\(\int \frac {(a+b x^2) (c+d x^2)^3}{(e+f x^2)^2} \, dx\) [216]

Optimal result

Integrand size = 26, antiderivative size = 178 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=-\frac {d \left (a d f (2 d e-3 c f)-3 b (d e-c f)^2\right ) x}{f^4}-\frac {d^2 (2 b d e-3 b c f-a d f) x^3}{3 f^3}+\frac {b d^3 x^5}{5 f^2}+\frac {(b e-a f) (d e-c f)^3 x}{2 e f^4 \left (e+f x^2\right )}-\frac {(d e-c f)^2 (b e (7 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{9/2}} \] Output:

-d*(a*d*f*(-3*c*f+2*d*e)-3*b*(-c*f+d*e)^2)*x/f^4-1/3*d^2*(-a*d*f-3*b*c*f+2 
*b*d*e)*x^3/f^3+1/5*b*d^3*x^5/f^2+1/2*(-a*f+b*e)*(-c*f+d*e)^3*x/e/f^4/(f*x 
^2+e)-1/2*(-c*f+d*e)^2*(b*e*(-c*f+7*d*e)-a*f*(c*f+5*d*e))*arctan(f^(1/2)*x 
/e^(1/2))/e^(3/2)/f^(9/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {d \left (3 b (d e-c f)^2+a d f (-2 d e+3 c f)\right ) x}{f^4}+\frac {d^2 (-2 b d e+3 b c f+a d f) x^3}{3 f^3}+\frac {b d^3 x^5}{5 f^2}+\frac {(b e-a f) (d e-c f)^3 x}{2 e f^4 \left (e+f x^2\right )}-\frac {(d e-c f)^2 (b e (7 d e-c f)-a f (5 d e+c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{2 e^{3/2} f^{9/2}} \] Input:

Integrate[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^2,x]
 

Output:

(d*(3*b*(d*e - c*f)^2 + a*d*f*(-2*d*e + 3*c*f))*x)/f^4 + (d^2*(-2*b*d*e + 
3*b*c*f + a*d*f)*x^3)/(3*f^3) + (b*d^3*x^5)/(5*f^2) + ((b*e - a*f)*(d*e - 
c*f)^3*x)/(2*e*f^4*(e + f*x^2)) - ((d*e - c*f)^2*(b*e*(7*d*e - c*f) - a*f* 
(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(9/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.43, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {401, 25, 403, 25, 403, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx\)

\(\Big \downarrow \) 401

\(\displaystyle -\frac {\int -\frac {\left (d x^2+c\right )^2 \left (d (7 b e-5 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (d x^2+c\right )^2 \left (d (7 b e-5 a f) x^2+c (b e+a f)\right )}{f x^2+e}dx}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\int -\frac {\left (d x^2+c\right ) \left (d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2+c (b e (7 d e-5 c f)-5 a f (d e+c f))\right )}{f x^2+e}dx}{5 f}+\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2+c (b e (7 d e-5 c f)-5 a f (d e+c f))\right )}{f x^2+e}dx}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\frac {\int \frac {d \left (5 a f \left (15 d^2 e^2-22 c d f e+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d f e+81 c^2 f^2\right )\right ) x^2+c \left (5 a f \left (5 d^2 e^2-6 c d f e-3 c^2 f^2\right )-b e \left (35 d^2 e^2-54 c d f e+15 c^2 f^2\right )\right )}{f x^2+e}dx}{3 f}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{3 f}}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\frac {\frac {15 (d e-c f)^2 (b e (7 d e-c f)-a f (c f+5 d e)) \int \frac {1}{f x^2+e}dx}{f}+\frac {d x \left (5 a f \left (3 c^2 f^2-22 c d e f+15 d^2 e^2\right )-b e \left (81 c^2 f^2-190 c d e f+105 d^2 e^2\right )\right )}{f}}{3 f}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{3 f}}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{5 f}-\frac {\frac {\frac {15 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^2 (b e (7 d e-c f)-a f (c f+5 d e))}{\sqrt {e} f^{3/2}}+\frac {d x \left (5 a f \left (3 c^2 f^2-22 c d e f+15 d^2 e^2\right )-b e \left (81 c^2 f^2-190 c d e f+105 d^2 e^2\right )\right )}{f}}{3 f}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{3 f}}{5 f}}{2 e f}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )}\)

Input:

Int[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^2,x]
 

Output:

