Integrand size = 28, antiderivative size = 340 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\frac {b^2 d^2 x}{f^4}+\frac {(b e-a f)^2 (d e-c f)^2 x}{6 e f^4 \left (e+f x^2\right )^3}-\frac {(b e-a f) (d e-c f) (b e (19 d e-7 c f)-a f (7 d e+5 c f)) x}{24 e^2 f^4 \left (e+f x^2\right )^2}-\frac {\left (2 a b e f \left (11 d^2 e^2-2 c d e f-c^2 f^2\right )-b^2 e^2 \left (29 d^2 e^2-22 c d e f+c^2 f^2\right )-a^2 f^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) x}{16 e^3 f^4 \left (e+f x^2\right )}-\frac {\left (b^2 e^2 \left (35 d^2 e^2-10 c d e f-c^2 f^2\right )-2 a b e f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-a^2 f^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{9/2}} \] Output:
b^2*d^2*x/f^4+1/6*(-a*f+b*e)^2*(-c*f+d*e)^2*x/e/f^4/(f*x^2+e)^3-1/24*(-a*f +b*e)*(-c*f+d*e)*(b*e*(-7*c*f+19*d*e)-a*f*(5*c*f+7*d*e))*x/e^2/f^4/(f*x^2+ e)^2-1/16*(2*a*b*e*f*(-c^2*f^2-2*c*d*e*f+11*d^2*e^2)-b^2*e^2*(c^2*f^2-22*c *d*e*f+29*d^2*e^2)-a^2*f^2*(5*c^2*f^2+2*c*d*e*f+d^2*e^2))*x/e^3/f^4/(f*x^2 +e)-1/16*(b^2*e^2*(-c^2*f^2-10*c*d*e*f+35*d^2*e^2)-2*a*b*e*f*(c^2*f^2+2*c* d*e*f+5*d^2*e^2)-a^2*f^2*(5*c^2*f^2+2*c*d*e*f+d^2*e^2))*arctan(f^(1/2)*x/e ^(1/2))/e^(7/2)/f^(9/2)
Time = 0.23 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\frac {b^2 d^2 x}{f^4}+\frac {(b e-a f)^2 (d e-c f)^2 x}{6 e f^4 \left (e+f x^2\right )^3}-\frac {(b e-a f) (d e-c f) (b e (19 d e-7 c f)-a f (7 d e+5 c f)) x}{24 e^2 f^4 \left (e+f x^2\right )^2}+\frac {\left (b^2 e^2 \left (29 d^2 e^2-22 c d e f+c^2 f^2\right )+2 a b e f \left (-11 d^2 e^2+2 c d e f+c^2 f^2\right )+a^2 f^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) x}{16 e^3 f^4 \left (e+f x^2\right )}-\frac {\left (b^2 e^2 \left (35 d^2 e^2-10 c d e f-c^2 f^2\right )-2 a b e f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )-a^2 f^2 \left (d^2 e^2+2 c d e f+5 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{9/2}} \] Input:
Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/(e + f*x^2)^4,x]
Output:
(b^2*d^2*x)/f^4 + ((b*e - a*f)^2*(d*e - c*f)^2*x)/(6*e*f^4*(e + f*x^2)^3) - ((b*e - a*f)*(d*e - c*f)*(b*e*(19*d*e - 7*c*f) - a*f*(7*d*e + 5*c*f))*x) /(24*e^2*f^4*(e + f*x^2)^2) + ((b^2*e^2*(29*d^2*e^2 - 22*c*d*e*f + c^2*f^2 ) + 2*a*b*e*f*(-11*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + a^2*f^2*(d^2*e^2 + 2*c *d*e*f + 5*c^2*f^2))*x)/(16*e^3*f^4*(e + f*x^2)) - ((b^2*e^2*(35*d^2*e^2 - 10*c*d*e*f - c^2*f^2) - 2*a*b*e*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) - a^2 *f^2*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e ^(7/2)*f^(9/2))
