\(\int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [30]

Optimal result

Integrand size = 35, antiderivative size = 232 \[ \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{9 d^2 x^9}-\frac {\left (9 b d^2+8 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x}}{63 d^4 x^7}-\frac {\left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{105 d^6 x^5}-\frac {4 e^2 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{315 d^8 x^3}-\frac {8 e^4 \left (21 c d^4+18 b d^2 e^2+16 a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{315 d^{10} x} \] Output:

-1/9*a*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^2/x^9-1/63*(8*a*e^2+9*b*d^2)*(-e*x+d 
)^(1/2)*(e*x+d)^(1/2)/d^4/x^7-1/105*(16*a*e^4+18*b*d^2*e^2+21*c*d^4)*(-e*x 
+d)^(1/2)*(e*x+d)^(1/2)/d^6/x^5-4/315*e^2*(16*a*e^4+18*b*d^2*e^2+21*c*d^4) 
*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^8/x^3-8/315*e^4*(16*a*e^4+18*b*d^2*e^2+21* 
c*d^4)*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^10/x
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.68 \[ \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (21 c d^4 x^4 \left (3 d^4+4 d^2 e^2 x^2+8 e^4 x^4\right )+9 b \left (5 d^8 x^2+6 d^6 e^2 x^4+8 d^4 e^4 x^6+16 d^2 e^6 x^8\right )+a \left (35 d^8+40 d^6 e^2 x^2+48 d^4 e^4 x^4+64 d^2 e^6 x^6+128 e^8 x^8\right )\right )}{315 d^{10} x^9} \] Input:

Integrate[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]
 

Output:

-1/315*(Sqrt[d - e*x]*Sqrt[d + e*x]*(21*c*d^4*x^4*(3*d^4 + 4*d^2*e^2*x^2 + 
 8*e^4*x^4) + 9*b*(5*d^8*x^2 + 6*d^6*e^2*x^4 + 8*d^4*e^4*x^6 + 16*d^2*e^6* 
x^8) + a*(35*d^8 + 40*d^6*e^2*x^2 + 48*d^4*e^4*x^4 + 64*d^2*e^6*x^6 + 128* 
e^8*x^8)))/(d^10*x^9)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1905, 1588, 25, 359, 245, 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.

\(\displaystyle \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx\)

\(\Big \downarrow \) 1905

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \frac {c x^4+b x^2+a}{x^{10} \sqrt {d^2-e^2 x^2}}dx}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 1588

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {\int -\frac {9 c x^2 d^2+9 b d^2+8 a e^2}{x^8 \sqrt {d^2-e^2 x^2}}dx}{9 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{9 d^2 x^9}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\int \frac {9 c x^2 d^2+9 b d^2+8 a e^2}{x^8 \sqrt {d^2-e^2 x^2}}dx}{9 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{9 d^2 x^9}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 359

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (16 a e^4+18 b d^2 e^2+21 c d^4\right ) \int \frac {1}{x^6 \sqrt {d^2-e^2 x^2}}dx}{7 d^2}-\frac {\sqrt {d^2-e^2 x^2} \left (\frac {8 a e^2}{d^2}+9 b\right )}{7 x^7}}{9 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{9 d^2 x^9}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (16 a e^4+18 b d^2 e^2+21 c d^4\right ) \left (\frac {4 e^2 \int \frac {1}{x^4 \sqrt {d^2-e^2 x^2}}dx}{5 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d^2 x^5}\right )}{7 d^2}-\frac {\sqrt {d^2-e^2 x^2} \left (\frac {8 a e^2}{d^2}+9 b\right )}{7 x^7}}{9 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{9 d^2 x^9}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (16 a e^4+18 b d^2 e^2+21 c d^4\right ) \left (\frac {4 e^2 \left (\frac {2 e^2 \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx}{3 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^2 x^3}\right )}{5 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d^2 x^5}\right )}{7 d^2}-\frac {\sqrt {d^2-e^2 x^2} \left (\frac {8 a e^2}{d^2}+9 b\right )}{7 x^7}}{9 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{9 d^2 x^9}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (\frac {4 e^2 \left (-\frac {\sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^4 x}\right )}{5 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{5 d^2 x^5}\right ) \left (16 a e^4+18 b d^2 e^2+21 c d^4\right )}{7 d^2}-\frac {\sqrt {d^2-e^2 x^2} \left (\frac {8 a e^2}{d^2}+9 b\right )}{7 x^7}}{9 d^2}-\frac {a \sqrt {d^2-e^2 x^2}}{9 d^2 x^9}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

