Integrand size = 35, antiderivative size = 216 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \sqrt {d-e x} \sqrt {d+e x}}{16 e^6}-\frac {\left (5 c d^2+6 b e^2\right ) x^3 \sqrt {d-e x} \sqrt {d+e x}}{24 e^4}+\frac {c x^5 (-d+e x) \sqrt {d+e x}}{6 e^2 \sqrt {d-e x}}+\frac {d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt {d^2-e^2 x^2} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^7 \sqrt {d-e x} \sqrt {d+e x}} \] Output:
-1/16*(8*a*e^4+6*b*d^2*e^2+5*c*d^4)*x*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/e^6-1/2 4*(6*b*e^2+5*c*d^2)*x^3*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/e^4+1/6*c*x^5*(e*x-d) *(e*x+d)^(1/2)/e^2/(-e*x+d)^(1/2)+1/16*d^2*(8*a*e^4+6*b*d^2*e^2+5*c*d^4)*( -e^2*x^2+d^2)^(1/2)*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^7/(-e*x+d)^(1/2)/(e *x+d)^(1/2)
Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.62 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {-e x \sqrt {d-e x} \sqrt {d+e x} \left (6 \left (3 b d^2 e^2+4 a e^4+2 b e^4 x^2\right )+c \left (15 d^4+10 d^2 e^2 x^2+8 e^4 x^4\right )\right )+6 d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{48 e^7} \] Input:
Integrate[(x^2*(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]*(6*(3*b*d^2*e^2 + 4*a*e^4 + 2*b*e^4*x^2 ) + c*(15*d^4 + 10*d^2*e^2*x^2 + 8*e^4*x^4))) + 6*d^2*(5*c*d^4 + 6*b*d^2*e ^2 + 8*a*e^4)*ArcTan[Sqrt[d + e*x]/Sqrt[d - e*x]])/(48*e^7)
Time = 0.39 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1905, 1590, 25, 363, 262, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \frac {x^2 \left (c x^4+b x^2+a\right )}{\sqrt {d^2-e^2 x^2}}dx}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {\int -\frac {x^2 \left (6 a e^2+\left (5 c d^2+6 b e^2\right ) x^2\right )}{\sqrt {d^2-e^2 x^2}}dx}{6 e^2}-\frac {c x^5 \sqrt {d^2-e^2 x^2}}{6 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\int \frac {x^2 \left (6 a e^2+\left (5 c d^2+6 b e^2\right ) x^2\right )}{\sqrt {d^2-e^2 x^2}}dx}{6 e^2}-\frac {c x^5 \sqrt {d^2-e^2 x^2}}{6 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (8 a e^4+6 b d^2 e^2+5 c d^4\right ) \int \frac {x^2}{\sqrt {d^2-e^2 x^2}}dx}{4 e^2}-\frac {1}{4} x^3 \sqrt {d^2-e^2 x^2} \left (6 b+\frac {5 c d^2}{e^2}\right )}{6 e^2}-\frac {c x^5 \sqrt {d^2-e^2 x^2}}{6 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (8 a e^4+6 b d^2 e^2+5 c d^4\right ) \left (\frac {d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^2}\right )}{4 e^2}-\frac {1}{4} x^3 \sqrt {d^2-e^2 x^2} \left (6 b+\frac {5 c d^2}{e^2}\right )}{6 e^2}-\frac {c x^5 \sqrt {d^2-e^2 x^2}}{6 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (8 a e^4+6 b d^2 e^2+5 c d^4\right ) \left (\frac {d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}}{2 e^2}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^2}\right )}{4 e^2}-\frac {1}{4} x^3 \sqrt {d^2-e^2 x^2} \left (6 b+\frac {5 c d^2}{e^2}\right )}{6 e^2}-\frac {c x^5 \sqrt {d^2-e^2 x^2}}{6 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (\frac {\frac {3 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^3}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^2}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{4 e^2}-\frac {1}{4} x^3 \sqrt {d^2-e^2 x^2} \left (6 b+\frac {5 c d^2}{e^2}\right )}{6 e^2}-\frac {c x^5 \sqrt {d^2-e^2 x^2}}{6 e^2}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\) |