Time = 0.74 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.48, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {425, 401, 25, 401, 25, 298, 218, 299, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\left (b x^2+a\right ) \left (d x^2+c\right )^2}{\left (f x^2+e\right )^3}dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right ) \left (d x^2+c\right )^2}{\left (f x^2+e\right )^4}dx}{f}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {b \left (-\frac {\int -\frac {\left (d x^2+c\right ) \left (d (5 b e-a f) x^2+c (b e+3 a f)\right )}{\left (f x^2+e\right )^2}dx}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (-\frac {\int -\frac {\left (d x^2+c\right ) \left (d (5 b e+a f) x^2+c (b e+5 a f)\right )}{\left (f x^2+e\right )^3}dx}{6 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {\left (d x^2+c\right ) \left (d (5 b e-a f) x^2+c (b e+3 a f)\right )}{\left (f x^2+e\right )^2}dx}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\int \frac {\left (d x^2+c\right ) \left (d (5 b e+a f) x^2+c (b e+5 a f)\right )}{\left (f x^2+e\right )^3}dx}{6 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {b \left (\frac {-\frac {\int \frac {c (a f (d e-3 c f)-b e (5 d e+c f))-d (b e (15 d e-c f)-3 a f (d e+c f)) x^2}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {-\frac {\int -\frac {d (b e (15 d e+c f)+a f (3 d e+5 c f)) x^2+c (d e (5 b e+a f)+3 c f (b e+5 a f))}{\left (f x^2+e\right )^2}dx}{4 e f}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{4 e f \left (e+f x^2\right )^2}}{6 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {-\frac {\int \frac {c (a f (d e-3 c f)-b e (5 d e+c f))-d (b e (15 d e-c f)-3 a f (d e+c f)) x^2}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {\int \frac {d (b e (15 d e+c f)+a f (3 d e+5 c f)) x^2+c (d e (5 b e+a f)+3 c f (b e+5 a f))}{\left (f x^2+e\right )^2}dx}{4 e f}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{4 e f \left (e+f x^2\right )^2}}{6 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {b \left (\frac {-\frac {\int \frac {c (a f (d e-3 c f)-b e (5 d e+c f))-d (b e (15 d e-c f)-3 a f (d e+c f)) x^2}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {\frac {3 \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{2 e f}-\frac {x \left (a f \left (-15 c^2 f^2+4 c d e f+3 d^2 e^2\right )+b e \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )\right )}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{4 e f \left (e+f x^2\right )^2}}{6 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b \left (\frac {-\frac {\int \frac {c (a f (d e-3 c f)-b e (5 d e+c f))-d (b e (15 d e-c f)-3 a f (d e+c f)) x^2}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {\frac {3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{2 e^{3/2} f^{3/2}}-\frac {x \left (a f \left (-15 c^2 f^2+4 c d e f+3 d^2 e^2\right )+b e \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )\right )}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{4 e f \left (e+f x^2\right )^2}}{6 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \left (\frac {-\frac {\frac {\left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right ) \int \frac {1}{f x^2+e}dx}{f}-\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{f}}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {\frac {3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{2 e^{3/2} f^{3/2}}-\frac {x \left (a f \left (-15 c^2 f^2+4 c d e f+3 d^2 e^2\right )+b e \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )\right )}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{4 e f \left (e+f x^2\right )^2}}{6 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b \left (\frac {-\frac {\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-6 c d e f+15 d^2 e^2\right )-a f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{\sqrt {e} f^{3/2}}-\frac {d x (b e (15 d e-c f)-3 a f (c f+d e))}{f}}{2 e f}-\frac {x \left (c+d x^2\right ) (b e (5 d e-c f)-a f (3 c f+d e))}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{4 e f \left (e+f x^2\right )^2}\right )}{f}-\frac {(b e-a f) \left (\frac {\frac {\frac {3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a f \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )+b e \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{2 e^{3/2} f^{3/2}}-\frac {x \left (a f \left (-15 c^2 f^2+4 c d e f+3 d^2 e^2\right )+b e \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )\right )}{2 e f \left (e+f x^2\right )}}{4 e f}-\frac {x \left (c+d x^2\right ) (d e (a f+5 b e)-c f (5 a f+b e))}{4 e f \left (e+f x^2\right )^2}}{6 e f}-\frac {x \left (c+d x^2\right )^2 (b e-a f)}{6 e f \left (e+f x^2\right )^3}\right )}{f}\) |
Input:
Int[((a + b*x^2)^2*(c + d*x^2)^2)/(e + f*x^2)^4,x]
Output:
(b*(-1/4*((b*e - a*f)*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)^2) + (-1/2*((b*e*( 5*d*e - c*f) - a*f*(d*e + 3*c*f))*x*(c + d*x^2))/(e*f*(e + f*x^2)) - (-((d *(b*e*(15*d*e - c*f) - 3*a*f*(d*e + c*f))*x)/f) + ((b*e*(15*d^2*e^2 - 6*c* d*e*f - c^2*f^2) - a*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*ArcTan[(Sqrt[f ]*x)/Sqrt[e]])/(Sqrt[e]*f^(3/2)))/(2*e*f))/(4*e*f)))/f - ((b*e - a*f)*(-1/ 6*((b*e - a*f)*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)^3) + (-1/4*((d*e*(5*b*e + a*f) - c*f*(b*e + 5*a*f))*x*(c + d*x^2))/(e*f*(e + f*x^2)^2) + (-1/2*((a* f*(3*d^2*e^2 + 4*c*d*e*f - 15*c^2*f^2) + b*e*(15*d^2*e^2 - 4*c*d*e*f - 3*c ^2*f^2))*x)/(e*f*(e + f*x^2)) + (3*(b*e*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + a*f*(d^2*e^2 + 2*c*d*e*f + 5*c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e ^(3/2)*f^(3/2)))/(4*e*f))/(6*e*f)))/f
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^2)^p*(c + d*x ^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt Q[p, 0] && GtQ[q, 0]
