\(\int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [23]

Optimal result

Integrand size = 35, antiderivative size = 180 \[ \int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{6 d^2 x^6}-\frac {\left (6 b d^2+5 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x}}{24 d^4 x^4}-\frac {\left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{16 d^6 x^2}-\frac {e^2 \left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right )}{16 d^7} \] Output:

-1/6*a*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^2/x^6-1/24*(5*a*e^2+6*b*d^2)*(-e*x+d 
)^(1/2)*(e*x+d)^(1/2)/d^4/x^4-1/16*(5*a*e^4+6*b*d^2*e^2+8*c*d^4)*(-e*x+d)^ 
(1/2)*(e*x+d)^(1/2)/d^6/x^2-1/16*e^2*(5*a*e^4+6*b*d^2*e^2+8*c*d^4)*arctanh 
((-e*x+d)^(1/2)*(e*x+d)^(1/2)/d)/d^7
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\frac {d \sqrt {d-e x} \sqrt {d+e x} \left (6 \left (2 b d^4 x^2+4 c d^4 x^4+3 b d^2 e^2 x^4\right )+a \left (8 d^4+10 d^2 e^2 x^2+15 e^4 x^4\right )\right )}{x^6}+6 e^2 \left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{48 d^7} \] Input:

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

Output:

-1/48*((d*Sqrt[d - e*x]*Sqrt[d + e*x]*(6*(2*b*d^4*x^2 + 4*c*d^4*x^4 + 3*b* 
d^2*e^2*x^4) + a*(8*d^4 + 10*d^2*e^2*x^2 + 15*e^4*x^4)))/x^6 + 6*e^2*(8*c* 
d^4 + 6*b*d^2*e^2 + 5*a*e^4)*ArcTanh[Sqrt[d + e*x]/Sqrt[d - e*x]])/d^7
 

Rubi [A] (warning: unable to verify)

Time = 0.43 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.26, 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, 1471, 25, 298, 215, 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.

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

\(\Big \downarrow \) 1905

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

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1192

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

\(\Big \downarrow \) 1471

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 298

\(\displaystyle -\frac {e^2 \sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (5 a e^4+6 b d^2 e^2+8 c d^4\right ) \int \frac {1}{\left (d^2-x^4\right )^2}d\sqrt {d^2-e^2 x^2}}{4 d^2}+\frac {e^2 \sqrt {d^2-e^2 x^2} \left (\frac {5 a e^2}{d^2}+6 b\right )}{4 \left (d^2-x^4\right )^2}}{6 d^2}+\frac {a e^4 \sqrt {d^2-e^2 x^2}}{6 d^2 \left (d^2-x^4\right )^3}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 215

\(\displaystyle -\frac {e^2 \sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (5 a e^4+6 b d^2 e^2+8 c d^4\right ) \left (\frac {\int \frac {1}{d^2-x^4}d\sqrt {d^2-e^2 x^2}}{2 d^2}+\frac {\sqrt {d^2-e^2 x^2}}{2 d^2 \left (d^2-x^4\right )}\right )}{4 d^2}+\frac {e^2 \sqrt {d^2-e^2 x^2} \left (\frac {5 a e^2}{d^2}+6 b\right )}{4 \left (d^2-x^4\right )^2}}{6 d^2}+\frac {a e^4 \sqrt {d^2-e^2 x^2}}{6 d^2 \left (d^2-x^4\right )^3}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((e^2*Sqrt[d^2 - e^2*x^2]*((a*e^4*Sqrt[d^2 - e^2*x^2])/(6*d^2*(d^2 - x^4) 
^3) + ((e^2*(6*b + (5*a*e^2)/d^2)*Sqrt[d^2 - e^2*x^2])/(4*(d^2 - x^4)^2) + 
 (3*(8*c*d^4 + 6*b*d^2*e^2 + 5*a*e^4)*(Sqrt[d^2 - e^2*x^2]/(2*d^2*(d^2 - x 
^4)) + ArcTanh[Sqrt[d^2 - e^2*x^2]/d]/(2*d^3)))/(4*d^2))/(6*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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 = 179, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (15 a \,e^{4} x^{4}+18 b \,d^{2} e^{2} x^{4}+24 c \,d^{4} x^{4}+10 a \,d^{2} e^{2} x^{2}+12 b \,d^{4} x^{2}+8 a \,d^{4}\right )}{48 d^{6} x^{6}}-\frac {e^{2} \left (5 a \,e^{4}+6 b \,d^{2} e^{2}+8 c \,d^{4}\right ) \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{16 d^{6} \sqrt {d^{2}}\, \sqrt {e x +d}\, \sqrt {-e x +d}}\)	\(179\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (15 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) a \,e^{6} x^{6}+18 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) b \,d^{2} e^{4} x^{6}+24 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right )+d \right )}{x}\right ) c \,d^{4} e^{2} x^{6}+15 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right ) d a \,e^{4} x^{4}+18 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right ) d^{3} b \,e^{2} x^{4}+24 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (d \right ) d^{5} c \,x^{4}+10 \,\operatorname {csgn}\left (d \right ) a \,d^{3} e^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+12 \,\operatorname {csgn}\left (d \right ) b \,d^{5} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+8 \,\operatorname {csgn}\left (d \right ) a \,d^{5} \sqrt {-e^{2} x^{2}+d^{2}}\right ) \operatorname {csgn}\left (d \right )}{48 d^{7} \sqrt {-e^{2} x^{2}+d^{2}}\, x^{6}}\)	\(306\)

