\(\int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [22]

Optimal result

Integrand size = 35, antiderivative size = 126 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{4 d^2 x^4}-\frac {\left (4 b d^2+3 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x}}{8 d^4 x^2}-\frac {\left (8 c d^4+4 b d^2 e^2+3 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right )}{8 d^5} \] Output:

-1/4*a*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^2/x^4-1/8*(3*a*e^2+4*b*d^2)*(-e*x+d) 
^(1/2)*(e*x+d)^(1/2)/d^4/x^2-1/8*(3*a*e^4+4*b*d^2*e^2+8*c*d^4)*arctanh((-e 
*x+d)^(1/2)*(e*x+d)^(1/2)/d)/d^5
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\frac {d \sqrt {d-e x} \sqrt {d+e x} \left (2 a d^2+4 b d^2 x^2+3 a e^2 x^2\right )}{x^4}+2 \left (8 c d^4+4 b d^2 e^2+3 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{8 d^5} \] Input:

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

Output:

-1/8*((d*Sqrt[d - e*x]*Sqrt[d + e*x]*(2*a*d^2 + 4*b*d^2*x^2 + 3*a*e^2*x^2) 
)/x^4 + 2*(8*c*d^4 + 4*b*d^2*e^2 + 3*a*e^4)*ArcTanh[Sqrt[d + e*x]/Sqrt[d - 
 e*x]])/d^5
 

Rubi [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.43, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {1905, 1578, 1192, 25, 1471, 25, 298, 219}

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.

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

\(\Big \downarrow \) 1905

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

\(\Big \downarrow \) 1578

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \frac {c x^4+b x^2+a}{x^6 \sqrt {d^2-e^2 x^2}}dx^2}{2 \sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 1192

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1471

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 298

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {\frac {\left (3 a e^4+4 b d^2 e^2+8 c d^4\right ) \int \frac {1}{d^2-x^4}d\sqrt {d^2-e^2 x^2}}{2 d^2}+\frac {e^2 \sqrt {d^2-e^2 x^2} \left (\frac {3 a e^2}{d^2}+4 b\right )}{2 \left (d^2-x^4\right )}}{4 d^2}-\frac {a e^4 \sqrt {d^2-e^2 x^2}}{4 d^2 \left (d^2-x^4\right )^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )}{2 d^3}+\frac {e^2 \sqrt {d^2-e^2 x^2} \left (\frac {3 a e^2}{d^2}+4 b\right )}{2 \left (d^2-x^4\right )}}{4 d^2}-\frac {a e^4 \sqrt {d^2-e^2 x^2}}{4 d^2 \left (d^2-x^4\right )^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

Input:

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

Output:

(Sqrt[d^2 - e^2*x^2]*(-1/4*(a*e^4*Sqrt[d^2 - e^2*x^2])/(d^2*(d^2 - x^4)^2) 
 - ((e^2*(4*b + (3*a*e^2)/d^2)*Sqrt[d^2 - e^2*x^2])/(2*(d^2 - x^4)) + ((8* 
c*d^4 + 4*b*d^2*e^2 + 3*a*e^4)*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^3))/(4 
*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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 298

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

rule 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1471

Int[((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, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

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]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (3 a \,e^{2} x^{2}+4 b \,d^{2} x^{2}+2 a \,d^{2}\right )}{8 d^{4} x^{4}}-\frac {\left (3 a \,e^{4}+4 b \,d^{2} e^{2}+8 c \,d^{4}\right ) \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{8 d^{4} \sqrt {d^{2}}\, \sqrt {e x +d}\, \sqrt {-e x +d}}\)	\(143\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (3 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) a \,e^{4} x^{4}+4 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) b \,d^{2} e^{2} x^{4}+8 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) c \,d^{4} x^{4}+3 \,\operatorname {csgn}\left (d \right ) a d \,e^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+4 \,\operatorname {csgn}\left (d \right ) b \,d^{3} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+2 \,\operatorname {csgn}\left (d \right ) a \,d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\right ) \operatorname {csgn}\left (d \right )}{8 d^{5} \sqrt {-e^{2} x^{2}+d^{2}}\, x^{4}}\)	\(222\)

