\(\int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\) [25]

Optimal result

Integrand size = 32, antiderivative size = 121 \[ \int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\left (3 c d^2+4 b e^2\right ) x \sqrt {d-e x} \sqrt {d+e x}}{8 e^4}-\frac {c x^3 \sqrt {d-e x} \sqrt {d+e x}}{4 e^2}+\frac {\left (3 c d^4+4 b d^2 e^2+8 a e^4\right ) \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{4 e^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {-e x \sqrt {d-e x} \sqrt {d+e x} \left (3 c d^2+4 b e^2+2 c e^2 x^2\right )+2 \left (3 c d^4+4 b d^2 e^2+8 a e^4\right ) \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{8 e^5} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1789, 1473, 25, 299, 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.

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

\(\Big \downarrow \) 1789

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

\(\Big \downarrow \) 1473

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

(Sqrt[d^2 - e^2*x^2]*(-1/4*(c*x^3*Sqrt[d^2 - e^2*x^2])/e^2 + (-1/2*((4*b + 
 (3*c*d^2)/e^2)*x*Sqrt[d^2 - e^2*x^2]) + ((3*c*d^4 + 4*b*d^2*e^2 + 8*a*e^4 
)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^3))/(4*e^2)))/(Sqrt[d - e*x]*Sqr 
t[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 299

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

rule 1473

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> Simp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))) 
, x] + Simp[1/(e*(4*p + 2*q + 1))   Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 
2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 
 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, b, c, d, e, q}, 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 1789

Int[((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.)*(x_)^(non2_.))^(q_ 
.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Simp[(d 
1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[q]/(d1*d2 + e1*e2*x 
^n)^FracPart[q])   Int[(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, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[ 
non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {x \left (2 c \,x^{2} e^{2}+4 b \,e^{2}+3 c \,d^{2}\right ) \sqrt {-e x +d}\, \sqrt {e x +d}}{8 e^{4}}+\frac {\left (8 a \,e^{4}+4 b \,d^{2} e^{2}+3 c \,d^{4}\right ) \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{8 e^{4} \sqrt {e^{2}}\, \sqrt {e x +d}\, \sqrt {-e x +d}}\)	\(128\)
default	\(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (2 \,\operatorname {csgn}\left (e \right ) c \,e^{3} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}+4 \,\operatorname {csgn}\left (e \right ) e^{3} \sqrt {-e^{2} x^{2}+d^{2}}\, b x +3 \,\operatorname {csgn}\left (e \right ) e \sqrt {-e^{2} x^{2}+d^{2}}\, c \,d^{2} x -8 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) a \,e^{4}-4 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) b \,d^{2} e^{2}-3 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) c \,d^{4}\right ) \operatorname {csgn}\left (e \right )}{8 e^{5} \sqrt {-e^{2} x^{2}+d^{2}}}\)	\(191\)

Input:

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

Output:

-1/8*x*(2*c*e^2*x^2+4*b*e^2+3*c*d^2)/e^4*(-e*x+d)^(1/2)*(e*x+d)^(1/2)+1/8* 
(8*a*e^4+4*b*d^2*e^2+3*c*d^4)/e^4/(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 = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (2 \, c e^{3} x^{3} + {\left (3 \, c d^{2} e + 4 \, b e^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {-e x + d} + 2 \, {\left (3 \, c d^{4} + 4 \, b d^{2} e^{2} + 8 \, a e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{e x}\right )}{8 \, e^{5}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

integrate((c*x**4+b*x**2+a)/(-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 = 146, normalized size of antiderivative = 1.21 \[ \int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{3}}{4 \, e^{2}} + \frac {a \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} + \frac {3 \, c d^{4} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}} e^{4}} + \frac {b d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}} e^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x}{8 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x}{2 \, e^{2}} \] Input:

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

Output:

-1/4*sqrt(-e^2*x^2 + d^2)*c*x^3/e^2 + a*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e 
^2) + 3/8*c*d^4*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^4) + 1/2*b*d^2*ar 
csin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^2) - 3/8*sqrt(-e^2*x^2 + d^2)*c*d^2 
*x/e^4 - 1/2*sqrt(-e^2*x^2 + d^2)*b*x/e^2
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95 \[ \int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {{\left (5 \, c d^{3} + 4 \, b d e^{2} - {\left (9 \, c d^{2} + 4 \, b e^{2} + 2 \, {\left ({\left (e x + d\right )} c - 3 \, c d\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} \sqrt {e x + d} \sqrt {-e x + d} + 2 \, {\left (3 \, c d^{4} + 4 \, b d^{2} e^{2} + 8 \, a e^{4}\right )} \arcsin \left (\frac {\sqrt {2} \sqrt {e x + d}}{2 \, \sqrt {d}}\right )}{8 \, e^{5}} \] Input:

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

Output:

1/8*((5*c*d^3 + 4*b*d*e^2 - (9*c*d^2 + 4*b*e^2 + 2*((e*x + d)*c - 3*c*d)*( 
e*x + d))*(e*x + d))*sqrt(e*x + d)*sqrt(-e*x + d) + 2*(3*c*d^4 + 4*b*d^2*e 
^2 + 8*a*e^4)*arcsin(1/2*sqrt(2)*sqrt(e*x + d)/sqrt(d)))/e^5
 