Input:

int((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBO 
SE)
 

Output:

-1/48*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(15*a*e^4*x^4+18*b*d^2*e^2*x^4+24*c*d^4 
*x^4+10*a*d^2*e^2*x^2+12*b*d^4*x^2+8*a*d^4)/d^6/x^6-1/16*e^2*(5*a*e^4+6*b* 
d^2*e^2+8*c*d^4)/d^6/(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 = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {3 \, {\left (8 \, c d^{4} e^{2} + 6 \, b d^{2} e^{4} + 5 \, a e^{6}\right )} x^{6} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (8 \, a d^{5} + 3 \, {\left (8 \, c d^{5} + 6 \, b d^{3} e^{2} + 5 \, a d e^{4}\right )} x^{4} + 2 \, {\left (6 \, b d^{5} + 5 \, a d^{3} e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{48 \, d^{7} x^{6}} \] Input:

integrate((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="f 
ricas")
 

Output:

1/48*(3*(8*c*d^4*e^2 + 6*b*d^2*e^4 + 5*a*e^6)*x^6*log((sqrt(e*x + d)*sqrt( 
-e*x + d) - d)/x) - (8*a*d^5 + 3*(8*c*d^5 + 6*b*d^3*e^2 + 5*a*d*e^4)*x^4 + 
 2*(6*b*d^5 + 5*a*d^3*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d))/(d^7*x^6)
 

Sympy [F(-1)]

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

integrate((c*x**4+b*x**2+a)/x**7/(-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 = 271, normalized size of antiderivative = 1.51 \[ \int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {c e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{3}} - \frac {3 \, b e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{5}} - \frac {5 \, a e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{7}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{2 \, d^{2} x^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} b e^{2}}{8 \, d^{4} x^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{4}}{16 \, d^{6} x^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{4 \, d^{2} x^{4}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{24 \, d^{4} x^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{6 \, d^{2} x^{6}} \] Input:

integrate((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="m 
axima")
 

Output:

-1/2*c*e^2*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 - 3/8*b 
*e^4*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^5 - 5/16*a*e^6* 
log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^7 - 1/2*sqrt(-e^2*x^ 
2 + d^2)*c/(d^2*x^2) - 3/8*sqrt(-e^2*x^2 + d^2)*b*e^2/(d^4*x^2) - 5/16*sqr 
t(-e^2*x^2 + d^2)*a*e^4/(d^6*x^2) - 1/4*sqrt(-e^2*x^2 + d^2)*b/(d^2*x^4) - 
 5/24*sqrt(-e^2*x^2 + d^2)*a*e^2/(d^4*x^4) - 1/6*sqrt(-e^2*x^2 + d^2)*a/(d 
^2*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1434 vs. \(2 (156) = 312\).

Time = 0.61 (sec) , antiderivative size = 1434, normalized size of antiderivative = 7.97 \[ \int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display} \] Input:

integrate((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="g 
iac")
 

Output:

-1/48*(3*(8*c*d^4*e^3 + 6*b*d^2*e^5 + 5*a*e^7)*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^7 - 3*(8*c*d^4*e^3 + 6*b*d^2*e^5 + 5*a*e^7)*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^7 - 4*(24*c*d^4*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) 
))^11 + 30*b*d^2*e^5*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - s 
qrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^11 + 33*a*e^7*((sqrt(2)*s 
qrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - 
sqrt(-e*x + d)))^11 - 288*c*d^4*e^3*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sq 
rt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^9 - 168*b* 
d^2*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)))^9 + 20*a*e^7*((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 + 768*c*d^4*e^3*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - s 
qrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^7 + 192*b*d^2*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)))^7 + 1440*a*e^7*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sq 
rt(e*x + d) - sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^7 + 3072*c 
*d^4*e^3*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqrt(e*x +...
 

