\(\int \frac {1}{x^4 (d+e x^2)^{3/2} (a+b x^2+c x^4)} \, dx\) [399]

Optimal result

Integrand size = 29, antiderivative size = 419 \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {1}{3 a d x^3 \sqrt {d+e x^2}}+\frac {3 b d+4 a e}{3 a^2 d^2 x \sqrt {d+e x^2}}+\frac {2 e (3 b d+4 a e) x}{3 a^2 d^3 \sqrt {d+e x^2}}-\frac {e \left (b c d-b^2 e+a c e\right ) x}{a^2 d \left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x^2}}+\frac {2 c^2 \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2}}+\frac {2 c^2 \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b+\sqrt {b^2-4 a c}} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 3.37 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.54, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1315, 277, 197, 6860, 270, 1706, 385, 211} \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\frac {c \left (\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac {c \left (-\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {d+e x^2} \left (a c e+b^2 (-e)+b c d\right )}{a^2 d x \left (a e^2-b d e+c d^2\right )}+\frac {2 e \sqrt {d+e x^2} (c d-b e)}{3 a d^2 x \left (a e^2-b d e+c d^2\right )}-\frac {e^2}{3 d x^3 \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x^2} (c d-b e)}{3 a d x^3 \left (a e^2-b d e+c d^2\right )}+\frac {4 e^3}{3 d^2 x \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}+\frac {8 e^4 x}{3 d^3 \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )} \]

[In]

Int[1/(x^4*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]

[Out]

-1/3*e^2/(d*(c*d^2 - b*d*e + a*e^2)*x^3*Sqrt[d + e*x^2]) + (4*e^3)/(3*d^2*(c*d^2 - b*d*e + a*e^2)*x*Sqrt[d + e
*x^2]) + (8*e^4*x)/(3*d^3*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x^2]) - ((c*d - b*e)*Sqrt[d + e*x^2])/(3*a*d*(c*d
^2 - b*d*e + a*e^2)*x^3) + (2*e*(c*d - b*e)*Sqrt[d + e*x^2])/(3*a*d^2*(c*d^2 - b*d*e + a*e^2)*x) + ((b*c*d - b
^2*e + a*c*e)*Sqrt[d + e*x^2])/(a^2*d*(c*d^2 - b*d*e + a*e^2)*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 - Sqr
t[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)) + (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*Sq
rt[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))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 1315

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[e^
2/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^m*(d + e*x^2)^q, x], x] + Dist[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]

Rule 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {c d-b e-c e x^2}{x^4 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{c d^2-b d e+a e^2}+\frac {e^2 \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2}} \, dx}{c d^2-b d e+a e^2} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {\int \left (\frac {c d-b e}{a x^4 \sqrt {d+e x^2}}+\frac {-b c d+b^2 e-a c e}{a^2 x^2 \sqrt {d+e x^2}}+\frac {b^2 c d-a c^2 d-b^3 e+2 a b c e+c \left (b c d-b^2 e+a c e\right ) x^2}{a^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )}\right ) \, dx}{c d^2-b d e+a e^2}-\frac {\left (4 e^3\right ) \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2}} \, dx}{3 d \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {\int \frac {b^2 c d-a c^2 d-b^3 e+2 a b c e+c \left (b c d-b^2 e+a c e\right ) x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (8 e^4\right ) \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d^2 \left (c d^2-b d e+a e^2\right )}+\frac {(c d-b e) \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx}{a \left (c d^2-b d e+a e^2\right )}-\frac {\left (b c d-b^2 e+a c e\right ) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{a^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {8 e^4 x}{3 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{3 a d \left (c d^2-b d e+a e^2\right ) x^3}+\frac {\left (b c d-b^2 e+a c e\right ) \sqrt {d+e x^2}}{a^2 d \left (c d^2-b d e+a e^2\right ) x}+\frac {\int \left (\frac {c \left (b c d-b^2 e+a c e\right )-\frac {c \left (-b^2 c d+2 a c^2 d+b^3 e-3 a b c e\right )}{\sqrt {b^2-4 a c}}}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}+\frac {c \left (b c d-b^2 e+a c e\right )+\frac {c \left (-b^2 c d+2 a c^2 d+b^3 e-3 a b c e\right )}{\sqrt {b^2-4 a c}}}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}\right ) \, dx}{a^2 \left (c d^2-b d e+a e^2\right )}-\frac {(2 e (c d-b e)) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{3 a d \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {8 e^4 x}{3 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{3 a d \left (c d^2-b d e+a e^2\right ) x^3}+\frac {2 e (c d-b e) \sqrt {d+e x^2}}{3 a d^2 \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (b c d-b^2 e+a c e\right ) \sqrt {d+e x^2}}{a^2 d \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (c \left (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{a^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{a^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {8 e^4 x}{3 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{3 a d \left (c d^2-b d e+a e^2\right ) x^3}+\frac {2 e (c d-b e) \sqrt {d+e x^2}}{3 a d^2 \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (b c d-b^2 e+a c e\right ) \sqrt {d+e x^2}}{a^2 d \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (c \left (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{a^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{a^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {8 e^4 x}{3 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{3 a d \left (c d^2-b d e+a e^2\right ) x^3}+\frac {2 e (c d-b e) \sqrt {d+e x^2}}{3 a d^2 \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (b c d-b^2 e+a c e\right ) \sqrt {d+e x^2}}{a^2 d \left (c d^2-b d e+a e^2\right ) x}+\frac {c \left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\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}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {c \left (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\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}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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 \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]

[In]

Integrate[1/(x^4*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x]

[Out]

(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))] + (30*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*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)))]] - (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 - (4
5*(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 + S
qrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b + Sqrt[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 + Sqrt[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))/(d*(-b + Sqrt[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)))])/d))/(15*a^2*(b - S
qrt[b^2 - 4*a*c])*d*(-(((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))))^(3/
2)*(1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b
+ Sqrt[b^2 - 4*a*c] - 2*c*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))] +
(30*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*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))]] - (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 + Sqrt[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 + S
qrt[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 + Sqrt[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)*Sqr
t[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[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))])/d))/(15*a^2*(b + Sqrt[b^2 - 4*
a*c])*d*(((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)))^(3/2)*(1 + (2*c*x^2)
/(b + Sqrt[b^2 - 4*a*c]))*Sqrt[d + e*x^2]*Sqrt[((b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(b + Sqrt[b^2 - 4*a*c]
+ 2*c*x^2))])

Maple [A] (verified)

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}}\)	\(637\)

[In]

int(1/x^4/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-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)
*arctanh(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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x^4/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^{4} \left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]

[In]

integrate(1/x**4/(e*x**2+d)**(3/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(1/(x**4*(d + e*x**2)**(3/2)*(a + b*x**2 + c*x**4)), x)

Maxima [F]

\[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate(1/((c*x^4 + b*x^2 + a)*(e*x^2 + d)^(3/2)*x^4), x)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x^4/(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^4\,{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]

[In]

int(1/(x^4*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)),x)

[Out]

int(1/(x^4*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)), x)