\(\int \frac {(a+b x^2)^2 (c+d x^2)^3}{e+f x^2} \, dx\) [232]

Optimal result

Integrand size = 28, antiderivative size = 290 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {\left (b^2 e (d e-c f)^3-2 a b f (d e-c f)^3+a^2 d f^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x}{f^5}-\frac {\left (a^2 d^2 f^2 (d e-3 c f)+b^2 (d e-c f)^3-2 a b d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^3}{3 f^4}+\frac {d \left (a^2 d^2 f^2-2 a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^5}{5 f^3}-\frac {b d^2 (b d e-3 b c f-2 a d f) x^7}{7 f^2}+\frac {b^2 d^3 x^9}{9 f}-\frac {(b e-a f)^2 (d e-c f)^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{11/2}} \] Output:

(b^2*e*(-c*f+d*e)^3-2*a*b*f*(-c*f+d*e)^3+a^2*d*f^2*(3*c^2*f^2-3*c*d*e*f+d^ 
2*e^2))*x/f^5-1/3*(a^2*d^2*f^2*(-3*c*f+d*e)+b^2*(-c*f+d*e)^3-2*a*b*d*f*(3* 
c^2*f^2-3*c*d*e*f+d^2*e^2))*x^3/f^4+1/5*d*(a^2*d^2*f^2-2*a*b*d*f*(-3*c*f+d 
*e)+b^2*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))*x^5/f^3-1/7*b*d^2*(-2*a*d*f-3*b*c*f 
+b*d*e)*x^7/f^2+1/9*b^2*d^3*x^9/f-(-a*f+b*e)^2*(-c*f+d*e)^3*arctan(f^(1/2) 
*x/e^(1/2))/e^(1/2)/f^(11/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{e+f x^2} \, dx=\frac {\left (b^2 e (d e-c f)^3+2 a b f (-d e+c f)^3+a^2 d f^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x}{f^5}+\frac {\left (-b^2 (d e-c f)^3+a^2 d^2 f^2 (-d e+3 c f)+2 a b d f \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^3}{3 f^4}+\frac {d \left (a^2 d^2 f^2-2 a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) x^5}{5 f^3}-\frac {b d^2 (b d e-3 b c f-2 a d f) x^7}{7 f^2}+\frac {b^2 d^3 x^9}{9 f}-\frac {(b e-a f)^2 (d e-c f)^3 \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{11/2}} \] Input:

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

Output:

((b^2*e*(d*e - c*f)^3 + 2*a*b*f*(-(d*e) + c*f)^3 + a^2*d*f^2*(d^2*e^2 - 3* 
c*d*e*f + 3*c^2*f^2))*x)/f^5 + ((-(b^2*(d*e - c*f)^3) + a^2*d^2*f^2*(-(d*e 
) + 3*c*f) + 2*a*b*d*f*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x^3)/(3*f^4) + ( 
d*(a^2*d^2*f^2 - 2*a*b*d*f*(d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*f + 3*c^ 
2*f^2))*x^5)/(5*f^3) - (b*d^2*(b*d*e - 3*b*c*f - 2*a*d*f)*x^7)/(7*f^2) + ( 
b^2*d^3*x^9)/(9*f) - ((b*e - a*f)^2*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[ 
e]])/(Sqrt[e]*f^(11/2))
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {420, 290, 403, 25, 403, 25, 403, 299, 218, 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.