Input:

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

Output:

-1/8*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(3*a*e^2*x^2+4*b*d^2*x^2+2*a*d^2)/d^4/x^ 
4-1/8/d^4*(3*a*e^4+4*b*d^2*e^2+8*c*d^4)/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2 
)*(-e^2*x^2+d^2)^(1/2))/x)*((e*x+d)*(-e*x+d))^(1/2)/(e*x+d)^(1/2)/(-e*x+d) 
^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {{\left (8 \, c d^{4} + 4 \, b d^{2} e^{2} + 3 \, a e^{4}\right )} x^{4} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (2 \, a d^{3} + {\left (4 \, b d^{3} + 3 \, a d e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{8 \, d^{5} x^{4}} \] Input:

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

Output:

1/8*((8*c*d^4 + 4*b*d^2*e^2 + 3*a*e^4)*x^4*log((sqrt(e*x + d)*sqrt(-e*x + 
d) - d)/x) - (2*a*d^3 + (4*b*d^3 + 3*a*d*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x 
 + d))/(d^5*x^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out} \] Input:

integrate((c*x**4+b*x**2+a)/x**5/(-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 = 193, normalized size of antiderivative = 1.53 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {c \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} - \frac {b e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{3}} - \frac {3 \, a e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{5}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{2 \, d^{2} x^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{8 \, d^{4} x^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{4 \, d^{2} x^{4}} \] Input:

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

Output:

-c*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d - 1/2*b*e^2*log(2 
*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 - 3/8*a*e^4*log(2*d^2/a 
bs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^5 - 1/2*sqrt(-e^2*x^2 + d^2)*b/ 
(d^2*x^2) - 3/8*sqrt(-e^2*x^2 + d^2)*a*e^2/(d^4*x^2) - 1/4*sqrt(-e^2*x^2 + 
 d^2)*a/(d^2*x^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (108) = 216\).

Time = 0.43 (sec) , antiderivative size = 767, normalized size of antiderivative = 6.09 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\frac {{\left (8 \, c d^{4} e + 4 \, b d^{2} e^{3} + 3 \, a e^{5}\right )} \log \left ({\left | -\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} + \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}} + 2 \right |}\right )}{d^{5}} - \frac {{\left (8 \, c d^{4} e + 4 \, b d^{2} e^{3} + 3 \, a e^{5}\right )} \log \left ({\left | -\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} + \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}} - 2 \right |}\right )}{d^{5}} - \frac {4 \, {\left (4 \, b d^{2} e^{3} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{7} + 5 \, a e^{5} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{7} - 16 \, b d^{2} e^{3} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} + 12 \, a e^{5} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} - 64 \, b d^{2} e^{3} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} + 48 \, a e^{5} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} + 256 \, b d^{2} e^{3} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )} + 320 \, a e^{5} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{2} - 4\right )}^{4} d^{5}}}{8 \, e} \] Input:

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

Output:

-1/8*((8*c*d^4*e + 4*b*d^2*e^3 + 3*a*e^5)*log(abs(-(sqrt(2)*sqrt(d) - sqrt 
(-e*x + d))/sqrt(e*x + d) + sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d 
)) + 2))/d^5 - (8*c*d^4*e + 4*b*d^2*e^3 + 3*a*e^5)*log(abs(-(sqrt(2)*sqrt( 
d) - sqrt(-e*x + d))/sqrt(e*x + d) + sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt 
(-e*x + d)) - 2))/d^5 - 4*(4*b*d^2*e^3*((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 + 5*a 
*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sq 
rt(2)*sqrt(d) - sqrt(-e*x + d)))^7 - 16*b*d^2*e^3*((sqrt(2)*sqrt(d) - sqrt 
(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d 
)))^5 + 12*a*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt( 
e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^5 - 64*b*d^2*e^3*((sqrt(2)*sq 
rt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - s 
qrt(-e*x + d)))^3 + 48*a*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x 
+ d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^3 + 256*b*d^2*e^3 
*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2 
)*sqrt(d) - sqrt(-e*x + d))) + 320*a*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d 
))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d))))/(((( 
sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*s 
qrt(d) - sqrt(-e*x + d)))^2 - 4)^4*d^5))/e
 

