Integrand size = 37, antiderivative size = 186 \[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\frac {(B d+A e) x \sqrt {a d^2-a e^2 x^4}}{2 a d^2 e \left (d-e x^2\right )}-\frac {(B d+A e) \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} e^{3/2} \sqrt {a d^2-a e^2 x^4}}+\frac {A \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {d} \sqrt {e} \sqrt {a d^2-a e^2 x^4}} \] Output:
1/2*(A*e+B*d)*x*(-a*e^2*x^4+a*d^2)^(1/2)/a/d^2/e/(-e*x^2+d)-1/2*(A*e+B*d)* (1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^(1/2)/e^(3/2)/(-a*e ^2*x^4+a*d^2)^(1/2)+A*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I) /d^(1/2)/e^(1/2)/(-a*e^2*x^4+a*d^2)^(1/2)
Result contains complex when optimal does not.
Time = 11.60 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=-\frac {\sqrt {-\frac {e}{d}} (B d+A e) x \left (d+e x^2\right )+i d (B d+A e) \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-2 i A d e \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{2 d^3 \left (-\frac {e}{d}\right )^{3/2} \sqrt {a \left (d^2-e^2 x^4\right )}} \] Input:
Integrate[(A + B*x^2)/((d - e*x^2)*Sqrt[a*d^2 - a*e^2*x^4]),x]
Output:
-1/2*(Sqrt[-(e/d)]*(B*d + A*e)*x*(d + e*x^2) + I*d*(B*d + A*e)*Sqrt[1 - (e ^2*x^4)/d^2]*EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] - (2*I)*A*d*e*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1])/(d^3*(-(e/d))^ (3/2)*Sqrt[a*(d^2 - e^2*x^4)])
Time = 0.46 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.35, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {1396, 402, 25, 27, 399, 289, 329, 327, 765, 762}
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.
\(\displaystyle \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \int \frac {B x^2+A}{\left (d-e x^2\right )^{3/2} \sqrt {a e x^2+a d}}dx}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {\int -\frac {a \left (e (B d+A e) x^2+d (B d-A e)\right )}{\sqrt {d-e x^2} \sqrt {a e x^2+a d}}dx}{2 a d^2 e}+\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}-\frac {\int \frac {a \left (e (B d+A e) x^2+d (B d-A e)\right )}{\sqrt {d-e x^2} \sqrt {a e x^2+a d}}dx}{2 a d^2 e}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}-\frac {\int \frac {e (B d+A e) x^2+d (B d-A e)}{\sqrt {d-e x^2} \sqrt {a e x^2+a d}}dx}{2 d^2 e}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}-\frac {\frac {(A e+B d) \int \frac {\sqrt {a e x^2+a d}}{\sqrt {d-e x^2}}dx}{a}-2 A d e \int \frac {1}{\sqrt {d-e x^2} \sqrt {a e x^2+a d}}dx}{2 d^2 e}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 289 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}-\frac {\frac {(A e+B d) \int \frac {\sqrt {a e x^2+a d}}{\sqrt {d-e x^2}}dx}{a}-\frac {2 A d e \sqrt {a d^2-a e^2 x^4} \int \frac {1}{\sqrt {a d^2-a e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {a d+a e x^2}}}{2 d^2 e}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 329 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}-\frac {\frac {d \sqrt {1-\frac {e^2 x^4}{d^2}} (A e+B d) \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {a d+a e x^2}}-\frac {2 A d e \sqrt {a d^2-a e^2 x^4} \int \frac {1}{\sqrt {a d^2-a e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {a d+a e x^2}}}{2 d^2 e}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}-\frac {\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} (A e+B d) E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {a d+a e x^2}}-\frac {2 A d e \sqrt {a d^2-a e^2 x^4} \int \frac {1}{\sqrt {a d^2-a e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {a d+a e x^2}}}{2 d^2 e}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}-\frac {\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} (A e+B d) E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {a d+a e x^2}}-\frac {2 A d e \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {a d+a e x^2}}}{2 d^2 e}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {a d+a e x^2} \left (\frac {x \sqrt {a d+a e x^2} (A e+B d)}{2 a d^2 e \sqrt {d-e x^2}}-\frac {\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} (A e+B d) E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {a d+a e x^2}}-\frac {2 A d^{3/2} \sqrt {e} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {d-e x^2} \sqrt {a d+a e x^2}}}{2 d^2 e}\right )}{\sqrt {a d^2-a e^2 x^4}}\) |
Input:
Int[(A + B*x^2)/((d - e*x^2)*Sqrt[a*d^2 - a*e^2*x^4]),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[a*d + a*e*x^2]*(((B*d + A*e)*x*Sqrt[a*d + a*e*x^2])/ (2*a*d^2*e*Sqrt[d - e*x^2]) - ((d^(3/2)*(B*d + A*e)*Sqrt[1 - (e^2*x^4)/d^2 ]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqr t[a*d + a*e*x^2]) - (2*A*d^(3/2)*Sqrt[e]*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF [ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[d - e*x^2]*Sqrt[a*d + a*e*x^2]))/ (2*d^2*e)))/Sqrt[a*d^2 - a*e^2*x^4]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart [p]) Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* c + a*d, 0] && !IntegerQ[p]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])) Int[Sqrt[1 + b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b *c + a*d, 0] && !(LtQ[a*c, 0] && GtQ[a*b, 0])