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

\(\Big \downarrow \) 420

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

\(\Big \downarrow \) 290

\(\displaystyle \frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \int \frac {\left (b x^2+a\right ) \left (d x^2+c\right )^3}{f x^2+e}dx}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {\int -\frac {\left (d x^2+c\right )^2 \left ((7 b d e-6 b c f-7 a d f) x^2+c (b e-7 a f)\right )}{f x^2+e}dx}{7 f}+\frac {b x \left (c+d x^2\right )^3}{7 f}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\int \frac {\left (d x^2+c\right )^2 \left ((7 b d e-6 b c f-7 a d f) x^2+c (b e-7 a f)\right )}{f x^2+e}dx}{7 f}\right )}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {\int -\frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}+\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}}{7 f}\right )}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\int \frac {\left (d x^2+c\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))-\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d f e+24 c^2 f^2\right )\right ) x^2\right )}{f x^2+e}dx}{5 f}}{7 f}\right )}{f}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\int \frac {\left (7 a d f \left (15 d^2 e^2-40 c d f e+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 f e^2+231 c^2 d f^2 e-48 c^3 f^3\right )\right ) x^2+c \left (7 a f \left (5 d^2 e^2-12 c d f e+15 c^2 f^2\right )-b e \left (35 d^2 e^2-84 c d f e+57 c^2 f^2\right )\right )}{f x^2+e}dx}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) (d e-c f)^3 \int \frac {1}{f x^2+e}dx}{f}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {b \int \left (b d^3 x^8+d^2 (3 b c+a d) x^6+3 c d (b c+a d) x^4+c^2 (b c+3 a d) x^2+a c^3\right )dx}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {1}{3} c^2 x^3 (3 a d+b c)+\frac {1}{7} d^2 x^7 (a d+3 b c)+\frac {3}{5} c d x^5 (a d+b c)+a c^3 x+\frac {1}{9} b d^3 x^9\right )}{f}-\frac {(b e-a f) \left (\frac {b x \left (c+d x^2\right )^3}{7 f}-\frac {\frac {x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{5 f}-\frac {\frac {\frac {105 (b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)^3}{\sqrt {e} f^{3/2}}+\frac {x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{f}}{3 f}-\frac {x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{3 f}}{5 f}}{7 f}\right )}{f}\)

Input:

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

Output:

(b*(a*c^3*x + (c^2*(b*c + 3*a*d)*x^3)/3 + (3*c*d*(b*c + a*d)*x^5)/5 + (d^2 
*(3*b*c + a*d)*x^7)/7 + (b*d^3*x^9)/9))/f - ((b*e - a*f)*((b*x*(c + d*x^2) 
^3)/(7*f) - (((7*b*d*e - 6*b*c*f - 7*a*d*f)*x*(c + d*x^2)^2)/(5*f) - (-1/3 
*((7*a*d*f*(5*d*e - 9*c*f) - b*(35*d^2*e^2 - 63*c*d*e*f + 24*c^2*f^2))*x*( 
c + d*x^2))/f + (((7*a*d*f*(15*d^2*e^2 - 40*c*d*e*f + 33*c^2*f^2) - b*(105 
*d^3*e^3 - 280*c*d^2*e^2*f + 231*c^2*d*e*f^2 - 48*c^3*f^3))*x)/f + (105*(b 
*e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(3/2)))/(3 
*f))/(5*f))/(7*f)))/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 290

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I 
nt[ExpandIntegrand[(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] && IGtQ[q, 0]
 

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 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]
 

rule 420

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(608\) vs. \(2(274)=548\).

Time = 0.56 (sec) , antiderivative size = 609, normalized size of antiderivative = 2.10

