\(\int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [29]

Optimal result

Integrand size = 35, antiderivative size = 178 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{7 d^2 x^7}-\frac {\left (7 b d^2+6 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x}}{35 d^4 x^5}-\frac {\left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{105 d^6 x^3}-\frac {2 e^2 \left (35 c d^4+28 b d^2 e^2+24 a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{105 d^8 x} \] Output:

-1/7*a*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^2/x^7-1/35*(6*a*e^2+7*b*d^2)*(-e*x+d 
)^(1/2)*(e*x+d)^(1/2)/d^4/x^5-1/105*(24*a*e^4+28*b*d^2*e^2+35*c*d^4)*(-e*x 
+d)^(1/2)*(e*x+d)^(1/2)/d^6/x^3-2/105*e^2*(24*a*e^4+28*b*d^2*e^2+35*c*d^4) 
*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^8/x
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.70 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (35 c d^4 x^4 \left (d^2+2 e^2 x^2\right )+7 b \left (3 d^6 x^2+4 d^4 e^2 x^4+8 d^2 e^4 x^6\right )+3 a \left (5 d^6+6 d^4 e^2 x^2+8 d^2 e^4 x^4+16 e^6 x^6\right )\right )}{105 d^8 x^7} \] Input:

Integrate[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]
 

Output:

-1/105*(Sqrt[d - e*x]*Sqrt[d + e*x]*(35*c*d^4*x^4*(d^2 + 2*e^2*x^2) + 7*b* 
(3*d^6*x^2 + 4*d^4*e^2*x^4 + 8*d^2*e^4*x^6) + 3*a*(5*d^6 + 6*d^4*e^2*x^2 + 
 8*d^2*e^4*x^4 + 16*e^6*x^6)))/(d^8*x^7)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1905, 1588, 25, 359, 245, 242}

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.

\(\displaystyle \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx\)

\(\Big \downarrow \) 1905

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \frac {c x^4+b x^2+a}{x^8 \sqrt {d^2-e^2 x^2}}dx}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 1588

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 359

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {\left (24 a e^4+28 b d^2 e^2+35 c d^4\right ) \int \frac {1}{x^4 \sqrt {d^2-e^2 x^2}}dx}{5 d^2}-\frac {\sqrt {d^2-e^2 x^2} \left (\frac {6 a e^2}{d^2}+7 b\right )}{5 x^5}}{7 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{7 d^2 x^7}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {\left (24 a e^4+28 b d^2 e^2+35 c d^4\right ) \left (\frac {2 e^2 \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx}{3 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^2 x^3}\right )}{5 d^2}-\frac {\sqrt {d^2-e^2 x^2} \left (\frac {6 a e^2}{d^2}+7 b\right )}{5 x^5}}{7 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{7 d^2 x^7}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {\left (-\frac {\sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^4 x}\right ) \left (24 a e^4+28 b d^2 e^2+35 c d^4\right )}{5 d^2}-\frac {\sqrt {d^2-e^2 x^2} \left (\frac {6 a e^2}{d^2}+7 b\right )}{5 x^5}}{7 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{7 d^2 x^7}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

Input:

Int[(a + b*x^2 + c*x^4)/(x^8*Sqrt[d - e*x]*Sqrt[d + e*x]),x]
 

Output:

(Sqrt[d^2 - e^2*x^2]*(-1/7*(a*Sqrt[d^2 - e^2*x^2])/(d^2*x^7) + (-1/5*((7*b 
 + (6*a*e^2)/d^2)*Sqrt[d^2 - e^2*x^2])/x^5 + ((35*c*d^4 + 28*b*d^2*e^2 + 2 
4*a*e^4)*(-1/3*Sqrt[d^2 - e^2*x^2]/(d^2*x^3) - (2*e^2*Sqrt[d^2 - e^2*x^2]) 
/(3*d^4*x)))/(5*d^2))/(7*d^2)))/(Sqrt[d - e*x]*Sqrt[d + e*x])
 

Defintions of rubi rules used

rule 25

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

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 245

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + 
 b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) 
   Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si 
mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 1588