Mupad [B] (verification not implemented)

Time = 11.17 (sec) , antiderivative size = 932, normalized size of antiderivative = 7.40 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:

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

Output:

((a*e^4)/4 + (6*a*e^4*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^ 
(1/2))^2 - (53*a*e^4*((d + e*x)^(1/2) - d^(1/2))^4)/(2*((d - e*x)^(1/2) - 
d^(1/2))^4) - (87*a*e^4*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) - 
d^(1/2))^6 + (657*a*e^4*((d + e*x)^(1/2) - d^(1/2))^8)/(4*((d - e*x)^(1/2) 
 - d^(1/2))^8) - (121*a*e^4*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/ 
2) - d^(1/2))^10)/((256*d^5*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x)^(1/2 
) - d^(1/2))^4 - (1024*d^5*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) 
 - d^(1/2))^6 + (1536*d^5*((d + e*x)^(1/2) - d^(1/2))^8)/((d - e*x)^(1/2) 
- d^(1/2))^8 - (1024*d^5*((d + e*x)^(1/2) - d^(1/2))^10)/((d - e*x)^(1/2) 
- d^(1/2))^10 + (256*d^5*((d + e*x)^(1/2) - d^(1/2))^12)/((d - e*x)^(1/2) 
- d^(1/2))^12) - ((b*e^2*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - 
 d^(1/2))^2 - (b*e^2)/2 + (15*b*e^2*((d + e*x)^(1/2) - d^(1/2))^4)/(2*((d 
- e*x)^(1/2) - d^(1/2))^4))/((16*d^3*((d + e*x)^(1/2) - d^(1/2))^2)/((d - 
e*x)^(1/2) - d^(1/2))^2 - (32*d^3*((d + e*x)^(1/2) - d^(1/2))^4)/((d - e*x 
)^(1/2) - d^(1/2))^4 + (16*d^3*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^( 
1/2) - d^(1/2))^6) + (c*(log(((d + e*x)^(1/2) - d^(1/2))^2/((d - e*x)^(1/2 
) - d^(1/2))^2 - 1) - log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - d 
^(1/2)))))/d - (3*a*e^4*log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - 
 d^(1/2))))/(8*d^5) - (b*e^2*log(((d + e*x)^(1/2) - d^(1/2))/((d - e*x)^(1 
/2) - d^(1/2))))/(2*d^3) + (3*a*e^4*log(((d + e*x)^(1/2) - d^(1/2))^2/(...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 505, normalized size of antiderivative = 4.01 \[ \int \frac {a+b x^2+c x^4}{x^5 \sqrt {d-e x} \sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(d + e*x)*sqrt(d - e*x)*a*d**3 - 3*sqrt(d + e*x)*sqrt(d - e*x)*a 
*d*e**2*x**2 - 4*sqrt(d + e*x)*sqrt(d - e*x)*b*d**3*x**2 - 3*log( - sqrt(2 
) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*a*e**4*x**4 - 4*log( 
 - sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*b*d**2*e**2 
*x**4 - 8*log( - sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 
1)*c*d**4*x**4 + 3*log( - sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2 
)))/2) + 1)*a*e**4*x**4 + 4*log( - sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt( 
d)*sqrt(2)))/2) + 1)*b*d**2*e**2*x**4 + 8*log( - sqrt(2) + tan(asin(sqrt(d 
 - e*x)/(sqrt(d)*sqrt(2)))/2) + 1)*c*d**4*x**4 - 3*log(sqrt(2) + tan(asin( 
sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*a*e**4*x**4 - 4*log(sqrt(2) + tan 
(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*b*d**2*e**2*x**4 - 8*log(sq 
rt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*c*d**4*x**4 + 3* 
log(sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) + 1)*a*e**4*x** 
4 + 4*log(sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) + 1)*b*d* 
*2*e**2*x**4 + 8*log(sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2 
) + 1)*c*d**4*x**4)/(8*d**5*x**4)