Input:
Int[(x^2*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]
Output:
(Sqrt[d^2 - e^2*x^2]*(-1/6*(c*x^5*Sqrt[d^2 - e^2*x^2])/e^2 + (-1/4*((6*b + (5*c*d^2)/e^2)*x^3*Sqrt[d^2 - e^2*x^2]) + (3*(5*c*d^4 + 6*b*d^2*e^2 + 8*a *e^4)*(-1/2*(x*Sqrt[d^2 - e^2*x^2])/e^2 + (d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2 *x^2]])/(2*e^3)))/(4*e^2))/(6*e^2)))/(Sqrt[d - e*x]*Sqrt[d + e*x])
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])
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]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ (q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q + 1)) Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
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]
Time = 0.86 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(-\frac {x \left (8 c \,x^{4} e^{4}+12 b \,e^{4} x^{2}+10 c \,d^{2} e^{2} x^{2}+24 a \,e^{4}+18 b \,d^{2} e^{2}+15 c \,d^{4}\right ) \sqrt {-e x +d}\, \sqrt {e x +d}}{48 e^{6}}+\frac {d^{2} \left (8 a \,e^{4}+6 b \,d^{2} e^{2}+5 c \,d^{4}\right ) \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{16 e^{6} \sqrt {e^{2}}\, \sqrt {e x +d}\, \sqrt {-e x +d}}\) | \(161\) |
| default | \(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (8 \,\operatorname {csgn}\left (e \right ) c \,e^{5} x^{5} \sqrt {-e^{2} x^{2}+d^{2}}+12 \,\operatorname {csgn}\left (e \right ) b \,e^{5} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}+10 \,\operatorname {csgn}\left (e \right ) c \,d^{2} e^{3} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}+24 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (e \right ) e^{5} a x +18 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (e \right ) e^{3} b \,d^{2} x +15 \sqrt {-e^{2} x^{2}+d^{2}}\, \operatorname {csgn}\left (e \right ) e c \,d^{4} x -24 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) a \,d^{2} e^{4}-18 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) b \,d^{4} e^{2}-15 \arctan \left (\frac {\operatorname {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) c \,d^{6}\right ) \operatorname {csgn}\left (e \right )}{48 e^{7} \sqrt {-e^{2} x^{2}+d^{2}}}\) | \(273\) |
Input:
int(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBO SE)
Output:
-1/48*x*(8*c*e^4*x^4+12*b*e^4*x^2+10*c*d^2*e^2*x^2+24*a*e^4+18*b*d^2*e^2+1 5*c*d^4)/e^6*(-e*x+d)^(1/2)*(e*x+d)^(1/2)+1/16*d^2*(8*a*e^4+6*b*d^2*e^2+5* c*d^4)/e^6/(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)