Time = 0.56 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(\frac {b^{2} d^{2} x}{f^{4}}+\frac {\frac {\frac {f^{2} \left (5 a^{2} c^{2} f^{4}+2 a^{2} c d e \,f^{3}+a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}+4 a b c d \,e^{2} f^{2}-22 a b \,d^{2} e^{3} f +b^{2} c^{2} e^{2} f^{2}-22 b^{2} c d \,e^{3} f +29 b^{2} d^{2} e^{4}\right ) x^{5}}{16 e^{3}}+\frac {f \left (5 a^{2} c^{2} f^{4}+2 a^{2} c d e \,f^{3}-a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}-4 a b c d \,e^{2} f^{2}-10 a b \,d^{2} e^{3} f -b^{2} c^{2} e^{2} f^{2}-10 b^{2} c d \,e^{3} f +17 b^{2} d^{2} e^{4}\right ) x^{3}}{6 e^{2}}+\frac {\left (11 a^{2} c^{2} f^{4}-2 a^{2} c d e \,f^{3}-a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}-4 a b c d \,e^{2} f^{2}-10 a b \,d^{2} e^{3} f -b^{2} c^{2} e^{2} f^{2}-10 b^{2} c d \,e^{3} f +19 b^{2} d^{2} e^{4}\right ) x}{16 e}}{\left (f \,x^{2}+e \right )^{3}}+\frac {\left (5 a^{2} c^{2} f^{4}+2 a^{2} c d e \,f^{3}+a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}+4 a b c d \,e^{2} f^{2}+10 a b \,d^{2} e^{3} f +b^{2} c^{2} e^{2} f^{2}+10 b^{2} c d \,e^{3} f -35 b^{2} d^{2} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 e^{3} \sqrt {e f}}}{f^{4}}\) | \(498\) |
| risch | \(\frac {b^{2} d^{2} x}{f^{4}}+\frac {\frac {f^{2} \left (5 a^{2} c^{2} f^{4}+2 a^{2} c d e \,f^{3}+a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}+4 a b c d \,e^{2} f^{2}-22 a b \,d^{2} e^{3} f +b^{2} c^{2} e^{2} f^{2}-22 b^{2} c d \,e^{3} f +29 b^{2} d^{2} e^{4}\right ) x^{5}}{16 e^{3}}+\frac {f \left (5 a^{2} c^{2} f^{4}+2 a^{2} c d e \,f^{3}-a^{2} d^{2} e^{2} f^{2}+2 a b \,c^{2} e \,f^{3}-4 a b c d \,e^{2} f^{2}-10 a b \,d^{2} e^{3} f -b^{2} c^{2} e^{2} f^{2}-10 b^{2} c d \,e^{3} f +17 b^{2} d^{2} e^{4}\right ) x^{3}}{6 e^{2}}+\frac {\left (11 a^{2} c^{2} f^{4}-2 a^{2} c d e \,f^{3}-a^{2} d^{2} e^{2} f^{2}-2 a b \,c^{2} e \,f^{3}-4 a b c d \,e^{2} f^{2}-10 a b \,d^{2} e^{3} f -b^{2} c^{2} e^{2} f^{2}-10 b^{2} c d \,e^{3} f +19 b^{2} d^{2} e^{4}\right ) x}{16 e}}{f^{4} \left (f \,x^{2}+e \right )^{3}}-\frac {5 \ln \left (f x +\sqrt {-e f}\right ) a^{2} c^{2}}{32 \sqrt {-e f}\, e^{3}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) a^{2} c d}{16 f \sqrt {-e f}\, e^{2}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) a^{2} d^{2}}{32 f^{2} \sqrt {-e f}\, e}-\frac {\ln \left (f x +\sqrt {-e f}\right ) a b \,c^{2}}{16 f \sqrt {-e f}\, e^{2}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) a b c d}{8 f^{2} \sqrt {-e f}\, e}-\frac {5 \ln \left (f x +\sqrt {-e f}\right ) a b \,d^{2}}{16 f^{3} \sqrt {-e f}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) b^{2} c^{2}}{32 f^{2} \sqrt {-e f}\, e}-\frac {5 \ln \left (f x +\sqrt {-e f}\right ) b^{2} c d}{16 f^{3} \sqrt {-e f}}+\frac {35 e \ln \left (f x +\sqrt {-e f}\right ) b^{2} d^{2}}{32 f^{4} \sqrt {-e f}}+\frac {5 \ln \left (-f x +\sqrt {-e f}\right ) a^{2} c^{2}}{32 \sqrt {-e f}\, e^{3}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) a^{2} c d}{16 f \sqrt {-e f}\, e^{2}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) a^{2} d^{2}}{32 f^{2} \sqrt {-e f}\, e}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) a b \,c^{2}}{16 f \sqrt {-e f}\, e^{2}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) a b c d}{8 f^{2} \sqrt {-e f}\, e}+\frac {5 \ln \left (-f x +\sqrt {-e f}\right ) a b \,d^{2}}{16 f^{3} \sqrt {-e f}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) b^{2} c^{2}}{32 f^{2} \sqrt {-e f}\, e}+\frac {5 \ln \left (-f x +\sqrt {-e f}\right ) b^{2} c d}{16 f^{3} \sqrt {-e f}}-\frac {35 e \ln \left (-f x +\sqrt {-e f}\right ) b^{2} d^{2}}{32 f^{4} \sqrt {-e f}}\) | \(905\) |
