Integrand size = 29, antiderivative size = 419 \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {1}{3 a d x^3 \sqrt {d+e x^2}}+\frac {3 b d+4 a e}{3 a^2 d^2 x \sqrt {d+e x^2}}+\frac {2 e (3 b d+4 a e) x}{3 a^2 d^3 \sqrt {d+e x^2}}-\frac {e \left (b c d-b^2 e+a c e\right ) x}{a^2 d \left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x^2}}+\frac {2 c^2 \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2}}+\frac {2 c^2 \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b+\sqrt {b^2-4 a c}} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2}} \]
-1/3/a/d/x^3/(e*x^2+d)^(1/2)+1/3*(4*a*e+3*b*d)/a^2/d^2/x/(e*x^2+d)^(1/2)+2 /3*e*(4*a*e+3*b*d)*x/a^2/d^3/(e*x^2+d)^(1/2)-e*(a*c*e-b^2*e+b*c*d)*x/a^2/d /(c*d^2+e*(a*e-b*d))/(e*x^2+d)^(1/2)+2*c^2*arctan(x*(2*c*d-e*(b-(-4*a*c+b^ 2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b+(-2*a*c+ b^2)/(-4*a*c+b^2)^(1/2))/a^2/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(3/2)/(b-(-4 *a*c+b^2)^(1/2))^(1/2)+2*c^2*arctan(x*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/ 2)/(e*x^2+d)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(b+(2*a*c-b^2)/(-4*a*c+b^ 2)^(1/2))/a^2/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(3/2)/(b+(-4*a*c+b^2)^(1/2) )^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 17.82 (sec) , antiderivative size = 2218, normalized size of antiderivative = 5.29 \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]
(b*(d + 2*e*x^2))/(a^2*d^2*x*Sqrt[d + e*x^2]) - (d^2 - 4*d*e*x^2 - 8*e^2*x ^4)/(3*a*d^3*x^3*Sqrt[d + e*x^2]) + ((b*c + (c*(b^2 - 2*a*c))/Sqrt[b^2 - 4 *a*c])*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4* a*c])*e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (3 0*e*x^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c]) *e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*A rcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c *d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^ 2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b + Sq rt[b^2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x ^4*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[ b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) + 4* (-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sq rt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4 *e*x^2*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4 *a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(...
Time = 2.45 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.30, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1624, 245, 245, 208, 2246, 2009}
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.
| \(\displaystyle \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1624 |
| \(\displaystyle \frac {e^2 \int \frac {1}{x^4 \left (e x^2+d\right )^{3/2}}dx}{a e^2-b d e+c d^2}+\frac {\int \frac {-c e x^2+c d-b e}{x^4 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\int \frac {-c e x^2+c d-b e}{x^4 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}+\frac {e^2 \left (-\frac {4 e \int \frac {1}{x^2 \left (e x^2+d\right )^{3/2}}dx}{3 d}-\frac {1}{3 d x^3 \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\int \frac {-c e x^2+c d-b e}{x^4 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}+\frac {e^2 \left (-\frac {4 e \left (-\frac {2 e \int \frac {1}{\left (e x^2+d\right )^{3/2}}dx}{d}-\frac {1}{d x \sqrt {d+e x^2}}\right )}{3 d}-\frac {1}{3 d x^3 \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\int \frac {-c e x^2+c d-b e}{x^4 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}+\frac {e^2 \left (-\frac {4 e \left (-\frac {2 e x}{d^2 \sqrt {d+e x^2}}-\frac {1}{d x \sqrt {d+e x^2}}\right )}{3 d}-\frac {1}{3 d x^3 \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 2246 |
| \(\displaystyle \frac {\int \left (\frac {c d-b e}{a x^4 \sqrt {e x^2+d}}+\frac {e b^2-c d b-a c e}{a^2 x^2 \sqrt {e x^2+d}}+\frac {-e b^3+c d b^2+2 a c e b+c \left (-e b^2+c d b+a c e\right ) x^2-a c^2 d}{a^2 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}\right )dx}{a e^2-b d e+c d^2}+\frac {e^2 \left (-\frac {4 e \left (-\frac {2 e x}{d^2 \sqrt {d+e x^2}}-\frac {1}{d x \sqrt {d+e x^2}}\right )}{3 d}-\frac {1}{3 d x^3 \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c \left (\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {c \left (-\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {d+e x^2} \left (a c e+b^2 (-e)+b c d\right )}{a^2 d x}+\frac {2 e \sqrt {d+e x^2} (c d-b e)}{3 a d^2 x}-\frac {\sqrt {d+e x^2} (c d-b e)}{3 a d x^3}}{a e^2-b d e+c d^2}+\frac {e^2 \left (-\frac {4 e \left (-\frac {2 e x}{d^2 \sqrt {d+e x^2}}-\frac {1}{d x \sqrt {d+e x^2}}\right )}{3 d}-\frac {1}{3 d x^3 \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}\) |
(e^2*(-1/3*1/(d*x^3*Sqrt[d + e*x^2]) - (4*e*(-(1/(d*x*Sqrt[d + e*x^2])) - (2*e*x)/(d^2*Sqrt[d + e*x^2])))/(3*d)))/(c*d^2 - b*d*e + a*e^2) + (-1/3*(( c*d - b*e)*Sqrt[d + e*x^2])/(a*d*x^3) + (2*e*(c*d - b*e)*Sqrt[d + e*x^2])/ (3*a*d^2*x) + ((b*c*d - b^2*e + a*c*e)*Sqrt[d + e*x^2])/(a^2*d*x) + (c*(b* c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt [b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(a^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2* c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (c*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])] )/(a^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e] ))/(c*d^2 - b*d*e + a*e^2)
3.4.99.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Simp[e^2/(c*d^2 - b*d*e + a*e^2) Int[(f*x)^m* (d + e*x^2)^q, x], x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(f*x)^m*(d + e *x^2)^(q + 1)*(Simp[c*d - b*e - c*e*x^2, x]/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && LtQ[q, -1]
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) ^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x ] && PolyQ[Px, x] && IntegerQ[p]
Time = 1.32 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.46
| method | result | size |
| pseudoelliptic | \(-\frac {-3 \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {2}\, d^{3} \left (\left (c \left (b e -\frac {c d}{2}\right ) a -\frac {b^{2} \left (b e -c d \right )}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (a^{2} c^{2} e +\left (-2 b^{2} c e +\frac {3}{2} b \,c^{2} d \right ) a +\frac {b^{3} \left (b e -c d \right )}{2}\right )\right ) x^{3} \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (-3 \sqrt {2}\, d^{3} \left (\left (\left (-e b c +\frac {1}{2} c^{2} d \right ) a +\frac {b^{2} \left (b e -c d \right )}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (a^{2} c^{2} e +\left (-2 b^{2} c e +\frac {3}{2} b \,c^{2} d \right ) a +\frac {b^{3} \left (b e -c d \right )}{2}\right )\right ) x^{3} \sqrt {e \,x^{2}+d}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (e^{2} \left (-8 e^{2} x^{4}-4 e d \,x^{2}+d^{2}\right ) a^{2}-\left (-c \,d^{2}+e \left (5 c \,x^{2}+b \right ) d -2 b \,e^{2} x^{2}\right ) \left (e \,x^{2}+d \right ) d a +3 b \,d^{2} x^{2} \left (e \,x^{2}+d \right ) \left (b e -c d \right )\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{3 \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, a^{2} x^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right ) d^{3}}\) | \(610\) |
| default | \(\frac {-\frac {1}{3 d \,x^{3} \sqrt {e \,x^{2}+d}}-\frac {4 e \left (-\frac {1}{d x \sqrt {e \,x^{2}+d}}-\frac {2 e x}{d^{2} \sqrt {e \,x^{2}+d}}\right )}{3 d}}{a}-\frac {b \left (-\frac {1}{d x \sqrt {e \,x^{2}+d}}-\frac {2 e x}{d^{2} \sqrt {e \,x^{2}+d}}\right )}{a^{2}}+\frac {\sqrt {2}\, d \left (\left (-\frac {d c \left (a c -b^{2}\right )}{2}+b e \left (a c -\frac {b^{2}}{2}\right )\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (\frac {\left (3 a b \,c^{2}-b^{3} c \right ) d}{2}+\left (-2 a \,b^{2} c +\frac {1}{2} b^{4}+a^{2} c^{2}\right ) e \right )\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\left (\sqrt {2}\, d \left (\left (\frac {d c \left (a c -b^{2}\right )}{2}-b e \left (a c -\frac {b^{2}}{2}\right )\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (\frac {\left (3 a b \,c^{2}-b^{3} c \right ) d}{2}+\left (-2 a \,b^{2} c +\frac {1}{2} b^{4}+a^{2} c^{2}\right ) e \right )\right ) \sqrt {e \,x^{2}+d}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-e \left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, x \right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{a^{2} \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(629\) |
| risch | \(-\frac {\sqrt {e \,x^{2}+d}\, \left (-5 a e \,x^{2}-3 b d \,x^{2}+d a \right )}{3 d^{3} a^{2} x^{3}}+\frac {\frac {e^{3} a^{2} \sqrt {\left (x -\frac {\sqrt {-e d}}{e}\right )^{2} e +2 \sqrt {-e d}\, \left (x -\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x -\frac {\sqrt {-e d}}{e}\right )}+\frac {e^{3} a^{2} \sqrt {\left (x +\frac {\sqrt {-e d}}{e}\right )^{2} e -2 \sqrt {-e d}\, \left (x +\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {\sqrt {-e d}}{e}\right )}+\frac {d^{2} \sqrt {2}\, \left (\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (c \left (b e -\frac {c d}{2}\right ) a -\frac {b^{2} \left (b e -c d \right )}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (a^{2} c^{2} e +\left (-2 b^{2} c e +\frac {3}{2} b \,c^{2} d \right ) a +\frac {b^{3} \left (b e -c d \right )}{2}\right )\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right ) \left (\left (\left (-e b c +\frac {1}{2} c^{2} d \right ) a +\frac {b^{2} \left (b e -c d \right )}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (a^{2} c^{2} e +\left (-2 b^{2} c e +\frac {3}{2} b \,c^{2} d \right ) a +\frac {b^{3} \left (b e -c d \right )}{2}\right )\right )\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}}{a^{2} d^{2}}\) | \(637\) |
-1/3*(-3*((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*2^(1/2)*d^3*( (c*(b*e-1/2*c*d)*a-1/2*b^2*(b*e-c*d))*(-4*d^2*(a*c-1/4*b^2))^(1/2)+d*(a^2* c^2*e+(-2*b^2*c*e+3/2*b*c^2*d)*a+1/2*b^3*(b*e-c*d)))*x^3*(e*x^2+d)^(1/2)*a rctanh(a/x*(e*x^2+d)^(1/2)*2^(1/2)/((2*a*e-b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2 ))*a)^(1/2))+((2*a*e-b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*(-3*2^(1/2 )*d^3*(((-e*b*c+1/2*c^2*d)*a+1/2*b^2*(b*e-c*d))*(-4*d^2*(a*c-1/4*b^2))^(1/ 2)+d*(a^2*c^2*e+(-2*b^2*c*e+3/2*b*c^2*d)*a+1/2*b^3*(b*e-c*d)))*x^3*(e*x^2+ d)^(1/2)*arctan(a/x*(e*x^2+d)^(1/2)*2^(1/2)/((-2*a*e+b*d+(-4*d^2*(a*c-1/4* b^2))^(1/2))*a)^(1/2))+((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2) *(e^2*(-8*e^2*x^4-4*d*e*x^2+d^2)*a^2-(-c*d^2+e*(5*c*x^2+b)*d-2*b*e^2*x^2)* (e*x^2+d)*d*a+3*b*d^2*x^2*(e*x^2+d)*(b*e-c*d))*(-4*d^2*(a*c-1/4*b^2))^(1/2 )))/((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)/(e*x^2+d)^(1/2)/(- 4*d^2*(a*c-1/4*b^2))^(1/2)/((2*a*e-b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1 /2)/a^2/x^3/(a*e^2-b*d*e+c*d^2)/d^3
Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^{4} \left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^4\,{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]