\(\int \frac {x^5 (a+b x^2+c x^4)}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\) [17]

Optimal result

Integrand size = 35, antiderivative size = 210 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {d^4 \left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{e^{10}}+\frac {d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) (d-e x)^{3/2} (d+e x)^{3/2}}{3 e^{10}}-\frac {\left (6 c d^4+3 b d^2 e^2+a e^4\right ) (d-e x)^{5/2} (d+e x)^{5/2}}{5 e^{10}}+\frac {\left (4 c d^2+b e^2\right ) (d-e x)^{7/2} (d+e x)^{7/2}}{7 e^{10}}-\frac {c (d-e x)^{9/2} (d+e x)^{9/2}}{9 e^{10}} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.71 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (21 a e^4 \left (8 d^4+4 d^2 e^2 x^2+3 e^4 x^4\right )+9 b \left (16 d^6 e^2+8 d^4 e^4 x^2+6 d^2 e^6 x^4+5 e^8 x^6\right )+c \left (128 d^8+64 d^6 e^2 x^2+48 d^4 e^4 x^4+40 d^2 e^6 x^6+35 e^8 x^8\right )\right )}{315 e^{10}} \] Input:

Integrate[(x^5*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1905, 1578, 1195, 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.

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

\(\Big \downarrow \) 1905

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

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^5*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]
 

Output:

(Sqrt[d^2 - e^2*x^2]*((-2*d^4*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d^2 - e^2*x 
^2])/e^10 + (2*d^2*(4*c*d^4 + 3*b*d^2*e^2 + 2*a*e^4)*(d^2 - e^2*x^2)^(3/2) 
)/(3*e^10) - (2*(6*c*d^4 + 3*b*d^2*e^2 + a*e^4)*(d^2 - e^2*x^2)^(5/2))/(5* 
e^10) + (2*(4*c*d^2 + b*e^2)*(d^2 - e^2*x^2)^(7/2))/(7*e^10) - (2*c*(d^2 - 
 e^2*x^2)^(9/2))/(9*e^10)))/(2*Sqrt[d - e*x]*Sqrt[d + e*x])
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

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]
 

rule 2009

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

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.69

method	result	size
gosper	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (35 c \,x^{8} e^{8}+45 b \,e^{8} x^{6}+40 c \,d^{2} e^{6} x^{6}+63 a \,e^{8} x^{4}+54 b \,d^{2} e^{6} x^{4}+48 c \,d^{4} e^{4} x^{4}+84 a \,d^{2} e^{6} x^{2}+72 b \,d^{4} e^{4} x^{2}+64 c \,d^{6} e^{2} x^{2}+168 a \,d^{4} e^{4}+144 b \,d^{6} e^{2}+128 c \,d^{8}\right )}{315 e^{10}}\)	\(145\)
default	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (35 c \,x^{8} e^{8}+45 b \,e^{8} x^{6}+40 c \,d^{2} e^{6} x^{6}+63 a \,e^{8} x^{4}+54 b \,d^{2} e^{6} x^{4}+48 c \,d^{4} e^{4} x^{4}+84 a \,d^{2} e^{6} x^{2}+72 b \,d^{4} e^{4} x^{2}+64 c \,d^{6} e^{2} x^{2}+168 a \,d^{4} e^{4}+144 b \,d^{6} e^{2}+128 c \,d^{8}\right )}{315 e^{10}}\)	\(145\)
risch	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (35 c \,x^{8} e^{8}+45 b \,e^{8} x^{6}+40 c \,d^{2} e^{6} x^{6}+63 a \,e^{8} x^{4}+54 b \,d^{2} e^{6} x^{4}+48 c \,d^{4} e^{4} x^{4}+84 a \,d^{2} e^{6} x^{2}+72 b \,d^{4} e^{4} x^{2}+64 c \,d^{6} e^{2} x^{2}+168 a \,d^{4} e^{4}+144 b \,d^{6} e^{2}+128 c \,d^{8}\right )}{315 e^{10}}\)	\(145\)
orering	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (35 c \,x^{8} e^{8}+45 b \,e^{8} x^{6}+40 c \,d^{2} e^{6} x^{6}+63 a \,e^{8} x^{4}+54 b \,d^{2} e^{6} x^{4}+48 c \,d^{4} e^{4} x^{4}+84 a \,d^{2} e^{6} x^{2}+72 b \,d^{4} e^{4} x^{2}+64 c \,d^{6} e^{2} x^{2}+168 a \,d^{4} e^{4}+144 b \,d^{6} e^{2}+128 c \,d^{8}\right )}{315 e^{10}}\)	\(145\)

Input:

int(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBO 
SE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.66 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (35 \, c e^{8} x^{8} + 128 \, c d^{8} + 144 \, b d^{6} e^{2} + 168 \, a d^{4} e^{4} + 5 \, {\left (8 \, c d^{2} e^{6} + 9 \, b e^{8}\right )} x^{6} + 3 \, {\left (16 \, c d^{4} e^{4} + 18 \, b d^{2} e^{6} + 21 \, a e^{8}\right )} x^{4} + 4 \, {\left (16 \, c d^{6} e^{2} + 18 \, b d^{4} e^{4} + 21 \, a d^{2} e^{6}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{315 \, e^{10}} \] Input:

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="f 
ricas")
 

Output:

-1/315*(35*c*e^8*x^8 + 128*c*d^8 + 144*b*d^6*e^2 + 168*a*d^4*e^4 + 5*(8*c* 
d^2*e^6 + 9*b*e^8)*x^6 + 3*(16*c*d^4*e^4 + 18*b*d^2*e^6 + 21*a*e^8)*x^4 + 
4*(16*c*d^6*e^2 + 18*b*d^4*e^4 + 21*a*d^2*e^6)*x^2)*sqrt(e*x + d)*sqrt(-e* 
x + d)/e^10
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 21.95 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.75 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=- \frac {i a d^{5} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {9}{4}, - \frac {7}{4} & -2, -2, - \frac {3}{2}, 1 \\- \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, - \frac {3}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {a d^{5} {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {11}{4}, - \frac {5}{2}, - \frac {9}{4}, -2, 1 & \\- \frac {11}{4}, - \frac {9}{4} & -3, - \frac {5}{2}, - \frac {5}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {i b d^{7} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {13}{4}, - \frac {11}{4} & -3, -3, - \frac {5}{2}, 1 \\- \frac {7}{2}, - \frac {13}{4}, -3, - \frac {11}{4}, - \frac {5}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{8}} - \frac {b d^{7} {G_{6, 6}^{2, 6}\left (\begin {matrix} -4, - \frac {15}{4}, - \frac {7}{2}, - \frac {13}{4}, -3, 1 & \\- \frac {15}{4}, - \frac {13}{4} & -4, - \frac {7}{2}, - \frac {7}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{8}} - \frac {i c d^{9} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {17}{4}, - \frac {15}{4} & -4, -4, - \frac {7}{2}, 1 \\- \frac {9}{2}, - \frac {17}{4}, -4, - \frac {15}{4}, - \frac {7}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{10}} - \frac {c d^{9} {G_{6, 6}^{2, 6}\left (\begin {matrix} -5, - \frac {19}{4}, - \frac {9}{2}, - \frac {17}{4}, -4, 1 & \\- \frac {19}{4}, - \frac {17}{4} & -5, - \frac {9}{2}, - \frac {9}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{10}} \] Input:

integrate(x**5*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
 

Output:

-I*a*d**5*meijerg(((-9/4, -7/4), (-2, -2, -3/2, 1)), ((-5/2, -9/4, -2, -7/ 
4, -3/2, 0), ()), d**2/(e**2*x**2))/(4*pi**(3/2)*e**6) - a*d**5*meijerg((( 
-3, -11/4, -5/2, -9/4, -2, 1), ()), ((-11/4, -9/4), (-3, -5/2, -5/2, 0)), 
d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**6) - I*b*d**7*meijerg 
(((-13/4, -11/4), (-3, -3, -5/2, 1)), ((-7/2, -13/4, -3, -11/4, -5/2, 0), 
()), d**2/(e**2*x**2))/(4*pi**(3/2)*e**8) - b*d**7*meijerg(((-4, -15/4, -7 
/2, -13/4, -3, 1), ()), ((-15/4, -13/4), (-4, -7/2, -7/2, 0)), d**2*exp_po 
lar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**8) - I*c*d**9*meijerg(((-17/4, - 
15/4), (-4, -4, -7/2, 1)), ((-9/2, -17/4, -4, -15/4, -7/2, 0), ()), d**2/( 
e**2*x**2))/(4*pi**(3/2)*e**10) - c*d**9*meijerg(((-5, -19/4, -9/2, -17/4, 
 -4, 1), ()), ((-19/4, -17/4), (-5, -9/2, -9/2, 0)), d**2*exp_polar(-2*I*p 
i)/(e**2*x**2))/(4*pi**(3/2)*e**10)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.40 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{8}}{9 \, e^{2}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x^{6}}{63 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x^{6}}{7 \, e^{2}} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{4} x^{4}}{105 \, e^{6}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{2} x^{4}}{35 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a x^{4}}{5 \, e^{2}} - \frac {64 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{6} x^{2}}{315 \, e^{8}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{4} x^{2}}{35 \, e^{6}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} a d^{2} x^{2}}{15 \, e^{4}} - \frac {128 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{8}}{315 \, e^{10}} - \frac {16 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{6}}{35 \, e^{8}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} a d^{4}}{15 \, e^{6}} \] Input:

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="m 
axima")
 

Output:

-1/9*sqrt(-e^2*x^2 + d^2)*c*x^8/e^2 - 8/63*sqrt(-e^2*x^2 + d^2)*c*d^2*x^6/ 
e^4 - 1/7*sqrt(-e^2*x^2 + d^2)*b*x^6/e^2 - 16/105*sqrt(-e^2*x^2 + d^2)*c*d 
^4*x^4/e^6 - 6/35*sqrt(-e^2*x^2 + d^2)*b*d^2*x^4/e^4 - 1/5*sqrt(-e^2*x^2 + 
 d^2)*a*x^4/e^2 - 64/315*sqrt(-e^2*x^2 + d^2)*c*d^6*x^2/e^8 - 8/35*sqrt(-e 
^2*x^2 + d^2)*b*d^4*x^2/e^6 - 4/15*sqrt(-e^2*x^2 + d^2)*a*d^2*x^2/e^4 - 12 
8/315*sqrt(-e^2*x^2 + d^2)*c*d^8/e^10 - 16/35*sqrt(-e^2*x^2 + d^2)*b*d^6/e 
^8 - 8/15*sqrt(-e^2*x^2 + d^2)*a*d^4/e^6
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.09 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (315 \, c d^{8} + 315 \, b d^{6} e^{2} + 315 \, a d^{4} e^{4} - {\left (840 \, c d^{7} + 630 \, b d^{5} e^{2} + 420 \, a d^{3} e^{4} - {\left (1932 \, c d^{6} + 1071 \, b d^{4} e^{2} + 462 \, a d^{2} e^{4} - {\left (2952 \, c d^{5} + 1116 \, b d^{3} e^{2} + 252 \, a d e^{4} - {\left (3098 \, c d^{4} + 729 \, b d^{2} e^{2} + 63 \, a e^{4} - 5 \, {\left (440 \, c d^{3} + 54 \, b d e^{2} - {\left (204 \, c d^{2} + 9 \, b e^{2} + 7 \, {\left ({\left (e x + d\right )} c - 8 \, c d\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{315 \, e^{10}} \] Input:

integrate(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="g 
iac")
 

Output:

-1/315*(315*c*d^8 + 315*b*d^6*e^2 + 315*a*d^4*e^4 - (840*c*d^7 + 630*b*d^5 
*e^2 + 420*a*d^3*e^4 - (1932*c*d^6 + 1071*b*d^4*e^2 + 462*a*d^2*e^4 - (295 
2*c*d^5 + 1116*b*d^3*e^2 + 252*a*d*e^4 - (3098*c*d^4 + 729*b*d^2*e^2 + 63* 
a*e^4 - 5*(440*c*d^3 + 54*b*d*e^2 - (204*c*d^2 + 9*b*e^2 + 7*((e*x + d)*c 
- 8*c*d)*(e*x + d))*(e*x + d))*(e*x + d))*(e*x + d))*(e*x + d))*(e*x + d)) 
*(e*x + d))*sqrt(e*x + d)*sqrt(-e*x + d)/e^10
 

Mupad [B] (verification not implemented)

Time = 4.14 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.37 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e\,x}\,\left (\frac {128\,c\,d^9+144\,b\,d^7\,e^2+168\,a\,d^5\,e^4}{315\,e^{10}}+\frac {x^7\,\left (40\,c\,d^2\,e^7+45\,b\,e^9\right )}{315\,e^{10}}+\frac {x^2\,\left (64\,c\,d^7\,e^2+72\,b\,d^5\,e^4+84\,a\,d^3\,e^6\right )}{315\,e^{10}}+\frac {x^3\,\left (64\,c\,d^6\,e^3+72\,b\,d^4\,e^5+84\,a\,d^2\,e^7\right )}{315\,e^{10}}+\frac {c\,x^9}{9\,e}+\frac {x^5\,\left (48\,c\,d^4\,e^5+54\,b\,d^2\,e^7+63\,a\,e^9\right )}{315\,e^{10}}+\frac {x\,\left (128\,c\,d^8\,e+144\,b\,d^6\,e^3+168\,a\,d^4\,e^5\right )}{315\,e^{10}}+\frac {x^6\,\left (40\,c\,d^3\,e^6+45\,b\,d\,e^8\right )}{315\,e^{10}}+\frac {x^4\,\left (48\,c\,d^5\,e^4+54\,b\,d^3\,e^6+63\,a\,d\,e^8\right )}{315\,e^{10}}+\frac {c\,d\,x^8}{9\,e^2}\right )}{\sqrt {d+e\,x}} \] Input:

int((x^5*(a + b*x^2 + c*x^4))/((d + e*x)^(1/2)*(d - e*x)^(1/2)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.68 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (-35 c \,e^{8} x^{8}-45 b \,e^{8} x^{6}-40 c \,d^{2} e^{6} x^{6}-63 a \,e^{8} x^{4}-54 b \,d^{2} e^{6} x^{4}-48 c \,d^{4} e^{4} x^{4}-84 a \,d^{2} e^{6} x^{2}-72 b \,d^{4} e^{4} x^{2}-64 c \,d^{6} e^{2} x^{2}-168 a \,d^{4} e^{4}-144 b \,d^{6} e^{2}-128 c \,d^{8}\right )}{315 e^{10}} \] Input:

int(x^5*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)
 

Output:

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