Integrand size = 34, antiderivative size = 306 \[ \int \frac {x^4}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}-\frac {2 b d \sqrt {c d-b e} \sqrt {1+\frac {e x^2}{d}} \sqrt {1-\frac {c e x^2}{c d-b e}} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )|-1+\frac {b e}{c d}\right )}{3 c^{3/2} e^{3/2} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {d \sqrt {c d-b e} (c d+b e) \sqrt {1+\frac {e x^2}{d}} \sqrt {1-\frac {c e x^2}{c d-b e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),-1+\frac {b e}{c d}\right )}{3 c^{3/2} e^{5/2} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}} \] Output:
1/3*x*(-d*(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2)/c-2/3*b*d*(-b*e+c*d)^(1/2)*(1+ e*x^2/d)^(1/2)*(1-c*e*x^2/(-b*e+c*d))^(1/2)*EllipticE(c^(1/2)*e^(1/2)*x/(- b*e+c*d)^(1/2),(-1+b*e/c/d)^(1/2))/c^(3/2)/e^(3/2)/(-d*(-b*e+c*d)/e^2+b*x^ 2+c*x^4)^(1/2)+1/3*d*(-b*e+c*d)^(1/2)*(b*e+c*d)*(1+e*x^2/d)^(1/2)*(1-c*e*x ^2/(-b*e+c*d))^(1/2)*EllipticF(c^(1/2)*e^(1/2)*x/(-b*e+c*d)^(1/2),(-1+b*e/ c/d)^(1/2))/c^(3/2)/e^(5/2)/(-d*(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2)
Result contains complex when optimal does not.
Time = 3.26 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {-c \sqrt {\frac {e}{d}} x \left (d+e x^2\right ) \left (-b e+c \left (d-e x^2\right )\right )+2 i b e (-c d+b e) \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|\frac {c d}{-c d+b e}\right )-i \left (c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),\frac {c d}{-c d+b e}\right )}{3 c^2 d^2 \left (\frac {e}{d}\right )^{5/2} \sqrt {\frac {\left (d+e x^2\right ) \left (-c d+b e+c e x^2\right )}{e^2}}} \] Input:
Integrate[x^4/Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x^2 + c*x^4],x]
Output:
(-(c*Sqrt[e/d]*x*(d + e*x^2)*(-(b*e) + c*(d - e*x^2))) + (2*I)*b*e*(-(c*d) + b*e)*Sqrt[(-(c*d) + b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + (e*x^2)/d]* EllipticE[I*ArcSinh[Sqrt[e/d]*x], (c*d)/(-(c*d) + b*e)] - I*(c^2*d^2 - 3*b *c*d*e + 2*b^2*e^2)*Sqrt[(-(c*d) + b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + (e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[e/d]*x], (c*d)/(-(c*d) + b*e)])/(3*c^ 2*d^2*(e/d)^(5/2)*Sqrt[((d + e*x^2)*(-(c*d) + b*e + c*e*x^2))/e^2])
Time = 0.74 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1442, 25, 27, 1514, 399, 321, 327}
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.
| \(\displaystyle \int \frac {x^4}{\sqrt {\frac {b d e-c d^2}{e^2}+b x^2+c x^4}} \, dx\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}-\frac {\int -\frac {d (c d-b e)-2 b e^2 x^2}{e^2 \sqrt {c x^4+b x^2-\frac {d (c d-b e)}{e^2}}}dx}{3 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d (c d-b e)-2 b e^2 x^2}{e^2 \sqrt {c x^4+b x^2-\frac {d (c d-b e)}{e^2}}}dx}{3 c}+\frac {x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {d (c d-b e)-2 b e^2 x^2}{\sqrt {c x^4+b x^2-\frac {d (c d-b e)}{e^2}}}dx}{3 c e^2}+\frac {x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 1514 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \int \frac {d (c d-b e)-2 b e^2 x^2}{\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}}}dx}{3 c e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (d (b e+c d) \int \frac {1}{\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}}}dx-2 b d e \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {c e x^2}{c d-b e}}}dx\right )}{3 c e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (\frac {d \sqrt {c d-b e} (b e+c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),\frac {b e}{c d}-1\right )}{\sqrt {c} \sqrt {e}}-2 b d e \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {c e x^2}{c d-b e}}}dx\right )}{3 c e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (\frac {d \sqrt {c d-b e} (b e+c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),\frac {b e}{c d}-1\right )}{\sqrt {c} \sqrt {e}}-\frac {2 b d \sqrt {e} \sqrt {c d-b e} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )|\frac {b e}{c d}-1\right )}{\sqrt {c}}\right )}{3 c e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}\) |
Input:
Int[x^4/Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x^2 + c*x^4],x]
Output:
(x*Sqrt[-((d*(c*d - b*e))/e^2) + b*x^2 + c*x^4])/(3*c) + (Sqrt[1 + (e*x^2) /d]*Sqrt[1 - (c*e*x^2)/(c*d - b*e)]*((-2*b*d*Sqrt[e]*Sqrt[c*d - b*e]*Ellip ticE[ArcSin[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]], -1 + (b*e)/(c*d)])/Sqrt[ c] + (d*Sqrt[c*d - b*e]*(c*d + b*e)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[e]*x)/S qrt[c*d - b*e]], -1 + (b*e)/(c*d)])/(Sqrt[c]*Sqrt[e])))/(3*c*e^2*Sqrt[-((d *(c*d - b*e))/e^2) + b*x^2 + c*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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
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[((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[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt [1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4]) Int[(d + e*x^2)/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
Time = 16.48 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(\frac {x \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{3 c}-\frac {\left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{3 c \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}+\frac {4 b \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{3 c \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\) | \(346\) |
| elliptic | \(\frac {x \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{3 c}-\frac {\left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{3 c \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}+\frac {4 b \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{3 c \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\) | \(346\) |
| risch | \(\frac {x \left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{3 c \,e^{2} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{e^{2}}}}-\frac {\left (\frac {b d e \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}}-\frac {c \,d^{2} \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}}-\frac {4 b \,e^{2} \left (b d e -c \,d^{2}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}\, \left (b \,e^{2}+e \left (b e -2 c d \right )\right )}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}}{3 c \,e^{2} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{e^{2}}}}\) | \(526\) |
Input:
int(x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3/c*x*(c*x^4+b*x^2+b*d/e-c*d^2/e^2)^(1/2)-1/3/c*(b*d/e-c*d^2/e^2)/(-c*e/ (b*e-c*d))^(1/2)*(1+c*e/(b*e-c*d)*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(c*x^4+b*x^ 2+b*d/e-c*d^2/e^2)^(1/2)*EllipticF(x*(-c*e/(b*e-c*d))^(1/2),(-1+b*e/c/d)^( 1/2))+4/3/c*b*(b*d/e-c*d^2/e^2)/(-c*e/(b*e-c*d))^(1/2)*(1+c*e/(b*e-c*d)*x^ 2)^(1/2)*(1+e*x^2/d)^(1/2)/(c*x^4+b*x^2+b*d/e-c*d^2/e^2)^(1/2)/(b+(b*e-2*c *d)/e)*(EllipticF(x*(-c*e/(b*e-c*d))^(1/2),(-1+b*e/c/d)^(1/2))-EllipticE(x *(-c*e/(b*e-c*d))^(1/2),(-1+b*e/c/d)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.56 \[ \int \frac {x^4}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {2 \, \sqrt {c e^{2}} b d x \sqrt {-\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{x}\right )\,|\,-\frac {c d - b e}{c d}) - \sqrt {c e^{2}} {\left ({\left (2 \, b - c\right )} d + b e\right )} x \sqrt {-\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{x}\right )\,|\,-\frac {c d - b e}{c d}) + {\left (c e^{2} x^{2} - 2 \, b e^{2}\right )} \sqrt {\frac {c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e}{e^{2}}}}{3 \, c^{2} e^{2} x} \] Input:
integrate(x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="fricas")
Output:
1/3*(2*sqrt(c*e^2)*b*d*x*sqrt(-d/e)*elliptic_e(arcsin(sqrt(-d/e)/x), -(c*d - b*e)/(c*d)) - sqrt(c*e^2)*((2*b - c)*d + b*e)*x*sqrt(-d/e)*elliptic_f(a rcsin(sqrt(-d/e)/x), -(c*d - b*e)/(c*d)) + (c*e^2*x^2 - 2*b*e^2)*sqrt((c*e ^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e)/e^2))/(c^2*e^2*x)
\[ \int \frac {x^4}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int \frac {x^{4}}{\sqrt {\left (\frac {d}{e} + x^{2}\right ) \left (b - \frac {c d}{e} + c x^{2}\right )}}\, dx \] Input:
integrate(x**4/((b*d*e-c*d**2)/e**2+b*x**2+c*x**4)**(1/2),x)
Output:
Integral(x**4/sqrt((d/e + x**2)*(b - c*d/e + c*x**2)), x)
\[ \int \frac {x^4}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int { \frac {x^{4}}{\sqrt {c x^{4} + b x^{2} - \frac {c d^{2} - b d e}{e^{2}}}} \,d x } \] Input:
integrate(x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="maxima")
Output:
integrate(x^4/sqrt(c*x^4 + b*x^2 - (c*d^2 - b*d*e)/e^2), x)
\[ \int \frac {x^4}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int { \frac {x^{4}}{\sqrt {c x^{4} + b x^{2} - \frac {c d^{2} - b d e}{e^{2}}}} \,d x } \] Input:
integrate(x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="giac")
Output:
integrate(x^4/sqrt(c*x^4 + b*x^2 - (c*d^2 - b*d*e)/e^2), x)
Timed out. \[ \int \frac {x^4}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int \frac {x^4}{\sqrt {b\,x^2-\frac {c\,d^2-b\,d\,e}{e^2}+c\,x^4}} \,d x \] Input:
int(x^4/(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(1/2),x)
Output:
int(x^4/(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(1/2), x)
\[ \int \frac {x^4}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}\, x -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}\, x^{2}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) b \,e^{2}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) b d e +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) c \,d^{2}}{3 c e} \] Input:
int(x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x)
Output:
(sqrt(d + e*x**2)*sqrt(b*e - c*d + c*e*x**2)*x - 2*int((sqrt(d + e*x**2)*s qrt(b*e - c*d + c*e*x**2)*x**2)/(b*d*e + b*e**2*x**2 - c*d**2 + c*e**2*x** 4),x)*b*e**2 - int((sqrt(d + e*x**2)*sqrt(b*e - c*d + c*e*x**2))/(b*d*e + b*e**2*x**2 - c*d**2 + c*e**2*x**4),x)*b*d*e + int((sqrt(d + e*x**2)*sqrt( b*e - c*d + c*e*x**2))/(b*d*e + b*e**2*x**2 - c*d**2 + c*e**2*x**4),x)*c*d **2)/(3*c*e)