\(\int \frac {(a e+c d x^2)^2 (d+e x^2)^2 (a-c x^4)}{(a+b x^2+c x^4)^{7/2}} \, dx\) [21]

Optimal result

Integrand size = 46, antiderivative size = 168 \[ \int \frac {\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2 \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^2 x \left (a \left (b^2-4 a c\right )+b \left (b^2-4 a c\right ) x^2\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{5/2}}+\frac {\left (c d^2-b d e+a e^2\right ) x \left (3 \left (c d^2-b d e+a e^2\right )+10 c d e x^2\right )}{15 c \left (a+b x^2+c x^4\right )^{3/2}}+\frac {d^2 e^2 x}{\sqrt {a+b x^2+c x^4}} \] Output:

-1/5*(a*e^2-b*d*e+c*d^2)^2*x*(a*(-4*a*c+b^2)+b*(-4*a*c+b^2)*x^2)/c/(-4*a*c 
+b^2)/(c*x^4+b*x^2+a)^(5/2)+1/15*(a*e^2-b*d*e+c*d^2)*x*(10*c*d*e*x^2+3*a*e 
^2-3*b*d*e+3*c*d^2)/c/(c*x^4+b*x^2+a)^(3/2)+d^2*e^2*x/(c*x^4+b*x^2+a)^(1/2 
)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2 \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx=\frac {a^2 e^2 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+2 a d e x^3 \left (2 b e \left (5 d+e x^2\right )+c \left (5 d^2+18 d e x^2+5 e^2 x^4\right )\right )+d^2 x^5 \left (8 b^2 e^2+4 b c e \left (d+5 e x^2\right )+c^2 \left (3 d^2+10 d e x^2+15 e^2 x^4\right )\right )}{15 \left (a+b x^2+c x^4\right )^{5/2}} \] Input:

Integrate[((a*e + c*d*x^2)^2*(d + e*x^2)^2*(a - c*x^4))/(a + b*x^2 + c*x^4 
)^(7/2),x]
 

Output:

(a^2*e^2*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + 2*a*d*e*x^3*(2*b*e*(5*d + e 
*x^2) + c*(5*d^2 + 18*d*e*x^2 + 5*e^2*x^4)) + d^2*x^5*(8*b^2*e^2 + 4*b*c*e 
*(d + 5*e*x^2) + c^2*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4)))/(15*(a + b*x^2 + 
c*x^4)^(5/2))
 

Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {2206, 25, 2206, 27, 2021}

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.

\(\displaystyle \int \frac {\left (a-c x^4\right ) \left (d+e x^2\right )^2 \left (a e+c d x^2\right )^2}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {\int -\frac {-5 a c^2 \left (b^2-4 a c\right ) d^2 e^2 x^8-5 a c \left (b^2-4 a c\right ) d e \left (2 c d^2-e (b d-2 a e)\right ) x^6-5 a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 x^4-\frac {a \left (b^2-4 a c\right ) \left (2 d^2 e^2 b^3-4 \left (c e d^3+a e^3 d\right ) b^2+\left (2 c^2 d^4+9 a c e^2 d^2+2 a^2 e^4\right ) b-10 a c d e \left (c d^2+a e^2\right )\right ) x^2}{c}+\frac {a^2 \left (b^2-4 a c\right ) \left (c^2 d^4-c e (2 b d-7 a e) d^2+e^2 (b d-a e)^2\right )}{c}}{\left (c x^4+b x^2+a\right )^{5/2}}dx}{5 a \left (b^2-4 a c\right )}-\frac {x \left (b x^2 \left (b^2-4 a c\right )+a \left (b^2-4 a c\right )\right ) \left (a e^2-b d e+c d^2\right )^2}{5 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-5 a c^2 \left (b^2-4 a c\right ) d^2 e^2 x^8-5 a c \left (b^2-4 a c\right ) d e \left (2 c d^2-e (b d-2 a e)\right ) x^6-5 a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 x^4-\frac {a \left (b^2-4 a c\right ) \left (2 d^2 e^2 b^3-4 \left (c e d^3+a e^3 d\right ) b^2+\left (2 c^2 d^4+9 a c e^2 d^2+2 a^2 e^4\right ) b-10 a c d e \left (c d^2+a e^2\right )\right ) x^2}{c}+\frac {a^2 \left (b^2-4 a c\right ) \left (c^2 d^4-c e (2 b d-7 a e) d^2+e^2 (b d-a e)^2\right )}{c}}{\left (c x^4+b x^2+a\right )^{5/2}}dx}{5 a \left (b^2-4 a c\right )}-\frac {x \left (b x^2 \left (b^2-4 a c\right )+a \left (b^2-4 a c\right )\right ) \left (a e^2-b d e+c d^2\right )^2}{5 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{5/2}}\)