Input:

Int[(a + b*x^2 + c*x^4)/(x^10*Sqrt[d - e*x]*Sqrt[d + e*x]),x]
 

Output:

(Sqrt[d^2 - e^2*x^2]*(-1/9*(a*Sqrt[d^2 - e^2*x^2])/(d^2*x^9) + (-1/7*((9*b 
 + (8*a*e^2)/d^2)*Sqrt[d^2 - e^2*x^2])/x^7 + (3*(21*c*d^4 + 18*b*d^2*e^2 + 
 16*a*e^4)*(-1/5*Sqrt[d^2 - e^2*x^2]/(d^2*x^5) + (4*e^2*(-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))/(9*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 = 1.01 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (128 a \,e^{8} x^{8}+144 b \,d^{2} e^{6} x^{8}+168 c \,d^{4} e^{4} x^{8}+64 a \,d^{2} e^{6} x^{6}+72 b \,d^{4} e^{4} x^{6}+84 c \,d^{6} e^{2} x^{6}+48 a \,d^{4} e^{4} x^{4}+54 b \,d^{6} e^{2} x^{4}+63 c \,d^{8} x^{4}+40 a \,d^{6} e^{2} x^{2}+45 b \,d^{8} x^{2}+35 a \,d^{8}\right )}{315 x^{9} d^{10}}\)	\(154\)
risch	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (128 a \,e^{8} x^{8}+144 b \,d^{2} e^{6} x^{8}+168 c \,d^{4} e^{4} x^{8}+64 a \,d^{2} e^{6} x^{6}+72 b \,d^{4} e^{4} x^{6}+84 c \,d^{6} e^{2} x^{6}+48 a \,d^{4} e^{4} x^{4}+54 b \,d^{6} e^{2} x^{4}+63 c \,d^{8} x^{4}+40 a \,d^{6} e^{2} x^{2}+45 b \,d^{8} x^{2}+35 a \,d^{8}\right )}{315 x^{9} d^{10}}\)	\(154\)
orering	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (128 a \,e^{8} x^{8}+144 b \,d^{2} e^{6} x^{8}+168 c \,d^{4} e^{4} x^{8}+64 a \,d^{2} e^{6} x^{6}+72 b \,d^{4} e^{4} x^{6}+84 c \,d^{6} e^{2} x^{6}+48 a \,d^{4} e^{4} x^{4}+54 b \,d^{6} e^{2} x^{4}+63 c \,d^{8} x^{4}+40 a \,d^{6} e^{2} x^{2}+45 b \,d^{8} x^{2}+35 a \,d^{8}\right )}{315 x^{9} d^{10}}\)	\(154\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \operatorname {csgn}\left (e \right )^{2} \left (128 a \,e^{8} x^{8}+144 b \,d^{2} e^{6} x^{8}+168 c \,d^{4} e^{4} x^{8}+64 a \,d^{2} e^{6} x^{6}+72 b \,d^{4} e^{4} x^{6}+84 c \,d^{6} e^{2} x^{6}+48 a \,d^{4} e^{4} x^{4}+54 b \,d^{6} e^{2} x^{4}+63 c \,d^{8} x^{4}+40 a \,d^{6} e^{2} x^{2}+45 b \,d^{8} x^{2}+35 a \,d^{8}\right )}{315 d^{10} x^{9}}\)	\(158\)