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 0.97 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.28
method | result | size |
elliptic | \(-\frac {\left (-a \,e^{2} x^{2}-a d e \right ) x \left (A e +B d \right )}{2 d^{2} a \,e^{2} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-a \,e^{2} x^{2}-a d e \right )}}+\frac {\left (-\frac {B}{e}+\frac {A e +B d}{2 d e}\right ) \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}}+\frac {\left (A e +B d \right ) \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}\, e}\) | \(239\) |
default | \(\frac {\left (A e +B d \right ) \left (-\frac {\left (-a \,e^{2} x^{2}-a d e \right ) x}{2 d^{2} a e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-a \,e^{2} x^{2}-a d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}}\right )}{e}-\frac {B \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{e \sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}}\) | \(283\) |
Input:
int((B*x^2+A)/(-e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(-a*e^2*x^2-a*d*e)/d^2*x/a/e^2*(A*e+B*d)/((x^2-d/e)*(-a*e^2*x^2-a*d*e ))^(1/2)+(-B/e+1/2/d/e*(A*e+B*d))/(1/d*e)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2 /d)^(1/2)/(-a*e^2*x^4+a*d^2)^(1/2)*EllipticF(x*(1/d*e)^(1/2),I)+1/2*(A*e+B *d)/d/(1/d*e)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-a*e^2*x^4+a*d^2) ^(1/2)/e*(EllipticF(x*(1/d*e)^(1/2),I)-EllipticE(x*(1/d*e)^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\frac {{\left (B d^{2} e + A d e^{2} - {\left (B d e^{2} + A e^{3}\right )} x^{2}\right )} \sqrt {a d^{2}} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (B d^{3} - {\left (A + B\right )} d^{2} e - A d e^{2} - {\left (B d^{2} e - {\left (A + B\right )} d e^{2} - A e^{3}\right )} x^{2}\right )} \sqrt {a d^{2}} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - \sqrt {-a e^{2} x^{4} + a d^{2}} {\left (B d^{2} e + A d e^{2}\right )} x}{2 \, {\left (a d^{3} e^{3} x^{2} - a d^{4} e^{2}\right )}} \] Input:
integrate((B*x^2+A)/(-e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x, algorithm="fric as")
Output:
1/2*((B*d^2*e + A*d*e^2 - (B*d*e^2 + A*e^3)*x^2)*sqrt(a*d^2)*sqrt(e/d)*ell iptic_e(arcsin(x*sqrt(e/d)), -1) + (B*d^3 - (A + B)*d^2*e - A*d*e^2 - (B*d ^2*e - (A + B)*d*e^2 - A*e^3)*x^2)*sqrt(a*d^2)*sqrt(e/d)*elliptic_f(arcsin (x*sqrt(e/d)), -1) - sqrt(-a*e^2*x^4 + a*d^2)*(B*d^2*e + A*d*e^2)*x)/(a*d^ 3*e^3*x^2 - a*d^4*e^2)
\[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=- \int \frac {A}{- d \sqrt {a d^{2} - a e^{2} x^{4}} + e x^{2} \sqrt {a d^{2} - a e^{2} x^{4}}}\, dx - \int \frac {B x^{2}}{- d \sqrt {a d^{2} - a e^{2} x^{4}} + e x^{2} \sqrt {a d^{2} - a e^{2} x^{4}}}\, dx \] Input:
integrate((B*x**2+A)/(-e*x**2+d)/(-a*e**2*x**4+a*d**2)**(1/2),x)
Output:
-Integral(A/(-d*sqrt(a*d**2 - a*e**2*x**4) + e*x**2*sqrt(a*d**2 - a*e**2*x **4)), x) - Integral(B*x**2/(-d*sqrt(a*d**2 - a*e**2*x**4) + e*x**2*sqrt(a *d**2 - a*e**2*x**4)), x)
Exception generated. \[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x^2+A)/(-e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x, algorithm="maxi ma")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\int { -\frac {B x^{2} + A}{\sqrt {-a e^{2} x^{4} + a d^{2}} {\left (e x^{2} - d\right )}} \,d x } \] Input:
integrate((B*x^2+A)/(-e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x, algorithm="giac ")
Output:
integrate(-(B*x^2 + A)/(sqrt(-a*e^2*x^4 + a*d^2)*(e*x^2 - d)), x)
Timed out. \[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {a\,d^2-a\,e^2\,x^4}\,\left (d-e\,x^2\right )} \,d x \] Input:
int((A + B*x^2)/((a*d^2 - a*e^2*x^4)^(1/2)*(d - e*x^2)),x)
Output:
int((A + B*x^2)/((a*d^2 - a*e^2*x^4)^(1/2)*(d - e*x^2)), x)
\[ \int \frac {A+B x^2}{\left (d-e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\frac {\sqrt {a}\, \left (\sqrt {-e^{2} x^{4}+d^{2}}\, b x +2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{3} x^{6}-d \,e^{2} x^{4}-d^{2} e \,x^{2}+d^{3}}d x \right ) a \,d^{2} e -2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{3} x^{6}-d \,e^{2} x^{4}-d^{2} e \,x^{2}+d^{3}}d x \right ) a d \,e^{2} x^{2}-\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e \,x^{2}+d}d x \right ) b d +\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e \,x^{2}+d}d x \right ) b e \,x^{2}\right )}{2 a d e \left (-e \,x^{2}+d \right )} \] Input:
int((B*x^2+A)/(-e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x)
Output:
(sqrt(a)*(sqrt(d**2 - e**2*x**4)*b*x + 2*int(sqrt(d**2 - e**2*x**4)/(d**3 - d**2*e*x**2 - d*e**2*x**4 + e**3*x**6),x)*a*d**2*e - 2*int(sqrt(d**2 - e **2*x**4)/(d**3 - d**2*e*x**2 - d*e**2*x**4 + e**3*x**6),x)*a*d*e**2*x**2 - int(sqrt(d**2 - e**2*x**4)/(d - e*x**2),x)*b*d + int(sqrt(d**2 - e**2*x* *4)/(d - e*x**2),x)*b*e*x**2))/(2*a*d*e*(d - e*x**2))