-1/2*((b*e - a*f)*x*(c + d*x^2)^3)/(e*f*(e + f*x^2)) + ((d*(7*b*e - 5*a*f) 
*x*(c + d*x^2)^2)/(5*f) - ((d*(b*e*(35*d*e - 33*c*f) - 5*a*f*(5*d*e - 3*c* 
f))*x*(c + d*x^2))/(3*f) + ((d*(5*a*f*(15*d^2*e^2 - 22*c*d*e*f + 3*c^2*f^2 
) - b*e*(105*d^2*e^2 - 190*c*d*e*f + 81*c^2*f^2))*x)/f + (15*(d*e - c*f)^2 
*(b*e*(7*d*e - c*f) - a*f*(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqr 
t[e]*f^(3/2)))/(3*f))/(5*f))/(2*e*f)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 218

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]
 

rule 299

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]
 

rule 401

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]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.73

method	result	size
default	\(\frac {d \left (\frac {1}{5} b \,d^{2} x^{5} f^{2}+\frac {1}{3} a \,d^{2} f^{2} x^{3}+b c d \,f^{2} x^{3}-\frac {2}{3} b \,d^{2} e f \,x^{3}+3 a c d \,f^{2} x -2 a \,d^{2} e f x +3 b \,c^{2} f^{2} x -6 b c d e f x +3 b \,d^{2} e^{2} x \right )}{f^{4}}+\frac {\frac {\left (c^{3} a \,f^{4}-3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-3 b c \,d^{2} e^{3} f +e^{4} b \,d^{3}\right ) x}{2 e \left (f \,x^{2}+e \right )}+\frac {\left (c^{3} a \,f^{4}+3 a \,c^{2} d e \,f^{3}-9 a c \,d^{2} e^{2} f^{2}+5 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}-9 b \,c^{2} d \,e^{2} f^{2}+15 b c \,d^{2} e^{3} f -7 e^{4} b \,d^{3}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 e \sqrt {e f}}}{f^{4}}\)	\(308\)
risch	\(-\frac {9 e \ln \left (-f x +\sqrt {-e f}\right ) a c \,d^{2}}{4 f^{2} \sqrt {-e f}}-\frac {9 e \ln \left (-f x +\sqrt {-e f}\right ) b \,c^{2} d}{4 f^{2} \sqrt {-e f}}+\frac {15 e^{2} \ln \left (-f x +\sqrt {-e f}\right ) b c \,d^{2}}{4 f^{3} \sqrt {-e f}}+\frac {d^{2} b c \,x^{3}}{f^{2}}-\frac {2 d^{3} b e \,x^{3}}{3 f^{3}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) b \,c^{3}}{4 f \sqrt {-e f}}-\frac {6 d^{2} b c e x}{f^{3}}+\frac {3 d^{2} a c x}{f^{2}}-\frac {2 d^{3} a e x}{f^{3}}+\frac {3 d b \,c^{2} x}{f^{2}}+\frac {3 d^{3} b \,e^{2} x}{f^{4}}+\frac {9 e \ln \left (f x +\sqrt {-e f}\right ) a c \,d^{2}}{4 f^{2} \sqrt {-e f}}+\frac {9 e \ln \left (f x +\sqrt {-e f}\right ) b \,c^{2} d}{4 f^{2} \sqrt {-e f}}-\frac {15 e^{2} \ln \left (f x +\sqrt {-e f}\right ) b c \,d^{2}}{4 f^{3} \sqrt {-e f}}+\frac {d^{3} a \,x^{3}}{3 f^{2}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) c^{3} a}{4 \sqrt {-e f}\, e}-\frac {\ln \left (f x +\sqrt {-e f}\right ) b \,c^{3}}{4 f \sqrt {-e f}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) c^{3} a}{4 \sqrt {-e f}\, e}+\frac {\left (c^{3} a \,f^{4}-3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-3 b c \,d^{2} e^{3} f +e^{4} b \,d^{3}\right ) x}{2 e \,f^{4} \left (f \,x^{2}+e \right )}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) a \,c^{2} d}{4 f \sqrt {-e f}}-\frac {5 e^{2} \ln \left (f x +\sqrt {-e f}\right ) a \,d^{3}}{4 f^{3} \sqrt {-e f}}+\frac {7 e^{3} \ln \left (f x +\sqrt {-e f}\right ) b \,d^{3}}{4 f^{4} \sqrt {-e f}}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) a \,c^{2} d}{4 f \sqrt {-e f}}+\frac {5 e^{2} \ln \left (-f x +\sqrt {-e f}\right ) a \,d^{3}}{4 f^{3} \sqrt {-e f}}-\frac {7 e^{3} \ln \left (-f x +\sqrt {-e f}\right ) b \,d^{3}}{4 f^{4} \sqrt {-e f}}+\frac {b \,d^{3} x^{5}}{5 f^{2}}\)	\(666\)