Time = 0.09 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.62 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (8 \, c e^{5} x^{5} + 2 \, {\left (5 \, c d^{2} e^{3} + 6 \, b e^{5}\right )} x^{3} + 3 \, {\left (5 \, c d^{4} e + 6 \, b d^{2} e^{3} + 8 \, a e^{5}\right )} x\right )} \sqrt {e x + d} \sqrt {-e x + d} + 6 \, {\left (5 \, c d^{6} + 6 \, b d^{4} e^{2} + 8 \, a d^{2} e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{e x}\right )}{48 \, e^{7}} \] Input:
integrate(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="f ricas")
Output:
-1/48*((8*c*e^5*x^5 + 2*(5*c*d^2*e^3 + 6*b*e^5)*x^3 + 3*(5*c*d^4*e + 6*b*d ^2*e^3 + 8*a*e^5)*x)*sqrt(e*x + d)*sqrt(-e*x + d) + 6*(5*c*d^6 + 6*b*d^4*e ^2 + 8*a*d^2*e^4)*arctan((sqrt(e*x + d)*sqrt(-e*x + d) - d)/(e*x)))/e^7
Timed out. \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Timed out} \] Input:
integrate(x**2*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{5}}{6 \, e^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x^{3}}{24 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x^{3}}{4 \, e^{2}} + \frac {5 \, c d^{6} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{16 \, \sqrt {e^{2}} e^{6}} + \frac {3 \, b d^{4} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}} e^{4}} + \frac {a d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}} e^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{4} x}{16 \, e^{6}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{2} x}{8 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a x}{2 \, e^{2}} \] Input:
integrate(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="m axima")
Output:
-1/6*sqrt(-e^2*x^2 + d^2)*c*x^5/e^2 - 5/24*sqrt(-e^2*x^2 + d^2)*c*d^2*x^3/ e^4 - 1/4*sqrt(-e^2*x^2 + d^2)*b*x^3/e^2 + 5/16*c*d^6*arcsin(e^2*x/(d*sqrt (e^2)))/(sqrt(e^2)*e^6) + 3/8*b*d^4*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2) *e^4) + 1/2*a*d^2*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^2) - 5/16*sqrt( -e^2*x^2 + d^2)*c*d^4*x/e^6 - 3/8*sqrt(-e^2*x^2 + d^2)*b*d^2*x/e^4 - 1/2*s qrt(-e^2*x^2 + d^2)*a*x/e^2
Time = 0.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {{\left (33 \, c d^{5} + 30 \, b d^{3} e^{2} + 24 \, a d e^{4} - {\left (85 \, c d^{4} + 54 \, b d^{2} e^{2} + 24 \, a e^{4} - 2 \, {\left (55 \, c d^{3} + 18 \, b d e^{2} - {\left (45 \, c d^{2} + 6 \, b e^{2} + 4 \, {\left ({\left (e x + d\right )} c - 5 \, c d\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} \sqrt {e x + d} \sqrt {-e x + d} + 6 \, {\left (5 \, c d^{6} + 6 \, b d^{4} e^{2} + 8 \, a d^{2} e^{4}\right )} \arcsin \left (\frac {\sqrt {2} \sqrt {e x + d}}{2 \, \sqrt {d}}\right )}{48 \, e^{7}} \] Input:
integrate(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="g iac")
Output:
1/48*((33*c*d^5 + 30*b*d^3*e^2 + 24*a*d*e^4 - (85*c*d^4 + 54*b*d^2*e^2 + 2 4*a*e^4 - 2*(55*c*d^3 + 18*b*d*e^2 - (45*c*d^2 + 6*b*e^2 + 4*((e*x + d)*c - 5*c*d)*(e*x + d))*(e*x + d))*(e*x + d))*(e*x + d))*sqrt(e*x + d)*sqrt(-e *x + d) + 6*(5*c*d^6 + 6*b*d^4*e^2 + 8*a*d^2*e^4)*arcsin(1/2*sqrt(2)*sqrt( e*x + d)/sqrt(d)))/e^7
