\(\int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [27]

Optimal result

Integrand size = 35, antiderivative size = 100 \[ \int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{3 d^2 x^3}-\frac {\left (3 b d^2+2 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x}}{3 d^4 x}+\frac {2 c \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (3 b d^2 x^2+a \left (d^2+2 e^2 x^2\right )\right )}{3 d^4 x^3}+\frac {2 c \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1905, 1588, 25, 358, 224, 216}

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.

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

\(\Big \downarrow \) 1905

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

\(\Big \downarrow \) 1588

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 358

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

(Sqrt[d^2 - e^2*x^2]*(-1/3*(a*Sqrt[d^2 - e^2*x^2])/(d^2*x^3) + (-(((3*b + 
(2*a*e^2)/d^2)*Sqrt[d^2 - e^2*x^2])/x) + (3*c*d^2*ArcTan[(e*x)/Sqrt[d^2 - 
e^2*x^2]])/e)/(3*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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 358

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] + S 
imp[d/e^2   Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, 
 -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 = 0.85 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07

method	result	size
risch	\(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (2 a \,e^{2} x^{2}+3 b \,d^{2} x^{2}+a \,d^{2}\right )}{3 d^{4} x^{3}}+\frac {c \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{\sqrt {e^{2}}\, \sqrt {e x +d}\, \sqrt {-e x +d}}\)	\(107\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (-3 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) c \,d^{4} x^{3}+2 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (e \right ) e^{3} a \,x^{2}+3 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (e \right ) e b \,d^{2} x^{2}+a \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} \operatorname {csgn}\left (e \right ) e \right ) \operatorname {csgn}\left (e \right )}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}\, x^{3} e}\)	\(146\)

Input:

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

Output:

-1/3*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(2*a*e^2*x^2+3*b*d^2*x^2+a*d^2)/d^4/x^3+ 
c/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))*((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 = 90, normalized size of antiderivative = 0.90 \[ \int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {6 \, c d^{4} x^{3} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{e x}\right ) + {\left (a d^{2} e + {\left (3 \, b d^{2} e + 2 \, a e^{3}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{3 \, d^{4} e x^{3}} \] Input:

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

Output:

-1/3*(6*c*d^4*x^3*arctan((sqrt(e*x + d)*sqrt(-e*x + d) - d)/(e*x)) + (a*d^ 
2*e + (3*b*d^2*e + 2*a*e^3)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d))/(d^4*e*x^3)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 19.15 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.57 \[ \int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {i a e^{3} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{4}, \frac {11}{4}, 1 & \frac {5}{2}, \frac {5}{2}, 3 \\2, \frac {9}{4}, \frac {5}{2}, \frac {11}{4}, 3 & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} + \frac {a e^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2}, 1 & \\\frac {7}{4}, \frac {9}{4} & \frac {3}{2}, 2, 2, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} + \frac {i b e {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {b e {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} - \frac {i c {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e} + \frac {c {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e} \] Input:

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

Output:

I*a*e**3*meijerg(((9/4, 11/4, 1), (5/2, 5/2, 3)), ((2, 9/4, 5/2, 11/4, 3), 
 (0,)), d**2/(e**2*x**2))/(4*pi**(3/2)*d**4) + a*e**3*meijerg(((3/2, 7/4, 
2, 9/4, 5/2, 1), ()), ((7/4, 9/4), (3/2, 2, 2, 0)), d**2*exp_polar(-2*I*pi 
)/(e**2*x**2))/(4*pi**(3/2)*d**4) + I*b*e*meijerg(((5/4, 7/4, 1), (3/2, 3/ 
2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), d**2/(e**2*x**2))/(4*pi**(3/2)*d**2 
) + b*e*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4, 5/4), (1/2, 1, 1, 
 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*d**2) - I*c*meijer 
g(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), d**2/(e* 
*2*x**2))/(4*pi**(3/2)*e) + c*meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), 
((-1/4, 1/4), (-1/2, 0, 0, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi 
**(3/2)*e)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {c \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{d^{2} x} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{3 \, d^{4} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{3 \, d^{2} x^{3}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (84) = 168\).

Time = 0.26 (sec) , antiderivative size = 530, normalized size of antiderivative = 5.30 \[ \int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {\sqrt {e x + d} {\left (\frac {{\left (\sqrt {2} \sqrt {d} - \sqrt {-e x + d}\right )}^{2}}{e x + d} - 1\right )}}{2 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {-e x + d}\right )}}\right )\right )} c - \frac {4 \, {\left (3 \, b d^{2} e^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} + 3 \, a e^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{5} - 24 \, b d^{2} e^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} - 8 \, a e^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{3} + 48 \, b d^{2} e^{2} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )} + 48 \, a e^{4} {\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}{\sqrt {e x + d}} - \frac {\sqrt {e x + d}}{\sqrt {2} \sqrt {d} - \sqrt {-e x + d}}\right )}^{2} - 4\right )}^{3} d^{4}}}{3 \, e} \] Input:

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

Output:

1/3*(3*(pi + 2*arctan(1/2*sqrt(e*x + d)*((sqrt(2)*sqrt(d) - sqrt(-e*x + d) 
)^2/(e*x + d) - 1)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d))))*c - 4*(3*b*d^2*e^2 
*((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 + 3*a*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)))^5 - 2 
4*b*d^2*e^2*((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 - 8*a*e^4*((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)))^3 + 48*b*d^2*e^2*((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - 
 sqrt(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d))) + 48*a*e^4*((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))))/((((sqrt(2)*sqrt(d) - sqrt(-e*x + d))/sqrt(e*x + d) - sqr 
t(e*x + d)/(sqrt(2)*sqrt(d) - sqrt(-e*x + d)))^2 - 4)^3*d^4))/e
 

Mupad [B] (verification not implemented)

Time = 4.50 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.38 \[ \int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {4\,c\,\mathrm {atan}\left (\frac {e\,\left (\sqrt {d-e\,x}-\sqrt {d}\right )}{\sqrt {e^2}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )}{\sqrt {e^2}}-\frac {\left (\frac {b}{d}+\frac {b\,e\,x}{d^2}\right )\,\sqrt {d-e\,x}}{x\,\sqrt {d+e\,x}}-\frac {\sqrt {d-e\,x}\,\left (\frac {a}{3\,d}+\frac {2\,a\,e^2\,x^2}{3\,d^3}+\frac {2\,a\,e^3\,x^3}{3\,d^4}+\frac {a\,e\,x}{3\,d^2}\right )}{x^3\,\sqrt {d+e\,x}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.03 \[ \int \frac {a+b x^2+c x^4}{x^4 \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {-6 \mathit {asin} \left (\frac {\sqrt {-e x +d}}{\sqrt {d}\, \sqrt {2}}\right ) c \,d^{4} x^{3}-\sqrt {e x +d}\, \sqrt {-e x +d}\, a \,d^{2} e -2 \sqrt {e x +d}\, \sqrt {-e x +d}\, a \,e^{3} x^{2}-3 \sqrt {e x +d}\, \sqrt {-e x +d}\, b \,d^{2} e \,x^{2}}{3 d^{4} e \,x^{3}} \] Input:

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

Output:

( - 6*asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))*c*d**4*x**3 - sqrt(d + e*x)*sq 
rt(d - e*x)*a*d**2*e - 2*sqrt(d + e*x)*sqrt(d - e*x)*a*e**3*x**2 - 3*sqrt( 
d + e*x)*sqrt(d - e*x)*b*d**2*e*x**2)/(3*d**4*e*x**3)