method	result	size
default	\(\frac {a^{2} c \,d^{2} f^{4} x^{3}+\frac {1}{9} b^{2} d^{3} x^{9} f^{4}+2 a b \,c^{3} f^{4} x -b^{2} c^{3} e \,f^{3} x +\frac {2}{7} a b \,d^{3} f^{4} x^{7}+3 b^{2} c^{2} d \,e^{2} f^{2} x -2 a b \,d^{3} e^{3} f x +\frac {3}{7} b^{2} c \,d^{2} f^{4} x^{7}-\frac {1}{7} b^{2} d^{3} e \,f^{3} x^{7}+\frac {3}{5} b^{2} c^{2} d \,f^{4} x^{5}+b^{2} c \,d^{2} e^{2} f^{2} x^{3}+b^{2} d^{3} e^{4} x +\frac {1}{5} a^{2} d^{3} f^{4} x^{5}+\frac {6}{5} a b c \,d^{2} f^{4} x^{5}-\frac {1}{3} a^{2} d^{3} e \,f^{3} x^{3}-\frac {1}{3} b^{2} d^{3} e^{3} f \,x^{3}+3 a^{2} c^{2} d \,f^{4} x +a^{2} d^{3} e^{2} f^{2} x -\frac {2}{5} a b \,d^{3} e \,f^{3} x^{5}-b^{2} c^{2} d e \,f^{3} x^{3}-3 a^{2} c \,d^{2} e \,f^{3} x -\frac {3}{5} b^{2} c \,d^{2} e \,f^{3} x^{5}-2 a b c \,d^{2} e \,f^{3} x^{3}+2 a b \,c^{2} d \,f^{4} x^{3}+\frac {2}{3} a b \,d^{3} e^{2} f^{2} x^{3}+\frac {1}{5} b^{2} d^{3} e^{2} f^{2} x^{5}-6 a b \,c^{2} d e \,f^{3} x +6 a b c \,d^{2} e^{2} f^{2} x -3 b^{2} c \,d^{2} e^{3} f x +\frac {1}{3} b^{2} c^{3} f^{4} x^{3}}{f^{5}}+\frac {\left (a^{2} c^{3} f^{5}-3 a^{2} c^{2} d e \,f^{4}+3 a^{2} c \,d^{2} e^{2} f^{3}-a^{2} d^{3} e^{3} f^{2}-2 a b \,c^{3} e \,f^{4}+6 a b \,c^{2} d \,e^{2} f^{3}-6 a b c \,d^{2} e^{3} f^{2}+2 a b \,d^{3} e^{4} f +b^{2} c^{3} e^{2} f^{3}-3 b^{2} c^{2} d \,e^{3} f^{2}+3 b^{2} c \,d^{2} e^{4} f -b^{2} d^{3} e^{5}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{f^{5} \sqrt {e f}}\)	\(609\)
risch	\(\text {Expression too large to display}\)	\(1178\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 1068, normalized size of antiderivative = 3.68 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{e+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/630*(70*b^2*d^3*e*f^5*x^9 - 90*(b^2*d^3*e^2*f^4 - (3*b^2*c*d^2 + 2*a*b* 
d^3)*e*f^5)*x^7 + 126*(b^2*d^3*e^3*f^3 - (3*b^2*c*d^2 + 2*a*b*d^3)*e^2*f^4 
 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e*f^5)*x^5 - 210*(b^2*d^3*e^4*f^2 
 - (3*b^2*c*d^2 + 2*a*b*d^3)*e^3*f^3 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^ 
3)*e^2*f^4 - (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e*f^5)*x^3 + 315*(b^2*d 
^3*e^5 - a^2*c^3*f^5 - (3*b^2*c*d^2 + 2*a*b*d^3)*e^4*f + (3*b^2*c^2*d + 6* 
a*b*c*d^2 + a^2*d^3)*e^3*f^2 - (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^2*f 
^3 + (2*a*b*c^3 + 3*a^2*c^2*d)*e*f^4)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f) 
*x - e)/(f*x^2 + e)) + 630*(b^2*d^3*e^5*f - (3*b^2*c*d^2 + 2*a*b*d^3)*e^4* 
f^2 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^3*f^3 - (b^2*c^3 + 6*a*b*c^2 
*d + 3*a^2*c*d^2)*e^2*f^4 + (2*a*b*c^3 + 3*a^2*c^2*d)*e*f^5)*x)/(e*f^6), 1 
/315*(35*b^2*d^3*e*f^5*x^9 - 45*(b^2*d^3*e^2*f^4 - (3*b^2*c*d^2 + 2*a*b*d^ 
3)*e*f^5)*x^7 + 63*(b^2*d^3*e^3*f^3 - (3*b^2*c*d^2 + 2*a*b*d^3)*e^2*f^4 + 
(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e*f^5)*x^5 - 105*(b^2*d^3*e^4*f^2 - 
(3*b^2*c*d^2 + 2*a*b*d^3)*e^3*f^3 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)* 
e^2*f^4 - (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e*f^5)*x^3 - 315*(b^2*d^3* 
e^5 - a^2*c^3*f^5 - (3*b^2*c*d^2 + 2*a*b*d^3)*e^4*f + (3*b^2*c^2*d + 6*a*b 
*c*d^2 + a^2*d^3)*e^3*f^2 - (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^2*f^3 
+ (2*a*b*c^3 + 3*a^2*c^2*d)*e*f^4)*sqrt(e*f)*arctan(sqrt(e*f)*x/e) + 315*( 
b^2*d^3*e^5*f - (3*b^2*c*d^2 + 2*a*b*d^3)*e^4*f^2 + (3*b^2*c^2*d + 6*a*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 911 vs. \(2 (291) = 582\).