Input:

int((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
 

Output:

d/f^4*(1/5*b*d^2*x^5*f^2+1/3*a*d^2*f^2*x^3+b*c*d*f^2*x^3-2/3*b*d^2*e*f*x^3 
+3*a*c*d*f^2*x-2*a*d^2*e*f*x+3*b*c^2*f^2*x-6*b*c*d*e*f*x+3*b*d^2*e^2*x)+1/ 
f^4*(1/2*(a*c^3*f^4-3*a*c^2*d*e*f^3+3*a*c*d^2*e^2*f^2-a*d^3*e^3*f-b*c^3*e* 
f^3+3*b*c^2*d*e^2*f^2-3*b*c*d^2*e^3*f+b*d^3*e^4)/e*x/(f*x^2+e)+1/2*(a*c^3* 
f^4+3*a*c^2*d*e*f^3-9*a*c*d^2*e^2*f^2+5*a*d^3*e^3*f+b*c^3*e*f^3-9*b*c^2*d* 
e^2*f^2+15*b*c*d^2*e^3*f-7*b*d^3*e^4)/e/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2) 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (162) = 324\).

Time = 0.10 (sec) , antiderivative size = 834, normalized size of antiderivative = 4.69 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e)^2,x, algorithm="fricas")
 

Output:

[1/60*(12*b*d^3*e^2*f^4*x^7 - 4*(7*b*d^3*e^3*f^3 - 5*(3*b*c*d^2 + a*d^3)*e 
^2*f^4)*x^5 + 20*(7*b*d^3*e^4*f^2 - 5*(3*b*c*d^2 + a*d^3)*e^3*f^3 + 9*(b*c 
^2*d + a*c*d^2)*e^2*f^4)*x^3 + 15*(7*b*d^3*e^5 - a*c^3*e*f^4 - 5*(3*b*c*d^ 
2 + a*d^3)*e^4*f + 9*(b*c^2*d + a*c*d^2)*e^3*f^2 - (b*c^3 + 3*a*c^2*d)*e^2 
*f^3 + (7*b*d^3*e^4*f - a*c^3*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^3*f^2 + 9*(b*c 
^2*d + a*c*d^2)*e^2*f^3 - (b*c^3 + 3*a*c^2*d)*e*f^4)*x^2)*sqrt(-e*f)*log(( 
f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) + 30*(7*b*d^3*e^5*f + a*c^3*e*f^5 
 - 5*(3*b*c*d^2 + a*d^3)*e^4*f^2 + 9*(b*c^2*d + a*c*d^2)*e^3*f^3 - (b*c^3 
+ 3*a*c^2*d)*e^2*f^4)*x)/(e^2*f^6*x^2 + e^3*f^5), 1/30*(6*b*d^3*e^2*f^4*x^ 
7 - 2*(7*b*d^3*e^3*f^3 - 5*(3*b*c*d^2 + a*d^3)*e^2*f^4)*x^5 + 10*(7*b*d^3* 
e^4*f^2 - 5*(3*b*c*d^2 + a*d^3)*e^3*f^3 + 9*(b*c^2*d + a*c*d^2)*e^2*f^4)*x 
^3 - 15*(7*b*d^3*e^5 - a*c^3*e*f^4 - 5*(3*b*c*d^2 + a*d^3)*e^4*f + 9*(b*c^ 
2*d + a*c*d^2)*e^3*f^2 - (b*c^3 + 3*a*c^2*d)*e^2*f^3 + (7*b*d^3*e^4*f - a* 
c^3*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^3*f^2 + 9*(b*c^2*d + a*c*d^2)*e^2*f^3 - 
(b*c^3 + 3*a*c^2*d)*e*f^4)*x^2)*sqrt(e*f)*arctan(sqrt(e*f)*x/e) + 15*(7*b* 
d^3*e^5*f + a*c^3*e*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^4*f^2 + 9*(b*c^2*d + a*c 
*d^2)*e^3*f^3 - (b*c^3 + 3*a*c^2*d)*e^2*f^4)*x)/(e^2*f^6*x^2 + e^3*f^5)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (167) = 334\).

