Integrand size = 52, antiderivative size = 254 \[ \int \frac {-a^2 e^2+c^2 d^2 x^4}{\left (a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4\right )^{3/2}} \, dx=\frac {a e^{3/2} \left (c d^2+a e^2\right ) \sqrt {\frac {d \left (a e+c d x^2\right )}{a e \left (d+e x^2\right )}} \left (d+e x^2\right ) E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|1-\frac {c d^2}{a e^2}\right )}{\sqrt {d} \left (c d^2-a e^2\right ) \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}}-\frac {2 a c d^{3/2} e^{3/2} \sqrt {\frac {d \left (a e+c d x^2\right )}{a e \left (d+e x^2\right )}} \left (d+e x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),1-\frac {c d^2}{a e^2}\right )}{\left (c d^2-a e^2\right ) \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}} \] Output:
a*e^(3/2)*(a*e^2+c*d^2)*(d*(c*d*x^2+a*e)/a/e/(e*x^2+d))^(1/2)*(e*x^2+d)*El lipticE(e^(1/2)*x/d^(1/2)/(1+e*x^2/d)^(1/2),(1-c*d^2/a/e^2)^(1/2))/d^(1/2) /(-a*e^2+c*d^2)/(a+(c*d/e+a*e/d)*x^2+c*x^4)^(1/2)-2*a*c*d^(3/2)*e^(3/2)*(d *(c*d*x^2+a*e)/a/e/(e*x^2+d))^(1/2)*(e*x^2+d)*InverseJacobiAM(arctan(e^(1/ 2)*x/d^(1/2)),(1-c*d^2/a/e^2)^(1/2))/(-a*e^2+c*d^2)/(a+(c*d/e+a*e/d)*x^2+c *x^4)^(1/2)
Result contains complex when optimal does not.
Time = 10.37 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.03 \[ \int \frac {-a^2 e^2+c^2 d^2 x^4}{\left (a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4\right )^{3/2}} \, dx=\frac {-\sqrt {\frac {c d}{a e}} e \left (c d^2+a e^2\right ) x \left (a e+c d x^2\right )-i c d^2 \left (c d^2+a e^2\right ) \sqrt {1+\frac {c d x^2}{a e}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {c d}{a e}} x\right )|\frac {a e^2}{c d^2}\right )-i c d^2 \left (-c d^2+a e^2\right ) \sqrt {1+\frac {c d x^2}{a e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {c d}{a e}} x\right ),\frac {a e^2}{c d^2}\right )}{\sqrt {\frac {c d}{a e}} \left (-c d^2+a e^2\right ) \sqrt {\frac {\left (a e+c d x^2\right ) \left (d+e x^2\right )}{d e}}} \] Input:
Integrate[(-(a^2*e^2) + c^2*d^2*x^4)/(a + ((c*d^2 + a*e^2)*x^2)/(d*e) + c* x^4)^(3/2),x]
Output:
(-(Sqrt[(c*d)/(a*e)]*e*(c*d^2 + a*e^2)*x*(a*e + c*d*x^2)) - I*c*d^2*(c*d^2 + a*e^2)*Sqrt[1 + (c*d*x^2)/(a*e)]*Sqrt[1 + (e*x^2)/d]*EllipticE[I*ArcSin h[Sqrt[(c*d)/(a*e)]*x], (a*e^2)/(c*d^2)] - I*c*d^2*(-(c*d^2) + a*e^2)*Sqrt [1 + (c*d*x^2)/(a*e)]*Sqrt[1 + (e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[(c*d)/( a*e)]*x], (a*e^2)/(c*d^2)])/(Sqrt[(c*d)/(a*e)]*(-(c*d^2) + a*e^2)*Sqrt[((a *e + c*d*x^2)*(d + e*x^2))/(d*e)])
Leaf count is larger than twice the leaf count of optimal. \(510\) vs. \(2(254)=508\).