Input:

int((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

-1/315*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(128*a*e^8*x^8+144*b*d^2*e^6*x^8+168*c 
*d^4*e^4*x^8+64*a*d^2*e^6*x^6+72*b*d^4*e^4*x^6+84*c*d^6*e^2*x^6+48*a*d^4*e 
^4*x^4+54*b*d^6*e^2*x^4+63*c*d^8*x^4+40*a*d^6*e^2*x^2+45*b*d^8*x^2+35*a*d^ 
8)/x^9/d^10
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.62 \[ \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (35 \, a d^{8} + 8 \, {\left (21 \, c d^{4} e^{4} + 18 \, b d^{2} e^{6} + 16 \, a e^{8}\right )} x^{8} + 4 \, {\left (21 \, c d^{6} e^{2} + 18 \, b d^{4} e^{4} + 16 \, a d^{2} e^{6}\right )} x^{6} + 3 \, {\left (21 \, c d^{8} + 18 \, b d^{6} e^{2} + 16 \, a d^{4} e^{4}\right )} x^{4} + 5 \, {\left (9 \, b d^{8} + 8 \, a d^{6} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{315 \, d^{10} x^{9}} \] Input:

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm=" 
fricas")
 

Output:

-1/315*(35*a*d^8 + 8*(21*c*d^4*e^4 + 18*b*d^2*e^6 + 16*a*e^8)*x^8 + 4*(21* 
c*d^6*e^2 + 18*b*d^4*e^4 + 16*a*d^2*e^6)*x^6 + 3*(21*c*d^8 + 18*b*d^6*e^2 
+ 16*a*d^4*e^4)*x^4 + 5*(9*b*d^8 + 8*a*d^6*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e 
*x + d)/(d^10*x^9)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out} \] Input:

integrate((c*x**4+b*x**2+a)/x**10/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.31 \[ \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} c e^{4}}{15 \, d^{6} x} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{6}}{35 \, d^{8} x} - \frac {128 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{8}}{315 \, d^{10} x} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} c e^{2}}{15 \, d^{4} x^{3}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{4}}{35 \, d^{6} x^{3}} - \frac {64 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{6}}{315 \, d^{8} x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{5 \, d^{2} x^{5}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{35 \, d^{4} x^{5}} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{105 \, d^{6} x^{5}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{7 \, d^{2} x^{7}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{63 \, d^{4} x^{7}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{9 \, d^{2} x^{9}} \] Input:

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm=" 
maxima")
 

Output:

-8/15*sqrt(-e^2*x^2 + d^2)*c*e^4/(d^6*x) - 16/35*sqrt(-e^2*x^2 + d^2)*b*e^ 
6/(d^8*x) - 128/315*sqrt(-e^2*x^2 + d^2)*a*e^8/(d^10*x) - 4/15*sqrt(-e^2*x 
^2 + d^2)*c*e^2/(d^4*x^3) - 8/35*sqrt(-e^2*x^2 + d^2)*b*e^4/(d^6*x^3) - 64 
/315*sqrt(-e^2*x^2 + d^2)*a*e^6/(d^8*x^3) - 1/5*sqrt(-e^2*x^2 + d^2)*c/(d^ 
2*x^5) - 6/35*sqrt(-e^2*x^2 + d^2)*b*e^2/(d^4*x^5) - 16/105*sqrt(-e^2*x^2 
+ d^2)*a*e^4/(d^6*x^5) - 1/7*sqrt(-e^2*x^2 + d^2)*b/(d^2*x^7) - 8/63*sqrt( 
-e^2*x^2 + d^2)*a*e^2/(d^4*x^7) - 1/9*sqrt(-e^2*x^2 + d^2)*a/(d^2*x^9)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1847 vs. \(2 (202) = 404\).

Time = 0.72 (sec) , antiderivative size = 1847, normalized size of antiderivative = 7.96 \[ \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm=" 
giac")
 

Output:

-4/315*(315*c*d^4*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)))^17 + 315*b*d^2*e^8*((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)))^17 + 315*a*e^10*((sqrt(2)*sqrt(d) - sqrt(-e*x + d)) 
/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^17 - 67 
20*c*d^4*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)))^15 - 5040*b*d^2*e^8*((sqrt(2)*sqr 
t(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sq 
rt(-e*x + d)))^15 - 3360*a*e^10*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e 
*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^15 + 76608*c*d 
^4*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)))^13 + 68544*b*d^2*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 + 76608*a*e^10*((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 - 580608*c*d^4* 
e^6*((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)))^11 - 509184*b*d^2*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 - 327168*a*e^10*((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 + 2892288*c*d...
 