Time = 1.94 (sec) , antiderivative size = 661, normalized size of antiderivative = 3.71 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\frac {b d^{3} x^{5}}{5 f^{2}} + x^{3} \left (\frac {a d^{3}}{3 f^{2}} + \frac {b c d^{2}}{f^{2}} - \frac {2 b d^{3} e}{3 f^{3}}\right ) + x \left (\frac {3 a c d^{2}}{f^{2}} - \frac {2 a d^{3} e}{f^{3}} + \frac {3 b c^{2} d}{f^{2}} - \frac {6 b c d^{2} e}{f^{3}} + \frac {3 b d^{3} e^{2}}{f^{4}}\right ) + \frac {x \left (a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}\right )}{2 e^{2} f^{4} + 2 e f^{5} x^{2}} - \frac {\sqrt {- \frac {1}{e^{3} f^{9}}} \left (c f - d e\right )^{2} \left (a c f^{2} + 5 a d e f + b c e f - 7 b d e^{2}\right ) \log {\left (- \frac {e^{2} f^{4} \sqrt {- \frac {1}{e^{3} f^{9}}} \left (c f - d e\right )^{2} \left (a c f^{2} + 5 a d e f + b c e f - 7 b d e^{2}\right )}{a c^{3} f^{4} + 3 a c^{2} d e f^{3} - 9 a c d^{2} e^{2} f^{2} + 5 a d^{3} e^{3} f + b c^{3} e f^{3} - 9 b c^{2} d e^{2} f^{2} + 15 b c d^{2} e^{3} f - 7 b d^{3} e^{4}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{e^{3} f^{9}}} \left (c f - d e\right )^{2} \left (a c f^{2} + 5 a d e f + b c e f - 7 b d e^{2}\right ) \log {\left (\frac {e^{2} f^{4} \sqrt {- \frac {1}{e^{3} f^{9}}} \left (c f - d e\right )^{2} \left (a c f^{2} + 5 a d e f + b c e f - 7 b d e^{2}\right )}{a c^{3} f^{4} + 3 a c^{2} d e f^{3} - 9 a c d^{2} e^{2} f^{2} + 5 a d^{3} e^{3} f + b c^{3} e f^{3} - 9 b c^{2} d e^{2} f^{2} + 15 b c d^{2} e^{3} f - 7 b d^{3} e^{4}} + x \right )}}{4} \] Input:

integrate((b*x**2+a)*(d*x**2+c)**3/(f*x**2+e)**2,x)
 

Output:

b*d**3*x**5/(5*f**2) + x**3*(a*d**3/(3*f**2) + b*c*d**2/f**2 - 2*b*d**3*e/ 
(3*f**3)) + x*(3*a*c*d**2/f**2 - 2*a*d**3*e/f**3 + 3*b*c**2*d/f**2 - 6*b*c 
*d**2*e/f**3 + 3*b*d**3*e**2/f**4) + x*(a*c**3*f**4 - 3*a*c**2*d*e*f**3 + 
3*a*c*d**2*e**2*f**2 - a*d**3*e**3*f - b*c**3*e*f**3 + 3*b*c**2*d*e**2*f** 
2 - 3*b*c*d**2*e**3*f + b*d**3*e**4)/(2*e**2*f**4 + 2*e*f**5*x**2) - sqrt( 
-1/(e**3*f**9))*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f + b*c*e*f - 7*b*d*e** 
2)*log(-e**2*f**4*sqrt(-1/(e**3*f**9))*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e* 
f + b*c*e*f - 7*b*d*e**2)/(a*c**3*f**4 + 3*a*c**2*d*e*f**3 - 9*a*c*d**2*e* 
*2*f**2 + 5*a*d**3*e**3*f + b*c**3*e*f**3 - 9*b*c**2*d*e**2*f**2 + 15*b*c* 
d**2*e**3*f - 7*b*d**3*e**4) + x)/4 + sqrt(-1/(e**3*f**9))*(c*f - d*e)**2* 
(a*c*f**2 + 5*a*d*e*f + b*c*e*f - 7*b*d*e**2)*log(e**2*f**4*sqrt(-1/(e**3* 
f**9))*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f + b*c*e*f - 7*b*d*e**2)/(a*c** 
3*f**4 + 3*a*c**2*d*e*f**3 - 9*a*c*d**2*e**2*f**2 + 5*a*d**3*e**3*f + b*c* 
*3*e*f**3 - 9*b*c**2*d*e**2*f**2 + 15*b*c*d**2*e**3*f - 7*b*d**3*e**4) + x 
)/4
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e)^2,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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (162) = 324\).