Time = 21.81 (sec) , antiderivative size = 1132, normalized size of antiderivative = 5.24 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\text {Too large to display} \] Input:
int((x^2*(a + b*x^2 + c*x^4))/((d + e*x)^(1/2)*(d - e*x)^(1/2)),x)
Output:
((14*a*d^2*((d + e*x)^(1/2) - d^(1/2))^3)/((d - e*x)^(1/2) - d^(1/2))^3 - (14*a*d^2*((d + e*x)^(1/2) - d^(1/2))^5)/((d - e*x)^(1/2) - d^(1/2))^5 + ( 2*a*d^2*((d + e*x)^(1/2) - d^(1/2))^7)/((d - e*x)^(1/2) - d^(1/2))^7 - (2* a*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) - ((175*c* d^6*((d + e*x)^(1/2) - d^(1/2))^3)/(12*((d - e*x)^(1/2) - d^(1/2))^3) + (3 11*c*d^6*((d + e*x)^(1/2) - d^(1/2))^5)/(4*((d - e*x)^(1/2) - d^(1/2))^5) - (8361*c*d^6*((d + e*x)^(1/2) - d^(1/2))^7)/(4*((d - e*x)^(1/2) - d^(1/2) )^7) + (42259*c*d^6*((d + e*x)^(1/2) - d^(1/2))^9)/(6*((d - e*x)^(1/2) - d ^(1/2))^9) - (25295*c*d^6*((d + e*x)^(1/2) - d^(1/2))^11)/(2*((d - e*x)^(1 /2) - d^(1/2))^11) + (25295*c*d^6*((d + e*x)^(1/2) - d^(1/2))^13)/(2*((d - e*x)^(1/2) - d^(1/2))^13) - (42259*c*d^6*((d + e*x)^(1/2) - d^(1/2))^15)/ (6*((d - e*x)^(1/2) - d^(1/2))^15) + (8361*c*d^6*((d + e*x)^(1/2) - d^(1/2 ))^17)/(4*((d - e*x)^(1/2) - d^(1/2))^17) - (311*c*d^6*((d + e*x)^(1/2) - d^(1/2))^19)/(4*((d - e*x)^(1/2) - d^(1/2))^19) - (175*c*d^6*((d + e*x)^(1 /2) - d^(1/2))^21)/(12*((d - e*x)^(1/2) - d^(1/2))^21) - (5*c*d^6*((d + e* x)^(1/2) - d^(1/2))^23)/(4*((d - e*x)^(1/2) - d^(1/2))^23) + (5*c*d^6*((d + e*x)^(1/2) - d^(1/2)))/(4*((d - e*x)^(1/2) - d^(1/2))))/(e^7*(((d + e*x) ^(1/2) - d^(1/2))^2/((d - e*x)^(1/2) - d^(1/2))^2 + 1)^12) - ((23*b*d^4*(( d + e*x)^(1/2) - d^(1/2))^3)/(2*((d - e*x)^(1/2) - d^(1/2))^3) - (333*b...
Time = 0.15 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {-48 \mathit {asin} \left (\frac {\sqrt {-e x +d}}{\sqrt {d}\, \sqrt {2}}\right ) a \,d^{2} e^{4}-36 \mathit {asin} \left (\frac {\sqrt {-e x +d}}{\sqrt {d}\, \sqrt {2}}\right ) b \,d^{4} e^{2}-30 \mathit {asin} \left (\frac {\sqrt {-e x +d}}{\sqrt {d}\, \sqrt {2}}\right ) c \,d^{6}-24 \sqrt {e x +d}\, \sqrt {-e x +d}\, a \,e^{5} x -18 \sqrt {e x +d}\, \sqrt {-e x +d}\, b \,d^{2} e^{3} x -12 \sqrt {e x +d}\, \sqrt {-e x +d}\, b \,e^{5} x^{3}-15 \sqrt {e x +d}\, \sqrt {-e x +d}\, c \,d^{4} e x -10 \sqrt {e x +d}\, \sqrt {-e x +d}\, c \,d^{2} e^{3} x^{3}-8 \sqrt {e x +d}\, \sqrt {-e x +d}\, c \,e^{5} x^{5}}{48 e^{7}} \] Input:
int(x^2*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)
Output:
( - 48*asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))*a*d**2*e**4 - 36*asin(sqrt(d - e*x)/(sqrt(d)*sqrt(2)))*b*d**4*e**2 - 30*asin(sqrt(d - e*x)/(sqrt(d)*sqr t(2)))*c*d**6 - 24*sqrt(d + e*x)*sqrt(d - e*x)*a*e**5*x - 18*sqrt(d + e*x) *sqrt(d - e*x)*b*d**2*e**3*x - 12*sqrt(d + e*x)*sqrt(d - e*x)*b*e**5*x**3 - 15*sqrt(d + e*x)*sqrt(d - e*x)*c*d**4*e*x - 10*sqrt(d + e*x)*sqrt(d - e* x)*c*d**2*e**3*x**3 - 8*sqrt(d + e*x)*sqrt(d - e*x)*c*e**5*x**5)/(48*e**7)