Input:
int((b*x^2+a)^2*(d*x^2+c)^2/(f*x^2+e)^4,x,method=_RETURNVERBOSE)
Output:
b^2*d^2*x/f^4+1/f^4*((1/16*f^2*(5*a^2*c^2*f^4+2*a^2*c*d*e*f^3+a^2*d^2*e^2* f^2+2*a*b*c^2*e*f^3+4*a*b*c*d*e^2*f^2-22*a*b*d^2*e^3*f+b^2*c^2*e^2*f^2-22* b^2*c*d*e^3*f+29*b^2*d^2*e^4)/e^3*x^5+1/6*f*(5*a^2*c^2*f^4+2*a^2*c*d*e*f^3 -a^2*d^2*e^2*f^2+2*a*b*c^2*e*f^3-4*a*b*c*d*e^2*f^2-10*a*b*d^2*e^3*f-b^2*c^ 2*e^2*f^2-10*b^2*c*d*e^3*f+17*b^2*d^2*e^4)/e^2*x^3+1/16*(11*a^2*c^2*f^4-2* a^2*c*d*e*f^3-a^2*d^2*e^2*f^2-2*a*b*c^2*e*f^3-4*a*b*c*d*e^2*f^2-10*a*b*d^2 *e^3*f-b^2*c^2*e^2*f^2-10*b^2*c*d*e^3*f+19*b^2*d^2*e^4)/e*x)/(f*x^2+e)^3+1 /16*(5*a^2*c^2*f^4+2*a^2*c*d*e*f^3+a^2*d^2*e^2*f^2+2*a*b*c^2*e*f^3+4*a*b*c *d*e^2*f^2+10*a*b*d^2*e^3*f+b^2*c^2*e^2*f^2+10*b^2*c*d*e^3*f-35*b^2*d^2*e^ 4)/e^3/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 788 vs. \(2 (322) = 644\).
Time = 0.14 (sec) , antiderivative size = 1596, normalized size of antiderivative = 4.69 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/(f*x^2+e)^4,x, algorithm="fricas")
Output:
[1/96*(96*b^2*d^2*e^4*f^4*x^7 + 6*(77*b^2*d^2*e^5*f^3 + 5*a^2*c^2*e*f^7 - 22*(b^2*c*d + a*b*d^2)*e^4*f^4 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^5 + 2*(a*b*c^2 + a^2*c*d)*e^2*f^6)*x^5 + 16*(35*b^2*d^2*e^6*f^2 + 5*a^2*c^2*e ^2*f^6 - 10*(b^2*c*d + a*b*d^2)*e^5*f^3 - (b^2*c^2 + 4*a*b*c*d + a^2*d^2)* e^4*f^4 + 2*(a*b*c^2 + a^2*c*d)*e^3*f^5)*x^3 + 3*(35*b^2*d^2*e^7 - 5*a^2*c ^2*e^3*f^4 - 10*(b^2*c*d + a*b*d^2)*e^6*f - (b^2*c^2 + 4*a*b*c*d + a^2*d^2 )*e^5*f^2 - 2*(a*b*c^2 + a^2*c*d)*e^4*f^3 + (35*b^2*d^2*e^4*f^3 - 5*a^2*c^ 2*f^7 - 10*(b^2*c*d + a*b*d^2)*e^3*f^4 - (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e ^2*f^5 - 2*(a*b*c^2 + a^2*c*d)*e*f^6)*x^6 + 3*(35*b^2*d^2*e^5*f^2 - 5*a^2* c^2*e*f^6 - 10*(b^2*c*d + a*b*d^2)*e^4*f^3 - (b^2*c^2 + 4*a*b*c*d + a^2*d^ 2)*e^3*f^4 - 2*(a*b*c^2 + a^2*c*d)*e^2*f^5)*x^4 + 3*(35*b^2*d^2*e^6*f - 5* a^2*c^2*e^2*f^5 - 10*(b^2*c*d + a*b*d^2)*e^5*f^2 - (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^4*f^3 - 2*(a*b*c^2 + a^2*c*d)*e^3*f^4)*x^2)*sqrt(-e*f)*log((f*x ^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) + 6*(35*b^2*d^2*e^7*f + 11*a^2*c^2*e ^3*f^5 - 10*(b^2*c*d + a*b*d^2)*e^6*f^2 - (b^2*c^2 + 4*a*b*c*d + a^2*d^2)* e^5*f^3 - 2*(a*b*c^2 + a^2*c*d)*e^4*f^4)*x)/(e^4*f^8*x^6 + 3*e^5*f^7*x^4 + 3*e^6*f^6*x^2 + e^7*f^5), 1/48*(48*b^2*d^2*e^4*f^4*x^7 + 3*(77*b^2*d^2*e^ 5*f^3 + 5*a^2*c^2*e*f^7 - 22*(b^2*c*d + a*b*d^2)*e^4*f^4 + (b^2*c^2 + 4*a* b*c*d + a^2*d^2)*e^3*f^5 + 2*(a*b*c^2 + a^2*c*d)*e^2*f^6)*x^5 + 8*(35*b^2* d^2*e^6*f^2 + 5*a^2*c^2*e^2*f^6 - 10*(b^2*c*d + a*b*d^2)*e^5*f^3 - (b^2...