Time = 0.77 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2206, 27, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {c^2 d^2 x^4-a^2 e^2}{\left (\frac {x^2 \left (a e^2+c d^2\right )}{d e}+a+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {e x \left (c d x^2 \left (c^2 d^4-a^2 e^4\right )+a e \left (c^2 d^4-a^2 e^4\right )\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {d^2 e^2 \int \frac {a c \left (\left (c^2 d^4-a^2 e^4\right ) x^2+2 a d e \left (c d^2-a e^2\right )\right )}{d e \sqrt {c x^4+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+a}}dx}{a \left (c d^2-a e^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x \left (c d x^2 \left (c^2 d^4-a^2 e^4\right )+a e \left (c^2 d^4-a^2 e^4\right )\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {c d e \int \frac {\left (c^2 d^4-a^2 e^4\right ) x^2+2 a d e \left (c d^2-a e^2\right )}{\sqrt {c x^4+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+a}}dx}{\left (c d^2-a e^2\right )^2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {e x \left (c d x^2 \left (c^2 d^4-a^2 e^4\right )+a e \left (c^2 d^4-a^2 e^4\right )\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {c d e \left (\frac {\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{\sqrt {c x^4+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (c^2 d^4-a^2 e^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+a}}dx}{\sqrt {c}}\right )}{\left (c d^2-a e^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e x \left (c d x^2 \left (c^2 d^4-a^2 e^4\right )+a e \left (c^2 d^4-a^2 e^4\right )\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {c d e \left (\frac {\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^3 \int \frac {1}{\sqrt {c x^4+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+a}}dx}{\sqrt {c}}-\frac {\left (c^2 d^4-a^2 e^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+a}}dx}{\sqrt {c}}\right )}{\left (c d^2-a e^2\right )^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {e x \left (c d x^2 \left (c^2 d^4-a^2 e^4\right )+a e \left (c^2 d^4-a^2 e^4\right )\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {c d e \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^3 \sqrt {\frac {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {\frac {c d}{e}+\frac {a e}{d}}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {\left (c^2 d^4-a^2 e^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+a}}dx}{\sqrt {c}}\right )}{\left (c d^2-a e^2\right )^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {e x \left (c d x^2 \left (c^2 d^4-a^2 e^4\right )+a e \left (c^2 d^4-a^2 e^4\right )\right )}{\left (c d^2-a e^2\right )^2 \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {c d e \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^3 \sqrt {\frac {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {\frac {c d}{e}+\frac {a e}{d}}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {\left (c^2 d^4-a^2 e^4\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {\frac {c d}{e}+\frac {a e}{d}}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {x \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{\left (c d^2-a e^2\right )^2}\) |
Input:
Int[(-(a^2*e^2) + c^2*d^2*x^4)/(a + ((c*d^2 + a*e^2)*x^2)/(d*e) + c*x^4)^( 3/2),x]
Output:
(e*x*(a*e*(c^2*d^4 - a^2*e^4) + c*d*(c^2*d^4 - a^2*e^4)*x^2))/((c*d^2 - a* e^2)^2*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4]) - (c*d*e*(-(((c^2*d^4 - a^2*e^4)*(-((x*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4])/(Sqrt[a] + Sqrt[ c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + ((c*d)/e + (a*e)/d)* x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^( 1/4)], (2 - ((c*d)/e + (a*e)/d)/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + ( (c*d)/e + (a*e)/d)*x^2 + c*x^4])))/Sqrt[c]) + (a^(1/4)*(Sqrt[c]*d - Sqrt[a ]*e)*(Sqrt[c]*d + Sqrt[a]*e)^3*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^( 1/4)*x)/a^(1/4)], (2 - ((c*d)/e + (a*e)/d)/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/ 4)*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4])))/(c*d^2 - a*e^2)^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Time = 3.65 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.68
| method | result | size |
| elliptic | \(-\frac {\left (c d \,x^{2}+a e \right ) e x \left (a \,e^{2}+c \,d^{2}\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\frac {\left (x^{2}+\frac {d}{e}\right ) \left (c d \,x^{2}+a e \right )}{d}}}+\frac {\left (-a \,e^{2}+\frac {a \,e^{2} \left (a \,e^{2}+c \,d^{2}\right )}{a \,e^{2}-c \,d^{2}}\right ) \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}-\frac {2 \left (a \,e^{2}+c \,d^{2}\right ) e c d a \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )\right )}{\left (a \,e^{2}-c \,d^{2}\right ) \sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}\, \left (\frac {c d}{e}+\frac {a e}{d}+\frac {a \,e^{2}-c \,d^{2}}{d e}\right )}\) | \(426\) |
| default | \(\text {Expression too large to display}\) | \(1044\) |
Input:
int((c^2*d^2*x^4-a^2*e^2)/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(3/2),x,method=_ RETURNVERBOSE)
Output:
-(c*d*x^2+a*e)*e/(a*e^2-c*d^2)*x*(a*e^2+c*d^2)/((x^2+d/e)*(c*d*x^2+a*e)/d) ^(1/2)+(-a*e^2+a*e^2/(a*e^2-c*d^2)*(a*e^2+c*d^2))/(-c*d/a/e)^(1/2)*(1+1/a* x^2*d/e*c)^(1/2)*(1+e*x^2/d)^(1/2)/(a+x^2/d*e*a+x^2*d/e*c+c*x^4)^(1/2)*Ell ipticF(x*(-c*d/a/e)^(1/2),(-1+(c*d/e+1/d*a*e)*e/d/c)^(1/2))-2*(a*e^2+c*d^2 )*e*c*d/(a*e^2-c*d^2)*a/(-c*d/a/e)^(1/2)*(1+1/a*x^2*d/e*c)^(1/2)*(1+e*x^2/ d)^(1/2)/(a+x^2/d*e*a+x^2*d/e*c+c*x^4)^(1/2)/(c*d/e+1/d*a*e+(a*e^2-c*d^2)/ d/e)*(EllipticF(x*(-c*d/a/e)^(1/2),(-1+(c*d/e+1/d*a*e)*e/d/c)^(1/2))-Ellip ticE(x*(-c*d/a/e)^(1/2),(-1+(c*d/e+1/d*a*e)*e/d/c)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.11 \[ \int \frac {-a^2 e^2+c^2 d^2 x^4}{\left (a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4\right )^{3/2}} \, dx=-\frac {{\left (c^{2} d^{5} + a c d^{3} e^{2} + {\left (c^{2} d^{4} e + a c d^{2} e^{3}\right )} x^{2}\right )} \sqrt {a} \sqrt {-\frac {c d}{a e}} E(\arcsin \left (x \sqrt {-\frac {c d}{a e}}\right )\,|\,\frac {a e^{2}}{c d^{2}}) - {\left (c^{2} d^{5} + a c d^{3} e^{2} + 2 \, a^{2} d^{2} e^{3} + {\left (c^{2} d^{4} e + a c d^{2} e^{3} + 2 \, a^{2} d e^{4}\right )} x^{2}\right )} \sqrt {a} \sqrt {-\frac {c d}{a e}} F(\arcsin \left (x \sqrt {-\frac {c d}{a e}}\right )\,|\,\frac {a e^{2}}{c d^{2}}) - {\left (a c d^{3} e^{2} + a^{2} d e^{4}\right )} x \sqrt {\frac {c d e x^{4} + a d e + {\left (c d^{2} + a e^{2}\right )} x^{2}}{d e}}}{a c d^{3} - a^{2} d e^{2} + {\left (a c d^{2} e - a^{2} e^{3}\right )} x^{2}} \] Input:
integrate((c^2*d^2*x^4-a^2*e^2)/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(3/2),x, a lgorithm="fricas")
Output:
-((c^2*d^5 + a*c*d^3*e^2 + (c^2*d^4*e + a*c*d^2*e^3)*x^2)*sqrt(a)*sqrt(-c* d/(a*e))*elliptic_e(arcsin(x*sqrt(-c*d/(a*e))), a*e^2/(c*d^2)) - (c^2*d^5 + a*c*d^3*e^2 + 2*a^2*d^2*e^3 + (c^2*d^4*e + a*c*d^2*e^3 + 2*a^2*d*e^4)*x^ 2)*sqrt(a)*sqrt(-c*d/(a*e))*elliptic_f(arcsin(x*sqrt(-c*d/(a*e))), a*e^2/( c*d^2)) - (a*c*d^3*e^2 + a^2*d*e^4)*x*sqrt((c*d*e*x^4 + a*d*e + (c*d^2 + a *e^2)*x^2)/(d*e)))/(a*c*d^3 - a^2*d*e^2 + (a*c*d^2*e - a^2*e^3)*x^2)
\[ \int \frac {-a^2 e^2+c^2 d^2 x^4}{\left (a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4\right )^{3/2}} \, dx=\int \frac {\left (- a e + c d x^{2}\right ) \left (a e + c d x^{2}\right )}{\left (a + \frac {a e x^{2}}{d} + \frac {c d x^{2}}{e} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((c**2*d**2*x**4-a**2*e**2)/(a+(a*e**2+c*d**2)*x**2/d/e+c*x**4)** (3/2),x)
Output:
Integral((-a*e + c*d*x**2)*(a*e + c*d*x**2)/(a + a*e*x**2/d + c*d*x**2/e + c*x**4)**(3/2), x)
\[ \int \frac {-a^2 e^2+c^2 d^2 x^4}{\left (a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4\right )^{3/2}} \, dx=\int { \frac {c^{2} d^{2} x^{4} - a^{2} e^{2}}{{\left (c x^{4} + a + \frac {{\left (c d^{2} + a e^{2}\right )} x^{2}}{d e}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c^2*d^2*x^4-a^2*e^2)/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(3/2),x, a lgorithm="maxima")
Output:
integrate((c^2*d^2*x^4 - a^2*e^2)/(c*x^4 + a + (c*d^2 + a*e^2)*x^2/(d*e))^ (3/2), x)
\[ \int \frac {-a^2 e^2+c^2 d^2 x^4}{\left (a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4\right )^{3/2}} \, dx=\int { \frac {c^{2} d^{2} x^{4} - a^{2} e^{2}}{{\left (c x^{4} + a + \frac {{\left (c d^{2} + a e^{2}\right )} x^{2}}{d e}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c^2*d^2*x^4-a^2*e^2)/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(3/2),x, a lgorithm="giac")
Output:
integrate((c^2*d^2*x^4 - a^2*e^2)/(c*x^4 + a + (c*d^2 + a*e^2)*x^2/(d*e))^ (3/2), x)
Timed out. \[ \int \frac {-a^2 e^2+c^2 d^2 x^4}{\left (a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4\right )^{3/2}} \, dx=\int -\frac {a^2\,e^2-c^2\,d^2\,x^4}{{\left (a+c\,x^4+\frac {x^2\,\left (c\,d^2+a\,e^2\right )}{d\,e}\right )}^{3/2}} \,d x \] Input:
int(-(a^2*e^2 - c^2*d^2*x^4)/(a + c*x^4 + (x^2*(a*e^2 + c*d^2))/(d*e))^(3/ 2),x)
Output:
int(-(a^2*e^2 - c^2*d^2*x^4)/(a + c*x^4 + (x^2*(a*e^2 + c*d^2))/(d*e))^(3/ 2), x)
\[ \int \frac {-a^2 e^2+c^2 d^2 x^4}{\left (a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4\right )^{3/2}} \, dx=\sqrt {e}\, \sqrt {d}\, d e \left (\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c d \,x^{2}+a e}\, x^{2}}{c d \,e^{2} x^{6}+a \,e^{3} x^{4}+2 c \,d^{2} e \,x^{4}+2 a d \,e^{2} x^{2}+c \,d^{3} x^{2}+a \,d^{2} e}d x \right ) c d -\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c d \,x^{2}+a e}}{c d \,e^{2} x^{6}+a \,e^{3} x^{4}+2 c \,d^{2} e \,x^{4}+2 a d \,e^{2} x^{2}+c \,d^{3} x^{2}+a \,d^{2} e}d x \right ) a e \right ) \] Input:
int((c^2*d^2*x^4-a^2*e^2)/(a+(a*e^2+c*d^2)*x^2/d/e+c*x^4)^(3/2),x)
Output:
sqrt(e)*sqrt(d)*d*e*(int((sqrt(d + e*x**2)*sqrt(a*e + c*d*x**2)*x**2)/(a*d **2*e + 2*a*d*e**2*x**2 + a*e**3*x**4 + c*d**3*x**2 + 2*c*d**2*e*x**4 + c* d*e**2*x**6),x)*c*d - int((sqrt(d + e*x**2)*sqrt(a*e + c*d*x**2))/(a*d**2* e + 2*a*d*e**2*x**2 + a*e**3*x**4 + c*d**3*x**2 + 2*c*d**2*e*x**4 + c*d*e* *2*x**6),x)*a*e)