\(\Big \downarrow \) 2206

\(\displaystyle \frac {\frac {a x \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right ) \left (3 \left (a e^2-b d e+c d^2\right )+10 c d e x^2\right )}{3 c \left (a+b x^2+c x^4\right )^{3/2}}-\frac {\int -\frac {15 a^2 \left (b^2-4 a c\right )^2 d^2 e^2 \left (a-c x^4\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{3 a \left (b^2-4 a c\right )}}{5 a \left (b^2-4 a c\right )}-\frac {x \left (b x^2 \left (b^2-4 a c\right )+a \left (b^2-4 a c\right )\right ) \left (a e^2-b d e+c d^2\right )^2}{5 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 a d^2 e^2 \left (b^2-4 a c\right ) \int \frac {a-c x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\frac {a x \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right ) \left (3 \left (a e^2-b d e+c d^2\right )+10 c d e x^2\right )}{3 c \left (a+b x^2+c x^4\right )^{3/2}}}{5 a \left (b^2-4 a c\right )}-\frac {x \left (b x^2 \left (b^2-4 a c\right )+a \left (b^2-4 a c\right )\right ) \left (a e^2-b d e+c d^2\right )^2}{5 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{5/2}}\)

\(\Big \downarrow \) 2021

\(\displaystyle \frac {\frac {5 a d^2 e^2 x \left (b^2-4 a c\right )}{\sqrt {a+b x^2+c x^4}}+\frac {a x \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right ) \left (3 \left (a e^2-b d e+c d^2\right )+10 c d e x^2\right )}{3 c \left (a+b x^2+c x^4\right )^{3/2}}}{5 a \left (b^2-4 a c\right )}-\frac {x \left (b x^2 \left (b^2-4 a c\right )+a \left (b^2-4 a c\right )\right ) \left (a e^2-b d e+c d^2\right )^2}{5 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{5/2}}\)

Input:

Int[((a*e + c*d*x^2)^2*(d + e*x^2)^2*(a - c*x^4))/(a + b*x^2 + c*x^4)^(7/2 
),x]
 

Output:

-1/5*((c*d^2 - b*d*e + a*e^2)^2*x*(a*(b^2 - 4*a*c) + b*(b^2 - 4*a*c)*x^2)) 
/(c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^(5/2)) + ((a*(b^2 - 4*a*c)*(c*d^2 - 
b*d*e + a*e^2)*x*(3*(c*d^2 - b*d*e + a*e^2) + 10*c*d*e*x^2))/(3*c*(a + b*x 
^2 + c*x^4)^(3/2)) + (5*a*(b^2 - 4*a*c)*d^2*e^2*x)/Sqrt[a + b*x^2 + c*x^4] 
)/(5*a*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2021