Mupad [B] (verification not implemented)

Time = 12.56 (sec) , antiderivative size = 651, normalized size of antiderivative = 5.38 \[ \int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {\frac {14\,b\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^3}-\frac {14\,b\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^5}+\frac {2\,b\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^7}-\frac {2\,b\,d^2\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}{\sqrt {d-e\,x}-\sqrt {d}}}{e^3\,{\left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}+1\right )}^4}-\frac {4\,a\,\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 {\frac {23\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^3}-\frac {333\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^5}+\frac {671\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^7}-\frac {671\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^9}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^9}+\frac {333\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{11}}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{11}}-\frac {23\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{13}}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{13}}-\frac {3\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{15}}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{15}}+\frac {3\,c\,d^4\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}{2\,\left (\sqrt {d-e\,x}-\sqrt {d}\right )}}{e^5\,{\left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}+1\right )}^8}+\frac {2\,b\,d^2\,\mathrm {atan}\left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{e^3}+\frac {3\,c\,d^4\,\mathrm {atan}\left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{2\,e^5} \] Input:

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

Output:

((14*b*d^2*((d + e*x)^(1/2) - d^(1/2))^3)/((d - e*x)^(1/2) - d^(1/2))^3 - 
(14*b*d^2*((d + e*x)^(1/2) - d^(1/2))^5)/((d - e*x)^(1/2) - d^(1/2))^5 + ( 
2*b*d^2*((d + e*x)^(1/2) - d^(1/2))^7)/((d - e*x)^(1/2) - d^(1/2))^7 - (2* 
b*d^2*((d + e*x)^(1/2) - d^(1/2)))/((d - e*x)^(1/2) - d^(1/2)))/(e^3*(((d 
+ e*x)^(1/2) - d^(1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 + 1)^4) - (4*a*ata 
n((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) - ((23*c*d^4*((d + e*x)^(1/2) - d^(1/2))^3)/(2*((d - e*x)^ 
(1/2) - d^(1/2))^3) - (333*c*d^4*((d + e*x)^(1/2) - d^(1/2))^5)/(2*((d - e 
*x)^(1/2) - d^(1/2))^5) + (671*c*d^4*((d + e*x)^(1/2) - d^(1/2))^7)/(2*((d 
 - e*x)^(1/2) - d^(1/2))^7) - (671*c*d^4*((d + e*x)^(1/2) - d^(1/2))^9)/(2 
*((d - e*x)^(1/2) - d^(1/2))^9) + (333*c*d^4*((d + e*x)^(1/2) - d^(1/2))^1 
1)/(2*((d - e*x)^(1/2) - d^(1/2))^11) - (23*c*d^4*((d + e*x)^(1/2) - d^(1/ 
2))^13)/(2*((d - e*x)^(1/2) - d^(1/2))^13) - (3*c*d^4*((d + e*x)^(1/2) - d 
^(1/2))^15)/(2*((d - e*x)^(1/2) - d^(1/2))^15) + (3*c*d^4*((d + e*x)^(1/2) 
 - d^(1/2)))/(2*((d - e*x)^(1/2) - d^(1/2))))/(e^5*(((d + e*x)^(1/2) - d^( 
1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 + 1)^8) + (2*b*d^2*atan(((d + e*x)^( 
1/2) - d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/e^3 + (3*c*d^4*atan(((d + e* 
x)^(1/2) - d^(1/2))/((d - e*x)^(1/2) - d^(1/2))))/(2*e^5)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {-16 \mathit {asin} \left (\frac {\sqrt {-e x +d}}{\sqrt {d}\, \sqrt {2}}\right ) a \,e^{4}-8 \mathit {asin} \left (\frac {\sqrt {-e x +d}}{\sqrt {d}\, \sqrt {2}}\right ) b \,d^{2} e^{2}-6 \mathit {asin} \left (\frac {\sqrt {-e x +d}}{\sqrt {d}\, \sqrt {2}}\right ) c \,d^{4}-4 \sqrt {e x +d}\, \sqrt {-e x +d}\, b \,e^{3} x -3 \sqrt {e x +d}\, \sqrt {-e x +d}\, c \,d^{2} e x -2 \sqrt {e x +d}\, \sqrt {-e x +d}\, c \,e^{3} x^{3}}{8 e^{5}} \] Input:

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

Output:

( - 16*asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))*a*e**4 - 8*asin(sqrt(d - e*x) 
/(sqrt(d)*sqrt(2)))*b*d**2*e**2 - 6*asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))* 
c*d**4 - 4*sqrt(d + e*x)*sqrt(d - e*x)*b*e**3*x - 3*sqrt(d + e*x)*sqrt(d - 
 e*x)*c*d**2*e*x - 2*sqrt(d + e*x)*sqrt(d - e*x)*c*e**3*x**3)/(8*e**5)