {2 x^{3} a b}{3 d}-\frac {x^{3} b^{2} c}{3 d^{2}}+\frac {a^{2} x}{d}-\frac {2 a b c x}{d^{2}}+\frac {b^{2} c^{2} x}{d^{3}}+\frac {\sqrt {-c d}\, \ln \left (-\sqrt {-c d}\, x -c \right ) a^{2}}{2 d^{2}}-\frac {\sqrt {-c d}\, \ln \left (-\sqrt {-c d}\, x -c \right ) a b c}{d^{3}}+\frac {\sqrt {-c d}\, \ln \left (-\sqrt {-c d}\, x -c \right ) b^{2} c^{2}}{2 d^{4}}-\frac {\sqrt {-c d}\, \ln \left (\sqrt {-c d}\, x -c \right ) a^{2}}{2 d^{2}}+\frac {\sqrt {-c d}\, \ln \left (\sqrt {-c d}\, x -c \right ) a b c}{d^{3}}-\frac {\sqrt {-c d}\, \ln \left (\sqrt {-c d}\, x -c \right ) b^{2} c^{2}}{2 d^{4}}\) | \(233\) |
Input:
int(x^2*(b*x^2+a)^2/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
1/d^3*(1/5*b^2*d^2*x^5+2/3*x^3*a*b*d^2-1/3*x^3*b^2*c*d+a^2*d^2*x-2*a*b*c*d *x+b^2*c^2*x)-c*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^3/(c*d)^(1/2)*arctan(x*d/(c* d)^(1/2))
Time = 0.18 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.75 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx=\left [\frac {6 \, b^{2} d^{2} x^{5} - 10 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{3} + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} - 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) + 30 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{30 \, d^{3}}, \frac {3 \, b^{2} d^{2} x^{5} - 5 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{3} - 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, d^{3}}\right ] \] Input:
integrate(x^2*(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")
Output:
[1/30*(6*b^2*d^2*x^5 - 10*(b^2*c*d - 2*a*b*d^2)*x^3 + 15*(b^2*c^2 - 2*a*b* c*d + a^2*d^2)*sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) + 30*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/d^3, 1/15*(3*b^2*d^2*x^5 - 5*(b^2* c*d - 2*a*b*d^2)*x^3 - 15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c/d)*arctan (d*x*sqrt(c/d)/c) + 15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/d^3]
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (73) = 146\).
Time = 0.24 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.34 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {b^{2} x^{5}}{5 d} + x^{3} \cdot \left (\frac {2 a b}{3 d} - \frac {b^{2} c}{3 d^{2}}\right ) + x \left (\frac {a^{2}}{d} - \frac {2 a b c}{d^{2}} + \frac {b^{2} c^{2}}{d^{3}}\right ) + \frac {\sqrt {- \frac {c}{d^{7}}} \left (a d - b c\right )^{2} \log {\left (- \frac {d^{3} \sqrt {- \frac {c}{d^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac {\sqrt {- \frac {c}{d^{7}}} \left (a d - b c\right )^{2} \log {\left (\frac {d^{3} \sqrt {- \frac {c}{d^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} \] Input:
integrate(x**2*(b*x**2+a)**2/(d*x**2+c),x)
Output:
b**2*x**5/(5*d) + x**3*(2*a*b/(3*d) - b**2*c/(3*d**2)) + x*(a**2/d - 2*a*b *c/d**2 + b**2*c**2/d**3) + sqrt(-c/d**7)*(a*d - b*c)**2*log(-d**3*sqrt(-c /d**7)*(a*d - b*c)**2/(a**2*d**2 - 2*a*b*c*d + b**2*c**2) + x)/2 - sqrt(-c /d**7)*(a*d - b*c)**2*log(d**3*sqrt(-c/d**7)*(a*d - b*c)**2/(a**2*d**2 - 2 *a*b*c*d + b**2*c**2) + x)/2
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.25 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{2} d^{2} x^{5} - 5 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} x^{3} + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, d^{3}} \] Input:
integrate(x^2*(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")
Output:
-(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*d^3) + 1/15*(3*b^2*d^2*x^5 - 5*(b^2*c*d - 2*a*b*d^2)*x^3 + 15*(b^2*c^2 - 2*a*b *c*d + a^2*d^2)*x)/d^3
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx=-\frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{2} d^{4} x^{5} - 5 \, b^{2} c d^{3} x^{3} + 10 \, a b d^{4} x^{3} + 15 \, b^{2} c^{2} d^{2} x - 30 \, a b c d^{3} x + 15 \, a^{2} d^{4} x}{15 \, d^{5}} \] Input:
integrate(x^2*(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")
Output:
-(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*d^3) + 1/15*(3*b^2*d^4*x^5 - 5*b^2*c*d^3*x^3 + 10*a*b*d^4*x^3 + 15*b^2*c^2*d^2 *x - 30*a*b*c*d^3*x + 15*a^2*d^4*x)/d^5
Time = 0.46 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.54 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx=x\,\left (\frac {a^2}{d}+\frac {c\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )}{d}\right )-x^3\,\left (\frac {b^2\,c}{3\,d^2}-\frac {2\,a\,b}{3\,d}\right )+\frac {b^2\,x^5}{5\,d}-\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^2}{a^2\,c\,d^2-2\,a\,b\,c^2\,d+b^2\,c^3}\right )\,{\left (a\,d-b\,c\right )}^2}{d^{7/2}} \] Input:
int((x^2*(a + b*x^2)^2)/(c + d*x^2),x)
Output:
x*(a^2/d + (c*((b^2*c)/d^2 - (2*a*b)/d))/d) - x^3*((b^2*c)/(3*d^2) - (2*a* b)/(3*d)) + (b^2*x^5)/(5*d) - (c^(1/2)*atan((c^(1/2)*d^(1/2)*x*(a*d - b*c) ^2)/(b^2*c^3 + a^2*c*d^2 - 2*a*b*c^2*d))*(a*d - b*c)^2)/d^(7/2)
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.65 \[ \int \frac {x^2 \left (a+b x^2\right )^2}{c+d x^2} \, dx=\frac {-15 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} d^{2}+30 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a b c d -15 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{2}+15 a^{2} d^{3} x -30 a b c \,d^{2} x +10 a b \,d^{3} x^{3}+15 b^{2} c^{2} d x -5 b^{2} c \,d^{2} x^{3}+3 b^{2} d^{3} x^{5}}{15 d^{4}} \] Input:
int(x^2*(b*x^2+a)^2/(d*x^2+c),x)
Output:
( - 15*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*d**2 + 30*sqrt(d )*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*b*c*d - 15*sqrt(d)*sqrt(c)*atan( (d*x)/(sqrt(d)*sqrt(c)))*b**2*c**2 + 15*a**2*d**3*x - 30*a*b*c*d**2*x + 10 *a*b*d**3*x**3 + 15*b**2*c**2*d*x - 5*b**2*c*d**2*x**3 + 3*b**2*d**3*x**5) /(15*d**4)