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c 
_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + 
c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, 
 Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f 
^2*(m + 1))   Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x 
) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne 
Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
 

rule 1905

Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) 
*(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x 
_Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ 
q]/(d1*d2 + e1*e2*x^n)^FracPart[q])   Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, 
q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (48 a \,e^{6} x^{6}+56 b \,d^{2} e^{4} x^{6}+70 c \,d^{4} e^{2} x^{6}+24 a \,d^{2} e^{4} x^{4}+28 b \,d^{4} e^{2} x^{4}+35 c \,d^{6} x^{4}+18 a \,d^{4} e^{2} x^{2}+21 b \,d^{6} x^{2}+15 a \,d^{6}\right )}{105 x^{7} d^{8}}\)	\(118\)
risch	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (48 a \,e^{6} x^{6}+56 b \,d^{2} e^{4} x^{6}+70 c \,d^{4} e^{2} x^{6}+24 a \,d^{2} e^{4} x^{4}+28 b \,d^{4} e^{2} x^{4}+35 c \,d^{6} x^{4}+18 a \,d^{4} e^{2} x^{2}+21 b \,d^{6} x^{2}+15 a \,d^{6}\right )}{105 x^{7} d^{8}}\)	\(118\)
orering	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (48 a \,e^{6} x^{6}+56 b \,d^{2} e^{4} x^{6}+70 c \,d^{4} e^{2} x^{6}+24 a \,d^{2} e^{4} x^{4}+28 b \,d^{4} e^{2} x^{4}+35 c \,d^{6} x^{4}+18 a \,d^{4} e^{2} x^{2}+21 b \,d^{6} x^{2}+15 a \,d^{6}\right )}{105 x^{7} d^{8}}\)	\(118\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \operatorname {csgn}\left (e \right )^{2} \left (48 a \,e^{6} x^{6}+56 b \,d^{2} e^{4} x^{6}+70 c \,d^{4} e^{2} x^{6}+24 a \,d^{2} e^{4} x^{4}+28 b \,d^{4} e^{2} x^{4}+35 c \,d^{6} x^{4}+18 a \,d^{4} e^{2} x^{2}+21 b \,d^{6} x^{2}+15 a \,d^{6}\right )}{105 d^{8} x^{7}}\)	\(122\)

Input:

int((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBO 
SE)
 

Output:

-1/105*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(48*a*e^6*x^6+56*b*d^2*e^4*x^6+70*c*d^ 
4*e^2*x^6+24*a*d^2*e^4*x^4+28*b*d^4*e^2*x^4+35*c*d^6*x^4+18*a*d^4*e^2*x^2+ 
21*b*d^6*x^2+15*a*d^6)/x^7/d^8
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.62 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (15 \, a d^{6} + 2 \, {\left (35 \, c d^{4} e^{2} + 28 \, b d^{2} e^{4} + 24 \, a e^{6}\right )} x^{6} + {\left (35 \, c d^{6} + 28 \, b d^{4} e^{2} + 24 \, a d^{2} e^{4}\right )} x^{4} + 3 \, {\left (7 \, b d^{6} + 6 \, a d^{4} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{105 \, d^{8} x^{7}} \] Input:

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="f 
ricas")
 

Output:

-1/105*(15*a*d^6 + 2*(35*c*d^4*e^2 + 28*b*d^2*e^4 + 24*a*e^6)*x^6 + (35*c* 
d^6 + 28*b*d^4*e^2 + 24*a*d^2*e^4)*x^4 + 3*(7*b*d^6 + 6*a*d^4*e^2)*x^2)*sq 
rt(e*x + d)*sqrt(-e*x + d)/(d^8*x^7)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out} \] Input:

integrate((c*x**4+b*x**2+a)/x**8/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.27 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} c e^{2}}{3 \, d^{4} x} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{4}}{15 \, d^{6} x} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{6}}{35 \, d^{8} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{3 \, d^{2} x^{3}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{15 \, d^{4} x^{3}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{35 \, d^{6} x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{5 \, d^{2} x^{5}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{35 \, d^{4} x^{5}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{7 \, d^{2} x^{7}} \] Input:

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="m 
axima")
 

Output:

-2/3*sqrt(-e^2*x^2 + d^2)*c*e^2/(d^4*x) - 8/15*sqrt(-e^2*x^2 + d^2)*b*e^4/ 
(d^6*x) - 16/35*sqrt(-e^2*x^2 + d^2)*a*e^6/(d^8*x) - 1/3*sqrt(-e^2*x^2 + d 
^2)*c/(d^2*x^3) - 4/15*sqrt(-e^2*x^2 + d^2)*b*e^2/(d^4*x^3) - 8/35*sqrt(-e 
^2*x^2 + d^2)*a*e^4/(d^6*x^3) - 1/5*sqrt(-e^2*x^2 + d^2)*b/(d^2*x^5) - 6/3 
5*sqrt(-e^2*x^2 + d^2)*a*e^2/(d^4*x^5) - 1/7*sqrt(-e^2*x^2 + d^2)*a/(d^2*x 
^7)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1451 vs. \(2 (154) = 308\).