Mupad [B] (verification not implemented)

Time = 4.66 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.25 \[ \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e\,x}\,\left (\frac {a}{9\,d}+\frac {x^2\,\left (45\,b\,d^9+40\,a\,d^7\,e^2\right )}{315\,d^{10}}+\frac {x^6\,\left (84\,c\,d^7\,e^2+72\,b\,d^5\,e^4+64\,a\,d^3\,e^6\right )}{315\,d^{10}}+\frac {x^7\,\left (84\,c\,d^6\,e^3+72\,b\,d^4\,e^5+64\,a\,d^2\,e^7\right )}{315\,d^{10}}+\frac {x^4\,\left (63\,c\,d^9+54\,b\,d^7\,e^2+48\,a\,d^5\,e^4\right )}{315\,d^{10}}+\frac {x^9\,\left (168\,c\,d^4\,e^5+144\,b\,d^2\,e^7+128\,a\,e^9\right )}{315\,d^{10}}+\frac {x^3\,\left (45\,b\,d^8\,e+40\,a\,d^6\,e^3\right )}{315\,d^{10}}+\frac {x^5\,\left (63\,c\,d^8\,e+54\,b\,d^6\,e^3+48\,a\,d^4\,e^5\right )}{315\,d^{10}}+\frac {x^8\,\left (168\,c\,d^5\,e^4+144\,b\,d^3\,e^6+128\,a\,d\,e^8\right )}{315\,d^{10}}+\frac {a\,e\,x}{9\,d^2}\right )}{x^9\,\sqrt {d+e\,x}} \] Input:

int((a + b*x^2 + c*x^4)/(x^10*(d + e*x)^(1/2)*(d - e*x)^(1/2)),x)
 

Output:

-((d - e*x)^(1/2)*(a/(9*d) + (x^2*(45*b*d^9 + 40*a*d^7*e^2))/(315*d^10) + 
(x^6*(64*a*d^3*e^6 + 72*b*d^5*e^4 + 84*c*d^7*e^2))/(315*d^10) + (x^7*(64*a 
*d^2*e^7 + 72*b*d^4*e^5 + 84*c*d^6*e^3))/(315*d^10) + (x^4*(63*c*d^9 + 48* 
a*d^5*e^4 + 54*b*d^7*e^2))/(315*d^10) + (x^9*(128*a*e^9 + 144*b*d^2*e^7 + 
168*c*d^4*e^5))/(315*d^10) + (x^3*(40*a*d^6*e^3 + 45*b*d^8*e))/(315*d^10) 
+ (x^5*(48*a*d^4*e^5 + 54*b*d^6*e^3 + 63*c*d^8*e))/(315*d^10) + (x^8*(144* 
b*d^3*e^6 + 168*c*d^5*e^4 + 128*a*d*e^8))/(315*d^10) + (a*e*x)/(9*d^2)))/( 
x^9*(d + e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x^2+c x^4}{x^{10} \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (-128 a \,e^{8} x^{8}-144 b \,d^{2} e^{6} x^{8}-168 c \,d^{4} e^{4} x^{8}-64 a \,d^{2} e^{6} x^{6}-72 b \,d^{4} e^{4} x^{6}-84 c \,d^{6} e^{2} x^{6}-48 a \,d^{4} e^{4} x^{4}-54 b \,d^{6} e^{2} x^{4}-63 c \,d^{8} x^{4}-40 a \,d^{6} e^{2} x^{2}-45 b \,d^{8} x^{2}-35 a \,d^{8}\right )}{315 d^{10} x^{9}} \] Input:

int((c*x^4+b*x^2+a)/x^10/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)
 

Output:

(sqrt(d + e*x)*sqrt(d - e*x)*( - 35*a*d**8 - 40*a*d**6*e**2*x**2 - 48*a*d* 
*4*e**4*x**4 - 64*a*d**2*e**6*x**6 - 128*a*e**8*x**8 - 45*b*d**8*x**2 - 54 
*b*d**6*e**2*x**4 - 72*b*d**4*e**4*x**6 - 144*b*d**2*e**6*x**8 - 63*c*d**8 
*x**4 - 84*c*d**6*e**2*x**6 - 168*c*d**4*e**4*x**8))/(315*d**10*x**9)