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**2*(d*x**2+c)**2/(f*x**2+e)**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/(f*x^2+e)^4,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.13 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\frac {b^{2} d^{2} x}{f^{4}} - \frac {{\left (35 \, b^{2} d^{2} e^{4} - 10 \, b^{2} c d e^{3} f - 10 \, a b d^{2} e^{3} f - b^{2} c^{2} e^{2} f^{2} - 4 \, a b c d e^{2} f^{2} - a^{2} d^{2} e^{2} f^{2} - 2 \, a b c^{2} e f^{3} - 2 \, a^{2} c d e f^{3} - 5 \, a^{2} c^{2} f^{4}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \, \sqrt {e f} e^{3} f^{4}} + \frac {87 \, b^{2} d^{2} e^{4} f^{2} x^{5} - 66 \, b^{2} c d e^{3} f^{3} x^{5} - 66 \, a b d^{2} e^{3} f^{3} x^{5} + 3 \, b^{2} c^{2} e^{2} f^{4} x^{5} + 12 \, a b c d e^{2} f^{4} x^{5} + 3 \, a^{2} d^{2} e^{2} f^{4} x^{5} + 6 \, a b c^{2} e f^{5} x^{5} + 6 \, a^{2} c d e f^{5} x^{5} + 15 \, a^{2} c^{2} f^{6} x^{5} + 136 \, b^{2} d^{2} e^{5} f x^{3} - 80 \, b^{2} c d e^{4} f^{2} x^{3} - 80 \, a b d^{2} e^{4} f^{2} x^{3} - 8 \, b^{2} c^{2} e^{3} f^{3} x^{3} - 32 \, a b c d e^{3} f^{3} x^{3} - 8 \, a^{2} d^{2} e^{3} f^{3} x^{3} + 16 \, a b c^{2} e^{2} f^{4} x^{3} + 16 \, a^{2} c d e^{2} f^{4} x^{3} + 40 \, a^{2} c^{2} e f^{5} x^{3} + 57 \, b^{2} d^{2} e^{6} x - 30 \, b^{2} c d e^{5} f x - 30 \, a b d^{2} e^{5} f x - 3 \, b^{2} c^{2} e^{4} f^{2} x - 12 \, a b c d e^{4} f^{2} x - 3 \, a^{2} d^{2} e^{4} f^{2} x - 6 \, a b c^{2} e^{3} f^{3} x - 6 \, a^{2} c d e^{3} f^{3} x + 33 \, a^{2} c^{2} e^{2} f^{4} x}{48 \, {\left (f x^{2} + e\right )}^{3} e^{3} f^{4}} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2/(f*x^2+e)^4,x, algorithm="giac")
Output:
b^2*d^2*x/f^4 - 1/16*(35*b^2*d^2*e^4 - 10*b^2*c*d*e^3*f - 10*a*b*d^2*e^3*f - b^2*c^2*e^2*f^2 - 4*a*b*c*d*e^2*f^2 - a^2*d^2*e^2*f^2 - 2*a*b*c^2*e*f^3 - 2*a^2*c*d*e*f^3 - 5*a^2*c^2*f^4)*arctan(f*x/sqrt(e*f))/(sqrt(e*f)*e^3*f ^4) + 1/48*(87*b^2*d^2*e^4*f^2*x^5 - 66*b^2*c*d*e^3*f^3*x^5 - 66*a*b*d^2*e ^3*f^3*x^5 + 3*b^2*c^2*e^2*f^4*x^5 + 12*a*b*c*d*e^2*f^4*x^5 + 3*a^2*d^2*e^ 2*f^4*x^5 + 6*a*b*c^2*e*f^5*x^5 + 6*a^2*c*d*e*f^5*x^5 + 15*a^2*c^2*f^6*x^5 + 136*b^2*d^2*e^5*f*x^3 - 80*b^2*c*d*e^4*f^2*x^3 - 80*a*b*d^2*e^4*f^2*x^3 - 8*b^2*c^2*e^3*f^3*x^3 - 32*a*b*c*d*e^3*f^3*x^3 - 8*a^2*d^2*e^3*f^3*x^3 + 16*a*b*c^2*e^2*f^4*x^3 + 16*a^2*c*d*e^2*f^4*x^3 + 40*a^2*c^2*e*f^5*x^3 + 57*b^2*d^2*e^6*x - 30*b^2*c*d*e^5*f*x - 30*a*b*d^2*e^5*f*x - 3*b^2*c^2*e^ 4*f^2*x - 12*a*b*c*d*e^4*f^2*x - 3*a^2*d^2*e^4*f^2*x - 6*a*b*c^2*e^3*f^3*x - 6*a^2*c*d*e^3*f^3*x + 33*a^2*c^2*e^2*f^4*x)/((f*x^2 + e)^3*e^3*f^4)
Time = 2.31 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx=\frac {b^2\,d^2\,x}{f^4}-\frac {\frac {x^3\,\left (-5\,a^2\,c^2\,f^5-2\,a^2\,c\,d\,e\,f^4+a^2\,d^2\,e^2\,f^3-2\,a\,b\,c^2\,e\,f^4+4\,a\,b\,c\,d\,e^2\,f^3+10\,a\,b\,d^2\,e^3\,f^2+b^2\,c^2\,e^2\,f^3+10\,b^2\,c\,d\,e^3\,f^2-17\,b^2\,d^2\,e^4\,f\right )}{6\,e^2}-\frac {x^5\,\left (5\,a^2\,c^2\,f^6+2\,a^2\,c\,d\,e\,f^5+a^2\,d^2\,e^2\,f^4+2\,a\,b\,c^2\,e\,f^5+4\,a\,b\,c\,d\,e^2\,f^4-22\,a\,b\,d^2\,e^3\,f^3+b^2\,c^2\,e^2\,f^4-22\,b^2\,c\,d\,e^3\,f^3+29\,b^2\,d^2\,e^4\,f^2\right )}{16\,e^3}+\frac {x\,\left (-11\,a^2\,c^2\,f^4+2\,a^2\,c\,d\,e\,f^3+a^2\,d^2\,e^2\,f^2+2\,a\,b\,c^2\,e\,f^3+4\,a\,b\,c\,d\,e^2\,f^2+10\,a\,b\,d^2\,e^3\,f+b^2\,c^2\,e^2\,f^2+10\,b^2\,c\,d\,e^3\,f-19\,b^2\,d^2\,e^4\right )}{16\,e}}{e^3\,f^4+3\,e^2\,f^5\,x^2+3\,e\,f^6\,x^4+f^7\,x^6}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (5\,a^2\,c^2\,f^4+2\,a^2\,c\,d\,e\,f^3+a^2\,d^2\,e^2\,f^2+2\,a\,b\,c^2\,e\,f^3+4\,a\,b\,c\,d\,e^2\,f^2+10\,a\,b\,d^2\,e^3\,f+b^2\,c^2\,e^2\,f^2+10\,b^2\,c\,d\,e^3\,f-35\,b^2\,d^2\,e^4\right )}{16\,e^{7/2}\,f^{9/2}} \] Input:
int(((a + b*x^2)^2*(c + d*x^2)^2)/(e + f*x^2)^4,x)
Output:
(b^2*d^2*x)/f^4 - ((x^3*(a^2*d^2*e^2*f^3 - 17*b^2*d^2*e^4*f - 5*a^2*c^2*f^ 5 + b^2*c^2*e^2*f^3 - 2*a*b*c^2*e*f^4 - 2*a^2*c*d*e*f^4 + 10*a*b*d^2*e^3*f ^2 + 10*b^2*c*d*e^3*f^2 + 4*a*b*c*d*e^2*f^3))/(6*e^2) - (x^5*(5*a^2*c^2*f^ 6 + a^2*d^2*e^2*f^4 + b^2*c^2*e^2*f^4 + 29*b^2*d^2*e^4*f^2 + 2*a*b*c^2*e*f ^5 + 2*a^2*c*d*e*f^5 - 22*a*b*d^2*e^3*f^3 - 22*b^2*c*d*e^3*f^3 + 4*a*b*c*d *e^2*f^4))/(16*e^3) + (x*(a^2*d^2*e^2*f^2 - 19*b^2*d^2*e^4 - 11*a^2*c^2*f^ 4 + b^2*c^2*e^2*f^2 + 2*a*b*c^2*e*f^3 + 10*a*b*d^2*e^3*f + 2*a^2*c*d*e*f^3 + 10*b^2*c*d*e^3*f + 4*a*b*c*d*e^2*f^2))/(16*e))/(e^3*f^4 + f^7*x^6 + 3*e *f^6*x^4 + 3*e^2*f^5*x^2) + (atan((f^(1/2)*x)/e^(1/2))*(5*a^2*c^2*f^4 - 35 *b^2*d^2*e^4 + a^2*d^2*e^2*f^2 + b^2*c^2*e^2*f^2 + 2*a*b*c^2*e*f^3 + 10*a* b*d^2*e^3*f + 2*a^2*c*d*e*f^3 + 10*b^2*c*d*e^3*f + 4*a*b*c*d*e^2*f^2))/(16 *e^(7/2)*f^(9/2))