Time = 1.66 (sec) , antiderivative size = 911, normalized size of antiderivative = 3.14 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{e+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

b**2*d**3*x**9/(9*f) + x**7*(2*a*b*d**3/(7*f) + 3*b**2*c*d**2/(7*f) - b**2 
*d**3*e/(7*f**2)) + x**5*(a**2*d**3/(5*f) + 6*a*b*c*d**2/(5*f) - 2*a*b*d** 
3*e/(5*f**2) + 3*b**2*c**2*d/(5*f) - 3*b**2*c*d**2*e/(5*f**2) + b**2*d**3* 
e**2/(5*f**3)) + x**3*(a**2*c*d**2/f - a**2*d**3*e/(3*f**2) + 2*a*b*c**2*d 
/f - 2*a*b*c*d**2*e/f**2 + 2*a*b*d**3*e**2/(3*f**3) + b**2*c**3/(3*f) - b* 
*2*c**2*d*e/f**2 + b**2*c*d**2*e**2/f**3 - b**2*d**3*e**3/(3*f**4)) + x*(3 
*a**2*c**2*d/f - 3*a**2*c*d**2*e/f**2 + a**2*d**3*e**2/f**3 + 2*a*b*c**3/f 
 - 6*a*b*c**2*d*e/f**2 + 6*a*b*c*d**2*e**2/f**3 - 2*a*b*d**3*e**3/f**4 - b 
**2*c**3*e/f**2 + 3*b**2*c**2*d*e**2/f**3 - 3*b**2*c*d**2*e**3/f**4 + b**2 
*d**3*e**4/f**5) - sqrt(-1/(e*f**11))*(a*f - b*e)**2*(c*f - d*e)**3*log(-e 
*f**5*sqrt(-1/(e*f**11))*(a*f - b*e)**2*(c*f - d*e)**3/(a**2*c**3*f**5 - 3 
*a**2*c**2*d*e*f**4 + 3*a**2*c*d**2*e**2*f**3 - a**2*d**3*e**3*f**2 - 2*a* 
b*c**3*e*f**4 + 6*a*b*c**2*d*e**2*f**3 - 6*a*b*c*d**2*e**3*f**2 + 2*a*b*d* 
*3*e**4*f + b**2*c**3*e**2*f**3 - 3*b**2*c**2*d*e**3*f**2 + 3*b**2*c*d**2* 
e**4*f - b**2*d**3*e**5) + x)/2 + sqrt(-1/(e*f**11))*(a*f - b*e)**2*(c*f - 
 d*e)**3*log(e*f**5*sqrt(-1/(e*f**11))*(a*f - b*e)**2*(c*f - d*e)**3/(a**2 
*c**3*f**5 - 3*a**2*c**2*d*e*f**4 + 3*a**2*c*d**2*e**2*f**3 - a**2*d**3*e* 
*3*f**2 - 2*a*b*c**3*e*f**4 + 6*a*b*c**2*d*e**2*f**3 - 6*a*b*c*d**2*e**3*f 
**2 + 2*a*b*d**3*e**4*f + b**2*c**3*e**2*f**3 - 3*b**2*c**2*d*e**3*f**2 + 
3*b**2*c*d**2*e**4*f - b**2*d**3*e**5) + x)/2
 

Maxima [F(-2)]

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

integrate((b*x^2+a)^2*(d*x^2+c)^3/(f*x^2+e),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. 623 vs. \(2 (274) = 548\).