Time = 0.12 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.88 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=-\frac {{\left (7 \, b d^{3} e^{4} - 15 \, b c d^{2} e^{3} f - 5 \, a d^{3} e^{3} f + 9 \, b c^{2} d e^{2} f^{2} + 9 \, a c d^{2} e^{2} f^{2} - b c^{3} e f^{3} - 3 \, a c^{2} d e f^{3} - a c^{3} f^{4}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{2 \, \sqrt {e f} e f^{4}} + \frac {b d^{3} e^{4} x - 3 \, b c d^{2} e^{3} f x - a d^{3} e^{3} f x + 3 \, b c^{2} d e^{2} f^{2} x + 3 \, a c d^{2} e^{2} f^{2} x - b c^{3} e f^{3} x - 3 \, a c^{2} d e f^{3} x + a c^{3} f^{4} x}{2 \, {\left (f x^{2} + e\right )} e f^{4}} + \frac {3 \, b d^{3} f^{8} x^{5} - 10 \, b d^{3} e f^{7} x^{3} + 15 \, b c d^{2} f^{8} x^{3} + 5 \, a d^{3} f^{8} x^{3} + 45 \, b d^{3} e^{2} f^{6} x - 90 \, b c d^{2} e f^{7} x - 30 \, a d^{3} e f^{7} x + 45 \, b c^{2} d f^{8} x + 45 \, a c d^{2} f^{8} x}{15 \, f^{10}} \] Input:

integrate((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e)^2,x, algorithm="giac")
 

Output:

-1/2*(7*b*d^3*e^4 - 15*b*c*d^2*e^3*f - 5*a*d^3*e^3*f + 9*b*c^2*d*e^2*f^2 + 
 9*a*c*d^2*e^2*f^2 - b*c^3*e*f^3 - 3*a*c^2*d*e*f^3 - a*c^3*f^4)*arctan(f*x 
/sqrt(e*f))/(sqrt(e*f)*e*f^4) + 1/2*(b*d^3*e^4*x - 3*b*c*d^2*e^3*f*x - a*d 
^3*e^3*f*x + 3*b*c^2*d*e^2*f^2*x + 3*a*c*d^2*e^2*f^2*x - b*c^3*e*f^3*x - 3 
*a*c^2*d*e*f^3*x + a*c^3*f^4*x)/((f*x^2 + e)*e*f^4) + 1/15*(3*b*d^3*f^8*x^ 
5 - 10*b*d^3*e*f^7*x^3 + 15*b*c*d^2*f^8*x^3 + 5*a*d^3*f^8*x^3 + 45*b*d^3*e 
^2*f^6*x - 90*b*c*d^2*e*f^7*x - 30*a*d^3*e*f^7*x + 45*b*c^2*d*f^8*x + 45*a 
*c*d^2*f^8*x)/f^10
 

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx=x^3\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{3\,f^2}-\frac {2\,b\,d^3\,e}{3\,f^3}\right )-x\,\left (\frac {2\,e\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{f^2}-\frac {2\,b\,d^3\,e}{f^3}\right )}{f}+\frac {b\,d^3\,e^2}{f^4}-\frac {3\,c\,d\,\left (a\,d+b\,c\right )}{f^2}\right )+\frac {b\,d^3\,x^5}{5\,f^2}+\frac {x\,\left (-b\,c^3\,e\,f^3+a\,c^3\,f^4+3\,b\,c^2\,d\,e^2\,f^2-3\,a\,c^2\,d\,e\,f^3-3\,b\,c\,d^2\,e^3\,f+3\,a\,c\,d^2\,e^2\,f^2+b\,d^3\,e^4-a\,d^3\,e^3\,f\right )}{2\,e\,\left (f^5\,x^2+e\,f^4\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,{\left (c\,f-d\,e\right )}^2\,\left (a\,c\,f^2-7\,b\,d\,e^2+5\,a\,d\,e\,f+b\,c\,e\,f\right )}{\sqrt {e}\,\left (b\,c^3\,e\,f^3+a\,c^3\,f^4-9\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3+15\,b\,c\,d^2\,e^3\,f-9\,a\,c\,d^2\,e^2\,f^2-7\,b\,d^3\,e^4+5\,a\,d^3\,e^3\,f\right )}\right )\,{\left (c\,f-d\,e\right )}^2\,\left (a\,c\,f^2-7\,b\,d\,e^2+5\,a\,d\,e\,f+b\,c\,e\,f\right )}{2\,e^{3/2}\,f^{9/2}} \] Input:

int(((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^2,x)
 