Time = 0.16 (sec) , antiderivative size = 1595, normalized size of antiderivative = 4.69 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^2*(d*x^2+c)^2/(f*x^2+e)^4,x)
Output:
(15*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*e**3*f**4 + 45 *sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*e**2*f**5*x**2 + 45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*e*f**6*x**4 + 1 5*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*f**7*x**6 + 6*sq rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d*e**4*f**3 + 18*sqrt(f )*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d*e**3*f**4*x**2 + 18*sqrt( f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d*e**2*f**5*x**4 + 6*sqrt( f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d*e*f**6*x**6 + 3*sqrt(f)* sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**2*e**5*f**2 + 9*sqrt(f)*sqrt (e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**2*e**4*f**3*x**2 + 9*sqrt(f)*sqr t(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**2*e**3*f**4*x**4 + 3*sqrt(f)*sq rt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**2*e**2*f**5*x**6 + 6*sqrt(f)*s qrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*e**4*f**3 + 18*sqrt(f)*sqrt( e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*e**3*f**4*x**2 + 18*sqrt(f)*sqrt (e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*e**2*f**5*x**4 + 6*sqrt(f)*sqrt (e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*e*f**6*x**6 + 12*sqrt(f)*sqrt(e )*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d*e**5*f**2 + 36*sqrt(f)*sqrt(e)*ata n((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d*e**4*f**3*x**2 + 36*sqrt(f)*sqrt(e)*ata n((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d*e**3*f**4*x**4 + 12*sqrt(f)*sqrt(e)*ata n((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d*e**2*f**5*x**6 + 30*sqrt(f)*sqrt(e)*...