Time = 0.12 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{e+f x^2} \, dx=-\frac {{\left (b^{2} d^{3} e^{5} - 3 \, b^{2} c d^{2} e^{4} f - 2 \, a b d^{3} e^{4} f + 3 \, b^{2} c^{2} d e^{3} f^{2} + 6 \, a b c d^{2} e^{3} f^{2} + a^{2} d^{3} e^{3} f^{2} - b^{2} c^{3} e^{2} f^{3} - 6 \, a b c^{2} d e^{2} f^{3} - 3 \, a^{2} c d^{2} e^{2} f^{3} + 2 \, a b c^{3} e f^{4} + 3 \, a^{2} c^{2} d e f^{4} - a^{2} c^{3} f^{5}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{\sqrt {e f} f^{5}} + \frac {35 \, b^{2} d^{3} f^{8} x^{9} - 45 \, b^{2} d^{3} e f^{7} x^{7} + 135 \, b^{2} c d^{2} f^{8} x^{7} + 90 \, a b d^{3} f^{8} x^{7} + 63 \, b^{2} d^{3} e^{2} f^{6} x^{5} - 189 \, b^{2} c d^{2} e f^{7} x^{5} - 126 \, a b d^{3} e f^{7} x^{5} + 189 \, b^{2} c^{2} d f^{8} x^{5} + 378 \, a b c d^{2} f^{8} x^{5} + 63 \, a^{2} d^{3} f^{8} x^{5} - 105 \, b^{2} d^{3} e^{3} f^{5} x^{3} + 315 \, b^{2} c d^{2} e^{2} f^{6} x^{3} + 210 \, a b d^{3} e^{2} f^{6} x^{3} - 315 \, b^{2} c^{2} d e f^{7} x^{3} - 630 \, a b c d^{2} e f^{7} x^{3} - 105 \, a^{2} d^{3} e f^{7} x^{3} + 105 \, b^{2} c^{3} f^{8} x^{3} + 630 \, a b c^{2} d f^{8} x^{3} + 315 \, a^{2} c d^{2} f^{8} x^{3} + 315 \, b^{2} d^{3} e^{4} f^{4} x - 945 \, b^{2} c d^{2} e^{3} f^{5} x - 630 \, a b d^{3} e^{3} f^{5} x + 945 \, b^{2} c^{2} d e^{2} f^{6} x + 1890 \, a b c d^{2} e^{2} f^{6} x + 315 \, a^{2} d^{3} e^{2} f^{6} x - 315 \, b^{2} c^{3} e f^{7} x - 1890 \, a b c^{2} d e f^{7} x - 945 \, a^{2} c d^{2} e f^{7} x + 630 \, a b c^{3} f^{8} x + 945 \, a^{2} c^{2} d f^{8} x}{315 \, f^{9}} \] Input:

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

Output:

-(b^2*d^3*e^5 - 3*b^2*c*d^2*e^4*f - 2*a*b*d^3*e^4*f + 3*b^2*c^2*d*e^3*f^2 
+ 6*a*b*c*d^2*e^3*f^2 + a^2*d^3*e^3*f^2 - b^2*c^3*e^2*f^3 - 6*a*b*c^2*d*e^ 
2*f^3 - 3*a^2*c*d^2*e^2*f^3 + 2*a*b*c^3*e*f^4 + 3*a^2*c^2*d*e*f^4 - a^2*c^ 
3*f^5)*arctan(f*x/sqrt(e*f))/(sqrt(e*f)*f^5) + 1/315*(35*b^2*d^3*f^8*x^9 - 
 45*b^2*d^3*e*f^7*x^7 + 135*b^2*c*d^2*f^8*x^7 + 90*a*b*d^3*f^8*x^7 + 63*b^ 
2*d^3*e^2*f^6*x^5 - 189*b^2*c*d^2*e*f^7*x^5 - 126*a*b*d^3*e*f^7*x^5 + 189* 
b^2*c^2*d*f^8*x^5 + 378*a*b*c*d^2*f^8*x^5 + 63*a^2*d^3*f^8*x^5 - 105*b^2*d 
^3*e^3*f^5*x^3 + 315*b^2*c*d^2*e^2*f^6*x^3 + 210*a*b*d^3*e^2*f^6*x^3 - 315 
*b^2*c^2*d*e*f^7*x^3 - 630*a*b*c*d^2*e*f^7*x^3 - 105*a^2*d^3*e*f^7*x^3 + 1 
05*b^2*c^3*f^8*x^3 + 630*a*b*c^2*d*f^8*x^3 + 315*a^2*c*d^2*f^8*x^3 + 315*b 
^2*d^3*e^4*f^4*x - 945*b^2*c*d^2*e^3*f^5*x - 630*a*b*d^3*e^3*f^5*x + 945*b 
^2*c^2*d*e^2*f^6*x + 1890*a*b*c*d^2*e^2*f^6*x + 315*a^2*d^3*e^2*f^6*x - 31 
5*b^2*c^3*e*f^7*x - 1890*a*b*c^2*d*e*f^7*x - 945*a^2*c*d^2*e*f^7*x + 630*a 
*b*c^3*f^8*x + 945*a^2*c^2*d*f^8*x)/f^9
 

Mupad [B] (verification not implemented)

Time = 2.08 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{e+f x^2} \, dx=x^3\,\left (\frac {3\,a^2\,c\,d^2+6\,a\,b\,c^2\,d+b^2\,c^3}{3\,f}-\frac {e\,\left (\frac {a^2\,d^3+6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d}{f}+\frac {e\,\left (\frac {b^2\,d^3\,e}{f^2}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{f}\right )}{f}\right )}{3\,f}\right )+x^5\,\left (\frac {a^2\,d^3+6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d}{5\,f}+\frac {e\,\left (\frac {b^2\,d^3\,e}{f^2}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{f}\right )}{5\,f}\right )-x^7\,\left (\frac {b^2\,d^3\,e}{7\,f^2}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{7\,f}\right )-x\,\left (\frac {e\,\left (\frac {3\,a^2\,c\,d^2+6\,a\,b\,c^2\,d+b^2\,c^3}{f}-\frac {e\,\left (\frac {a^2\,d^3+6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d}{f}+\frac {e\,\left (\frac {b^2\,d^3\,e}{f^2}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{f}\right )}{f}\right )}{f}\right )}{f}-\frac {a\,c^2\,\left (3\,a\,d+2\,b\,c\right )}{f}\right )+\frac {b^2\,d^3\,x^9}{9\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,{\left (a\,f-b\,e\right )}^2\,{\left (c\,f-d\,e\right )}^3}{\sqrt {e}\,\left (a^2\,c^3\,f^5-3\,a^2\,c^2\,d\,e\,f^4+3\,a^2\,c\,d^2\,e^2\,f^3-a^2\,d^3\,e^3\,f^2-2\,a\,b\,c^3\,e\,f^4+6\,a\,b\,c^2\,d\,e^2\,f^3-6\,a\,b\,c\,d^2\,e^3\,f^2+2\,a\,b\,d^3\,e^4\,f+b^2\,c^3\,e^2\,f^3-3\,b^2\,c^2\,d\,e^3\,f^2+3\,b^2\,c\,d^2\,e^4\,f-b^2\,d^3\,e^5\right )}\right )\,{\left (a\,f-b\,e\right )}^2\,{\left (c\,f-d\,e\right )}^3}{\sqrt {e}\,f^{11/2}} \] Input:

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

Output:

x^3*((b^2*c^3 + 3*a^2*c*d^2 + 6*a*b*c^2*d)/(3*f) - (e*((a^2*d^3 + 3*b^2*c^ 
2*d + 6*a*b*c*d^2)/f + (e*((b^2*d^3*e)/f^2 - (b*d^2*(2*a*d + 3*b*c))/f))/f 
))/(3*f)) + x^5*((a^2*d^3 + 3*b^2*c^2*d + 6*a*b*c*d^2)/(5*f) + (e*((b^2*d^ 
3*e)/f^2 - (b*d^2*(2*a*d + 3*b*c))/f))/(5*f)) - x^7*((b^2*d^3*e)/(7*f^2) - 
 (b*d^2*(2*a*d + 3*b*c))/(7*f)) - x*((e*((b^2*c^3 + 3*a^2*c*d^2 + 6*a*b*c^ 
2*d)/f - (e*((a^2*d^3 + 3*b^2*c^2*d + 6*a*b*c*d^2)/f + (e*((b^2*d^3*e)/f^2 
 - (b*d^2*(2*a*d + 3*b*c))/f))/f))/f))/f - (a*c^2*(3*a*d + 2*b*c))/f) + (b 
^2*d^3*x^9)/(9*f) + (atan((f^(1/2)*x*(a*f - b*e)^2*(c*f - d*e)^3)/(e^(1/2) 
*(a^2*c^3*f^5 - b^2*d^3*e^5 - a^2*d^3*e^3*f^2 + b^2*c^3*e^2*f^3 - 2*a*b*c^ 
3*e*f^4 + 2*a*b*d^3*e^4*f - 3*a^2*c^2*d*e*f^4 + 3*b^2*c*d^2*e^4*f + 3*a^2* 
c*d^2*e^2*f^3 - 3*b^2*c^2*d*e^3*f^2 - 6*a*b*c*d^2*e^3*f^2 + 6*a*b*c^2*d*e^ 
2*f^3)))*(a*f - b*e)^2*(c*f - d*e)^3)/(e^(1/2)*f^(11/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.85 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{e+f x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(315*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*f**5 - 945*sq 
rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e*f**4 + 945*sqrt( 
f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**2*f**3 - 315*sqrt( 
f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**3*f**2 - 630*sqrt(f) 
*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e*f**4 + 1890*sqrt(f)*sqrt 
(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*d*e**2*f**3 - 1890*sqrt(f)*sqrt 
(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d**2*e**3*f**2 + 630*sqrt(f)*sqrt( 
e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*d**3*e**4*f + 315*sqrt(f)*sqrt(e)*ata 
n((f*x)/(sqrt(f)*sqrt(e)))*b**2*c**3*e**2*f**3 - 945*sqrt(f)*sqrt(e)*atan( 
(f*x)/(sqrt(f)*sqrt(e)))*b**2*c**2*d*e**3*f**2 + 945*sqrt(f)*sqrt(e)*atan( 
(f*x)/(sqrt(f)*sqrt(e)))*b**2*c*d**2*e**4*f - 315*sqrt(f)*sqrt(e)*atan((f* 
x)/(sqrt(f)*sqrt(e)))*b**2*d**3*e**5 + 945*a**2*c**2*d*e*f**5*x - 945*a**2 
*c*d**2*e**2*f**4*x + 315*a**2*c*d**2*e*f**5*x**3 + 315*a**2*d**3*e**3*f** 
3*x - 105*a**2*d**3*e**2*f**4*x**3 + 63*a**2*d**3*e*f**5*x**5 + 630*a*b*c* 
*3*e*f**5*x - 1890*a*b*c**2*d*e**2*f**4*x + 630*a*b*c**2*d*e*f**5*x**3 + 1 
890*a*b*c*d**2*e**3*f**3*x - 630*a*b*c*d**2*e**2*f**4*x**3 + 378*a*b*c*d** 
2*e*f**5*x**5 - 630*a*b*d**3*e**4*f**2*x + 210*a*b*d**3*e**3*f**3*x**3 - 1 
26*a*b*d**3*e**2*f**4*x**5 + 90*a*b*d**3*e*f**5*x**7 - 315*b**2*c**3*e**2* 
f**4*x + 105*b**2*c**3*e*f**5*x**3 + 945*b**2*c**2*d*e**3*f**3*x - 315*b** 
2*c**2*d*e**2*f**4*x**3 + 189*b**2*c**2*d*e*f**5*x**5 - 945*b**2*c*d**2...