Mupad [B] (verification not implemented)

Time = 19.64 (sec) , antiderivative size = 1621, normalized size of antiderivative = 9.01 \[ \int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display} \] Input:

int((a + b*x^2 + c*x^4)/(x^7*(d + e*x)^(1/2)*(d - e*x)^(1/2)),x)
 

Output:

((b*e^4)/4 + (6*b*e^4*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - d^ 
(1/2))^2 - (53*b*e^4*((d + e*x)^(1/2) - d^(1/2))^4)/(2*((d - e*x)^(1/2) - 
d^(1/2))^4) - (87*b*e^4*((d + e*x)^(1/2) - d^(1/2))^6)/((d - e*x)^(1/2) - 
d^(1/2))^6 + (657*b*e^4*((d + e*x)^(1/2) - d^(1/2))^8)/(4*((d - e*x)^(1/2) 
 - d^(1/2))^8) - (121*b*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) - ((c*e^2*((d + e*x)^(1/2) - d^(1/2))^2)/((d - e*x)^(1/2) - 
 d^(1/2))^2 - (c*e^2)/2 + (15*c*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) + ((a*e^6)/6 + (4*a*e^6*((d + e*x)^(1/2) - d^(1/2))^2)/ 
((d - e*x)^(1/2) - d^(1/2))^2 + (71*a*e^6*((d + e*x)^(1/2) - d^(1/2))^4)/( 
(d - e*x)^(1/2) - d^(1/2))^4 - (1558*a*e^6*((d + e*x)^(1/2) - d^(1/2))^6)/ 
(3*((d - e*x)^(1/2) - d^(1/2))^6) - (540*a*e^6*((d + e*x)^(1/2) - d^(1/2)) 
^8)/((d - e*x)^(1/2) - d^(1/2))^8 + (4248*a*e^6*((d + e*x)^(1/2) - d^(1...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 589, normalized size of antiderivative = 3.27 \[ \int \frac {a+b x^2+c x^4}{x^7 \sqrt {d-e x} \sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:

int((c*x^4+b*x^2+a)/x^7/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)
 

Output:

( - 8*sqrt(d + e*x)*sqrt(d - e*x)*a*d**5 - 10*sqrt(d + e*x)*sqrt(d - e*x)* 
a*d**3*e**2*x**2 - 15*sqrt(d + e*x)*sqrt(d - e*x)*a*d*e**4*x**4 - 12*sqrt( 
d + e*x)*sqrt(d - e*x)*b*d**5*x**2 - 18*sqrt(d + e*x)*sqrt(d - e*x)*b*d**3 
*e**2*x**4 - 24*sqrt(d + e*x)*sqrt(d - e*x)*c*d**5*x**4 - 15*log( - sqrt(2 
) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*a*e**6*x**6 - 18*log 
( - sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*b*d**2*e** 
4*x**6 - 24*log( - sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) 
- 1)*c*d**4*e**2*x**6 + 15*log( - sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d 
)*sqrt(2)))/2) + 1)*a*e**6*x**6 + 18*log( - sqrt(2) + tan(asin(sqrt(d - e* 
x)/(sqrt(d)*sqrt(2)))/2) + 1)*b*d**2*e**4*x**6 + 24*log( - sqrt(2) + tan(a 
sin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) + 1)*c*d**4*e**2*x**6 - 15*log(sqr 
t(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*a*e**6*x**6 - 18* 
log(sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1)*b*d**2*e** 
4*x**6 - 24*log(sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))/2) - 1 
)*c*d**4*e**2*x**6 + 15*log(sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt(d)*sqrt 
(2)))/2) + 1)*a*e**6*x**6 + 18*log(sqrt(2) + tan(asin(sqrt(d - e*x)/(sqrt( 
d)*sqrt(2)))/2) + 1)*b*d**2*e**4*x**6 + 24*log(sqrt(2) + tan(asin(sqrt(d - 
 e*x)/(sqrt(d)*sqrt(2)))/2) + 1)*c*d**4*e**2*x**6)/(48*d**7*x**6)