Output:

x^3*((a*d^3 + 3*b*c*d^2)/(3*f^2) - (2*b*d^3*e)/(3*f^3)) - x*((2*e*((a*d^3 
+ 3*b*c*d^2)/f^2 - (2*b*d^3*e)/f^3))/f + (b*d^3*e^2)/f^4 - (3*c*d*(a*d + b 
*c))/f^2) + (b*d^3*x^5)/(5*f^2) + (x*(a*c^3*f^4 + b*d^3*e^4 - a*d^3*e^3*f 
- b*c^3*e*f^3 - 3*a*c^2*d*e*f^3 - 3*b*c*d^2*e^3*f + 3*a*c*d^2*e^2*f^2 + 3* 
b*c^2*d*e^2*f^2))/(2*e*(e*f^4 + f^5*x^2)) + (atan((f^(1/2)*x*(c*f - d*e)^2 
*(a*c*f^2 - 7*b*d*e^2 + 5*a*d*e*f + b*c*e*f))/(e^(1/2)*(a*c^3*f^4 - 7*b*d^ 
3*e^4 + 5*a*d^3*e^3*f + b*c^3*e*f^3 + 3*a*c^2*d*e*f^3 + 15*b*c*d^2*e^3*f - 
 9*a*c*d^2*e^2*f^2 - 9*b*c^2*d*e^2*f^2)))*(c*f - d*e)^2*(a*c*f^2 - 7*b*d*e 
^2 + 5*a*d*e*f + b*c*e*f))/(2*e^(3/2)*f^(9/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 723, normalized size of antiderivative = 4.06 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)*(d*x^2+c)^3/(f*x^2+e)^2,x)
 

Output:

(15*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*c**3*e*f**4 + 15*sqrt( 
f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*c**3*f**5*x**2 + 45*sqrt(f)*sqr 
t(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*c**2*d*e**2*f**3 + 45*sqrt(f)*sqrt(e) 
*atan((f*x)/(sqrt(f)*sqrt(e)))*a*c**2*d*e*f**4*x**2 - 135*sqrt(f)*sqrt(e)* 
atan((f*x)/(sqrt(f)*sqrt(e)))*a*c*d**2*e**3*f**2 - 135*sqrt(f)*sqrt(e)*ata 
n((f*x)/(sqrt(f)*sqrt(e)))*a*c*d**2*e**2*f**3*x**2 + 75*sqrt(f)*sqrt(e)*at 
an((f*x)/(sqrt(f)*sqrt(e)))*a*d**3*e**4*f + 75*sqrt(f)*sqrt(e)*atan((f*x)/ 
(sqrt(f)*sqrt(e)))*a*d**3*e**3*f**2*x**2 + 15*sqrt(f)*sqrt(e)*atan((f*x)/( 
sqrt(f)*sqrt(e)))*b*c**3*e**2*f**3 + 15*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f 
)*sqrt(e)))*b*c**3*e*f**4*x**2 - 135*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*s 
qrt(e)))*b*c**2*d*e**3*f**2 - 135*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt 
(e)))*b*c**2*d*e**2*f**3*x**2 + 225*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sq 
rt(e)))*b*c*d**2*e**4*f + 225*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)) 
)*b*c*d**2*e**3*f**2*x**2 - 105*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e 
)))*b*d**3*e**5 - 105*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b*d**3 
*e**4*f*x**2 + 15*a*c**3*e*f**5*x - 45*a*c**2*d*e**2*f**4*x + 135*a*c*d**2 
*e**3*f**3*x + 90*a*c*d**2*e**2*f**4*x**3 - 75*a*d**3*e**4*f**2*x - 50*a*d 
**3*e**3*f**3*x**3 + 10*a*d**3*e**2*f**4*x**5 - 15*b*c**3*e**2*f**4*x + 13 
5*b*c**2*d*e**3*f**3*x + 90*b*c**2*d*e**2*f**4*x**3 - 225*b*c*d**2*e**4*f* 
*2*x - 150*b*c*d**2*e**3*f**3*x**3 + 30*b*c*d**2*e**2*f**4*x**5 + 105*b...