Time = 0.49 (sec) , antiderivative size = 1451, normalized size of antiderivative = 8.15 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="g 
iac")
 

Output:

-4/105*(105*c*d^4*e^4*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - 
sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^13 + 105*b*d^2*e^6*((sqr 
t(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt 
(d) - sqrt(-e*x + d)))^13 + 105*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/ 
sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^13 - 196 
0*c*d^4*e^4*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + 
 d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^11 - 1400*b*d^2*e^6*((sqrt(2)*sqrt 
(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqr 
t(-e*x + d)))^11 - 840*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x 
+ d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^11 + 16240*c*d^4* 
e^4*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqr 
t(2)*sqrt(d) - sqrt(-e*x + d)))^9 + 12656*b*d^2*e^6*((sqrt(2)*sqrt(d) - sq 
rt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + 
 d)))^9 + 14448*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - 
sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^9 - 80640*c*d^4*e^4*((sq 
rt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqr 
t(d) - sqrt(-e*x + d)))^7 - 69888*b*d^2*e^6*((sqrt(2)*sqrt(d) - sqrt(-e*x 
+ d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^7 
- 40704*a*e^8*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x 
 + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^7 + 259840*c*d^4*e^4*((sqrt(2...
 

Mupad [B] (verification not implemented)

Time = 4.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.22 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e\,x}\,\left (\frac {a}{7\,d}+\frac {x^2\,\left (21\,b\,d^7+18\,a\,d^5\,e^2\right )}{105\,d^8}+\frac {x^4\,\left (35\,c\,d^7+28\,b\,d^5\,e^2+24\,a\,d^3\,e^4\right )}{105\,d^8}+\frac {x^7\,\left (70\,c\,d^4\,e^3+56\,b\,d^2\,e^5+48\,a\,e^7\right )}{105\,d^8}+\frac {x^3\,\left (21\,b\,d^6\,e+18\,a\,d^4\,e^3\right )}{105\,d^8}+\frac {x^5\,\left (35\,c\,d^6\,e+28\,b\,d^4\,e^3+24\,a\,d^2\,e^5\right )}{105\,d^8}+\frac {x^6\,\left (70\,c\,d^5\,e^2+56\,b\,d^3\,e^4+48\,a\,d\,e^6\right )}{105\,d^8}+\frac {a\,e\,x}{7\,d^2}\right )}{x^7\,\sqrt {d+e\,x}} \] Input:

int((a + b*x^2 + c*x^4)/(x^8*(d + e*x)^(1/2)*(d - e*x)^(1/2)),x)
 

Output:

-((d - e*x)^(1/2)*(a/(7*d) + (x^2*(21*b*d^7 + 18*a*d^5*e^2))/(105*d^8) + ( 
x^4*(35*c*d^7 + 24*a*d^3*e^4 + 28*b*d^5*e^2))/(105*d^8) + (x^7*(48*a*e^7 + 
 56*b*d^2*e^5 + 70*c*d^4*e^3))/(105*d^8) + (x^3*(18*a*d^4*e^3 + 21*b*d^6*e 
))/(105*d^8) + (x^5*(24*a*d^2*e^5 + 28*b*d^4*e^3 + 35*c*d^6*e))/(105*d^8) 
+ (x^6*(56*b*d^3*e^4 + 70*c*d^5*e^2 + 48*a*d*e^6))/(105*d^8) + (a*e*x)/(7* 
d^2)))/(x^7*(d + e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x^2+c x^4}{x^8 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (-48 a \,e^{6} x^{6}-56 b \,d^{2} e^{4} x^{6}-70 c \,d^{4} e^{2} x^{6}-24 a \,d^{2} e^{4} x^{4}-28 b \,d^{4} e^{2} x^{4}-35 c \,d^{6} x^{4}-18 a \,d^{4} e^{2} x^{2}-21 b \,d^{6} x^{2}-15 a \,d^{6}\right )}{105 d^{8} x^{7}} \] Input:

int((c*x^4+b*x^2+a)/x^8/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)
 

Output:

(sqrt(d + e*x)*sqrt(d - e*x)*( - 15*a*d**6 - 18*a*d**4*e**2*x**2 - 24*a*d* 
*2*e**4*x**4 - 48*a*e**6*x**6 - 21*b*d**6*x**2 - 28*b*d**4*e**2*x**4 - 56* 
b*d**2*e**4*x**6 - 35*c*d**6*x**4 - 70*c*d**4*e**2*x**6))/(105*d**8*x**7)