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x 
]}, Simp[Coeff[Pp, x, p]*x^(p - q + 1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, 
 x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp 
, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; Free 
Q[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {\left (\frac {c^{2} d^{4} x^{4}}{5}+\frac {2 c e \left (c \,x^{4}+\frac {2}{5} b \,x^{2}+a \right ) x^{2} d^{3}}{3}+e^{2} \left (a^{2}+\frac {4 x^{2} \left (\frac {9 c \,x^{2}}{5}+b \right ) a}{3}+c^{2} x^{8}+\frac {4 b c \,x^{6}}{3}+\frac {8 b^{2} x^{4}}{15}\right ) d^{2}+\frac {2 a \,e^{3} \left (c \,x^{4}+\frac {2}{5} b \,x^{2}+a \right ) x^{2} d}{3}+\frac {a^{2} e^{4} x^{4}}{5}\right ) x}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {5}{2}}}\)	\(133\)
elliptic	\(\frac {\left (\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) d e \sqrt {2}\, x^{3}}{3 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {d^{2} e^{2} \sqrt {2}\, x}{\sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \sqrt {2}\, x^{5}}{5 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {5}{2}}}\right ) \sqrt {2}}{2}\)	\(145\)
gosper	\(\frac {x \left (15 d^{2} e^{2} c^{2} x^{8}+10 a c d \,e^{3} x^{6}+20 b \,x^{6} d^{2} e^{2} c +10 c^{2} d^{3} e \,x^{6}+3 a^{2} e^{4} x^{4}+4 a b d \,e^{3} x^{4}+36 a c \,d^{2} e^{2} x^{4}+8 b^{2} d^{2} e^{2} x^{4}+4 b c \,d^{3} e \,x^{4}+3 c^{2} d^{4} x^{4}+10 a^{2} d \,e^{3} x^{2}+20 a b \,x^{2} d^{2} e^{2}+10 a c \,d^{3} e \,x^{2}+15 a^{2} d^{2} e^{2}\right )}{15 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {5}{2}}}\)	\(187\)
trager	\(\frac {x \left (15 d^{2} e^{2} c^{2} x^{8}+10 a c d \,e^{3} x^{6}+20 b \,x^{6} d^{2} e^{2} c +10 c^{2} d^{3} e \,x^{6}+3 a^{2} e^{4} x^{4}+4 a b d \,e^{3} x^{4}+36 a c \,d^{2} e^{2} x^{4}+8 b^{2} d^{2} e^{2} x^{4}+4 b c \,d^{3} e \,x^{4}+3 c^{2} d^{4} x^{4}+10 a^{2} d \,e^{3} x^{2}+20 a b \,x^{2} d^{2} e^{2}+10 a c \,d^{3} e \,x^{2}+15 a^{2} d^{2} e^{2}\right )}{15 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {5}{2}}}\)	\(187\)
orering	\(\frac {x \left (15 d^{2} e^{2} c^{2} x^{8}+10 a c d \,e^{3} x^{6}+20 b \,x^{6} d^{2} e^{2} c +10 c^{2} d^{3} e \,x^{6}+3 a^{2} e^{4} x^{4}+4 a b d \,e^{3} x^{4}+36 a c \,d^{2} e^{2} x^{4}+8 b^{2} d^{2} e^{2} x^{4}+4 b c \,d^{3} e \,x^{4}+3 c^{2} d^{4} x^{4}+10 a^{2} d \,e^{3} x^{2}+20 a b \,x^{2} d^{2} e^{2}+10 a c \,d^{3} e \,x^{2}+15 a^{2} d^{2} e^{2}\right )}{15 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {5}{2}}}\)	\(187\)
default	\(\text {Expression too large to display}\)	\(4733\)

Input:

int((c*d*x^2+a*e)^2*(e*x^2+d)^2*(-c*x^4+a)/(c*x^4+b*x^2+a)^(7/2),x,method= 
_RETURNVERBOSE)
 

Output:

(1/5*c^2*d^4*x^4+2/3*c*e*(c*x^4+2/5*b*x^2+a)*x^2*d^3+e^2*(a^2+4/3*x^2*(9/5 
*c*x^2+b)*a+c^2*x^8+4/3*b*c*x^6+8/15*b^2*x^4)*d^2+2/3*a*e^3*(c*x^4+2/5*b*x 
^2+a)*x^2*d+1/5*a^2*e^4*x^4)*x/(c*x^4+b*x^2+a)^(5/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2 \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx=\frac {{\left (15 \, c^{2} d^{2} e^{2} x^{9} + 10 \, {\left (c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + a c d e^{3}\right )} x^{7} + 15 \, a^{2} d^{2} e^{2} x + {\left (3 \, c^{2} d^{4} + 4 \, b c d^{3} e + 4 \, a b d e^{3} + 3 \, a^{2} e^{4} + 4 \, {\left (2 \, b^{2} + 9 \, a c\right )} d^{2} e^{2}\right )} x^{5} + 10 \, {\left (a c d^{3} e + 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{15 \, {\left (c^{3} x^{12} + 3 \, b c^{2} x^{10} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + {\left (b^{3} + 6 \, a b c\right )} x^{6} + 3 \, a^{2} b x^{2} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{4} + a^{3}\right )}} \] Input:

integrate((c*d*x^2+a*e)^2*(e*x^2+d)^2*(-c*x^4+a)/(c*x^4+b*x^2+a)^(7/2),x, 
algorithm="fricas")
 

Output:

1/15*(15*c^2*d^2*e^2*x^9 + 10*(c^2*d^3*e + 2*b*c*d^2*e^2 + a*c*d*e^3)*x^7 
+ 15*a^2*d^2*e^2*x + (3*c^2*d^4 + 4*b*c*d^3*e + 4*a*b*d*e^3 + 3*a^2*e^4 + 
4*(2*b^2 + 9*a*c)*d^2*e^2)*x^5 + 10*(a*c*d^3*e + 2*a*b*d^2*e^2 + a^2*d*e^3 
)*x^3)*sqrt(c*x^4 + b*x^2 + a)/(c^3*x^12 + 3*b*c^2*x^10 + 3*(b^2*c + a*c^2 
)*x^8 + (b^3 + 6*a*b*c)*x^6 + 3*a^2*b*x^2 + 3*(a*b^2 + a^2*c)*x^4 + a^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2 \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((c*d*x**2+a*e)**2*(e*x**2+d)**2*(-c*x**4+a)/(c*x**4+b*x**2+a)**( 
7/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2 \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx=\frac {{\left (15 \, c^{2} d^{2} e^{2} x^{9} + 10 \, {\left (c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + a c d e^{3}\right )} x^{7} + 15 \, a^{2} d^{2} e^{2} x + {\left (3 \, c^{2} d^{4} + 4 \, b c d^{3} e + 8 \, b^{2} d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (9 \, c d^{2} e^{2} + b d e^{3}\right )} a\right )} x^{5} + 10 \, {\left (a^{2} d e^{3} + {\left (c d^{3} e + 2 \, b d^{2} e^{2}\right )} a\right )} x^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{15 \, {\left (c^{3} x^{12} + 3 \, b c^{2} x^{10} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + {\left (b^{3} + 6 \, a b c\right )} x^{6} + 3 \, a^{2} b x^{2} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{4} + a^{3}\right )}} \] Input:

integrate((c*d*x^2+a*e)^2*(e*x^2+d)^2*(-c*x^4+a)/(c*x^4+b*x^2+a)^(7/2),x, 
algorithm="maxima")
 

Output:

1/15*(15*c^2*d^2*e^2*x^9 + 10*(c^2*d^3*e + 2*b*c*d^2*e^2 + a*c*d*e^3)*x^7 
+ 15*a^2*d^2*e^2*x + (3*c^2*d^4 + 4*b*c*d^3*e + 8*b^2*d^2*e^2 + 3*a^2*e^4 
+ 4*(9*c*d^2*e^2 + b*d*e^3)*a)*x^5 + 10*(a^2*d*e^3 + (c*d^3*e + 2*b*d^2*e^ 
2)*a)*x^3)*sqrt(c*x^4 + b*x^2 + a)/(c^3*x^12 + 3*b*c^2*x^10 + 3*(b^2*c + a 
*c^2)*x^8 + (b^3 + 6*a*b*c)*x^6 + 3*a^2*b*x^2 + 3*(a*b^2 + a^2*c)*x^4 + a^ 
3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1828 vs. \(2 (157) = 314\).

Time = 1.23 (sec) , antiderivative size = 1828, normalized size of antiderivative = 10.88 \[ \int \frac {\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2 \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((c*d*x^2+a*e)^2*(e*x^2+d)^2*(-c*x^4+a)/(c*x^4+b*x^2+a)^(7/2),x, 
algorithm="giac")
 

Output:

1/15*(((5*(3*(a^2*b^12*c^4*d^2*e^2 - 24*a^3*b^10*c^5*d^2*e^2 + 240*a^4*b^8 
*c^6*d^2*e^2 - 1280*a^5*b^6*c^7*d^2*e^2 + 3840*a^6*b^4*c^8*d^2*e^2 - 6144* 
a^7*b^2*c^9*d^2*e^2 + 4096*a^8*c^10*d^2*e^2)*x^2/(a^2*b^12*c^2 - 24*a^3*b^ 
10*c^3 + 240*a^4*b^8*c^4 - 1280*a^5*b^6*c^5 + 3840*a^6*b^4*c^6 - 6144*a^7* 
b^2*c^7 + 4096*a^8*c^8) + 2*(a^2*b^12*c^4*d^3*e - 24*a^3*b^10*c^5*d^3*e + 
240*a^4*b^8*c^6*d^3*e - 1280*a^5*b^6*c^7*d^3*e + 3840*a^6*b^4*c^8*d^3*e - 
6144*a^7*b^2*c^9*d^3*e + 4096*a^8*c^10*d^3*e + 2*a^2*b^13*c^3*d^2*e^2 - 48 
*a^3*b^11*c^4*d^2*e^2 + 480*a^4*b^9*c^5*d^2*e^2 - 2560*a^5*b^7*c^6*d^2*e^2 
 + 7680*a^6*b^5*c^7*d^2*e^2 - 12288*a^7*b^3*c^8*d^2*e^2 + 8192*a^8*b*c^9*d 
^2*e^2 + a^3*b^12*c^3*d*e^3 - 24*a^4*b^10*c^4*d*e^3 + 240*a^5*b^8*c^5*d*e^ 
3 - 1280*a^6*b^6*c^6*d*e^3 + 3840*a^7*b^4*c^7*d*e^3 - 6144*a^8*b^2*c^8*d*e 
^3 + 4096*a^9*c^9*d*e^3)/(a^2*b^12*c^2 - 24*a^3*b^10*c^3 + 240*a^4*b^8*c^4 
 - 1280*a^5*b^6*c^5 + 3840*a^6*b^4*c^6 - 6144*a^7*b^2*c^7 + 4096*a^8*c^8)) 
*x^2 + (3*a^2*b^12*c^4*d^4 - 72*a^3*b^10*c^5*d^4 + 720*a^4*b^8*c^6*d^4 - 3 
840*a^5*b^6*c^7*d^4 + 11520*a^6*b^4*c^8*d^4 - 18432*a^7*b^2*c^9*d^4 + 1228 
8*a^8*c^10*d^4 + 4*a^2*b^13*c^3*d^3*e - 96*a^3*b^11*c^4*d^3*e + 960*a^4*b^ 
9*c^5*d^3*e - 5120*a^5*b^7*c^6*d^3*e + 15360*a^6*b^5*c^7*d^3*e - 24576*a^7 
*b^3*c^8*d^3*e + 16384*a^8*b*c^9*d^3*e + 8*a^2*b^14*c^2*d^2*e^2 - 156*a^3* 
b^12*c^3*d^2*e^2 + 1056*a^4*b^10*c^4*d^2*e^2 - 1600*a^5*b^8*c^5*d^2*e^2 - 
15360*a^6*b^6*c^6*d^2*e^2 + 89088*a^7*b^4*c^7*d^2*e^2 - 188416*a^8*b^2*...
 

Mupad [B] (verification not implemented)

Time = 33.28 (sec) , antiderivative size = 13164, normalized size of antiderivative = 78.36 \[ \int \frac {\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2 \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

int(((a - c*x^4)*(d + e*x^2)^2*(a*e + c*d*x^2)^2)/(a + b*x^2 + c*x^4)^(7/2 
),x)
 

Output:

(c*d^4*x)/(15*(a + b*x^2 + c*x^4)^(3/2)) + (b^2*d^4*x)/(10*(a + b*x^2 + c* 
x^4)^(5/2)) - (3*b*c*d^4*x^3)/(10*(a + b*x^2 + c*x^4)^(5/2)) + (2*a*d^2*e^ 
2*x)/(15*(a + b*x^2 + c*x^4)^(3/2)) - (b^5*d^4*x^3)/(30*(4*a^3*c - a^2*b^2 
)*(a + b*x^2 + c*x^4)^(3/2)) + (b^5*e^4*x^3)/(30*(4*a*c^3 - b^2*c^2)*(a + 
b*x^2 + c*x^4)^(3/2)) - (b^2*d^4*x)/(30*a*(a + b*x^2 + c*x^4)^(3/2)) + (a^ 
2*e^4*x)/(15*c*(a + b*x^2 + c*x^4)^(3/2)) - (a^3*e^4*x)/(5*c*(a + b*x^2 + 
c*x^4)^(5/2)) - (2*a^2*d^2*e^2*x)/(5*(a + b*x^2 + c*x^4)^(5/2)) - (14*a^4* 
e^4*x)/(15*(a*b^2 - 4*a^2*c)*(a + b*x^2 + c*x^4)^(3/2)) - (b*d^2*e^2*x^3)/ 
(5*(a + b*x^2 + c*x^4)^(3/2)) + (3*b^2*d^3*e*x^3)/(5*(a + b*x^2 + c*x^4)^( 
5/2)) - (a*c*d^4*x)/(5*(a + b*x^2 + c*x^4)^(5/2)) - (2*a^3*e^4*x)/(15*(4*a 
*c - b^2)*(a + b*x^2 + c*x^4)^(3/2)) - (b*d^3*e*x)/(15*(a + b*x^2 + c*x^4) 
^(3/2)) - (b^3*d^4*x^3)/(30*a^2*(a + b*x^2 + c*x^4)^(3/2)) + (b^3*d^4*x^3) 
/(10*a*(a + b*x^2 + c*x^4)^(5/2)) - (b^3*e^4*x^3)/(30*c^2*(a + b*x^2 + c*x 
^4)^(3/2)) - (b^5*d^4*x^3)/(10*(a*b^2 - 4*a^2*c)*(a + b*x^2 + c*x^4)^(5/2) 
) - (2*b^7*d^4*x^3)/(15*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)*(a + b*x^2 + 
c*x^4)^(3/2)) - (2*a*c^2*d^4*x)/(15*(4*a*c - b^2)*(a + b*x^2 + c*x^4)^(3/2 
)) - (2*a^2*b^5*e^4*x^3)/(15*(4*a^3*c^3 - a^2*b^2*c^2)*(a + b*x^2 + c*x^4) 
^(3/2)) - (b^2*c*d^4*x)/(30*(4*a*c - b^2)*(a + b*x^2 + c*x^4)^(3/2)) + (2* 
a*b*d^3*e*x)/(5*(a + b*x^2 + c*x^4)^(5/2)) + (b^3*d^3*e*x)/(15*(4*a*c - b^ 
2)*(a + b*x^2 + c*x^4)^(3/2)) - (b^7*d^2*e^2*x^3)/(30*(4*a^3*c^3 - a^2*...
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a e+c d x^2\right )^2 \left (d+e x^2\right )^2 \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{7/2}} \, dx=\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x \left (15 c^{2} d^{2} e^{2} x^{8}+10 a c d \,e^{3} x^{6}+20 b c \,d^{2} e^{2} x^{6}+10 c^{2} d^{3} e \,x^{6}+3 a^{2} e^{4} x^{4}+4 a b d \,e^{3} x^{4}+36 a c \,d^{2} e^{2} x^{4}+8 b^{2} d^{2} e^{2} x^{4}+4 b c \,d^{3} e \,x^{4}+3 c^{2} d^{4} x^{4}+10 a^{2} d \,e^{3} x^{2}+20 a b \,d^{2} e^{2} x^{2}+10 a c \,d^{3} e \,x^{2}+15 a^{2} d^{2} e^{2}\right )}{15 c^{3} x^{12}+45 b \,c^{2} x^{10}+45 a \,c^{2} x^{8}+45 b^{2} c \,x^{8}+90 a b c \,x^{6}+15 b^{3} x^{6}+45 a^{2} c \,x^{4}+45 a \,b^{2} x^{4}+45 a^{2} b \,x^{2}+15 a^{3}} \] Input:

int((c*d*x^2+a*e)^2*(e*x^2+d)^2*(-c*x^4+a)/(c*x^4+b*x^2+a)^(7/2),x)
 

Output:

(sqrt(a + b*x**2 + c*x**4)*x*(15*a**2*d**2*e**2 + 10*a**2*d*e**3*x**2 + 3* 
a**2*e**4*x**4 + 20*a*b*d**2*e**2*x**2 + 4*a*b*d*e**3*x**4 + 10*a*c*d**3*e 
*x**2 + 36*a*c*d**2*e**2*x**4 + 10*a*c*d*e**3*x**6 + 8*b**2*d**2*e**2*x**4 
 + 4*b*c*d**3*e*x**4 + 20*b*c*d**2*e**2*x**6 + 3*c**2*d**4*x**4 + 10*c**2* 
d**3*e*x**6 + 15*c**2*d**2*e**2*x**8))/(15*(a**3 + 3*a**2*b*x**2 + 3*a**2* 
c*x**4 + 3*a*b**2*x**4 + 6*a*b*c*x**6 + 3*a*c**2*x**8 + b**3*x**6 + 3*b**2 
*c*x**8 + 3*b*c**2*x**10 + c**3*x**12))