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

Optimal. Leaf size=419 \[ \frac{2 c^2 \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2}}+\frac{2 c^2 \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2}}-\frac{e x \left (a c e+b^2 (-e)+b c d\right )}{a^2 d \sqrt{d+e x^2} \left (e (a e-b d)+c d^2\right )}+\frac{2 e x (4 a e+3 b d)}{3 a^2 d^3 \sqrt{d+e x^2}}+\frac{4 a e+3 b d}{3 a^2 d^2 x \sqrt{d+e x^2}}-\frac{1}{3 a d x^3 \sqrt{d+e x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 5.56641, antiderivative size = 647, normalized size of antiderivative = 1.54, number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {1301, 271, 191, 6728, 264, 1692, 377, 205} \[ \frac{c \left (\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac{c \left (-\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{d+e x^2} \left (a c e+b^2 (-e)+b c d\right )}{a^2 d x \left (a e^2-b d e+c d^2\right )}+\frac{8 e^4 x}{3 d^3 \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}+\frac{4 e^3}{3 d^2 x \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac{e^2}{3 d x^3 \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}+\frac{2 e \sqrt{d+e x^2} (c d-b e)}{3 a d^2 x \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x^2} (c d-b e)}{3 a d x^3 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-e^2/(3*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 - 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)) + (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))

Rule 1301

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 271

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 191

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 6728

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]

Rule 264

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 1692

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 377

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 205

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

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx &=\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 ) \operatorname{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 ) \operatorname{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]  time = 8.59638, size = 2218, normalized size = 5.29 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[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))])

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Maple [C]  time = 0.036, size = 541, normalized size = 1.3 \begin{align*} -8\,{\frac{{e}^{3/2}c}{a \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}+8\,{\frac{{e}^{3/2}{b}^{2}}{{a}^{2} \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}-8\,{\frac{\sqrt{e}bcd}{{a}^{2} \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}-2\,{\frac{\sqrt{e}}{{a}^{2} \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) }\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,deb+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{ \left ( c \left ( ace-{b}^{2}e+bcd \right ){{\it \_R}}^{2}+2\, \left ( 4\,abc{e}^{2}-3\,a{c}^{2}de-2\,{b}^{3}{e}^{2}+3\,{b}^{2}cde-b{c}^{2}{d}^{2} \right ){\it \_R}+a{c}^{2}{d}^{2}e-{b}^{2}c{d}^{2}e+b{c}^{2}{d}^{3} \right ) \ln \left ( \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{2}-{\it \_R} \right ) }{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}}}+{\frac{b}{{a}^{2}dx}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+2\,{\frac{bex}{{a}^{2}{d}^{2}\sqrt{e{x}^{2}+d}}}-{\frac{1}{3\,ad{x}^{3}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+{\frac{4\,e}{3\,a{d}^{2}x}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+{\frac{8\,{e}^{2}x}{3\,a{d}^{3}}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/a*e^(3/2)/(4*a*e^2-4*b*d*e+4*c*d^2)/(2*e*x^2-2*e^(1/2)*(e*x^2+d)^(1/2)*x+2*d)*c+8/a^2*e^(3/2)/(4*a*e^2-4*b*
d*e+4*c*d^2)/(2*e*x^2-2*e^(1/2)*(e*x^2+d)^(1/2)*x+2*d)*b^2-8/a^2*e^(1/2)/(4*a*e^2-4*b*d*e+4*c*d^2)/(2*e*x^2-2*
e^(1/2)*(e*x^2+d)^(1/2)*x+2*d)*b*c*d-2/a^2*e^(1/2)/(4*a*e^2-4*b*d*e+4*c*d^2)*sum((c*(a*c*e-b^2*e+b*c*d)*_R^2+2
*(4*a*b*c*e^2-3*a*c^2*d*e-2*b^3*e^2+3*b^2*c*d*e-b*c^2*d^2)*_R+a*c^2*d^2*e-b^2*c*d^2*e+b*c^2*d^3)/(_R^3*c+3*_R^
2*b*e-3*_R^2*c*d+8*_R*a*e^2-4*_R*b*d*e+3*_R*c*d^2+b*d^2*e-c*d^3)*ln(((e*x^2+d)^(1/2)-e^(1/2)*x)^2-_R),_R=RootO
f(c*_Z^4+(4*b*e-4*c*d)*_Z^3+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z+c*d^4))+1/a^2*b/d/x/(e*x^2+
d)^(1/2)+2/a^2*b*e/d^2*x/(e*x^2+d)^(1/2)-1/3/a/d/x^3/(e*x^2+d)^(1/2)+4/3/a*e/d^2/x/(e*x^2+d)^(1/2)+8/3/a*e^2/d
^3*x/(e*x^2+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError