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

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

[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)

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Rubi [A]
time = 3.95, 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 = {1315, 277, 197, 6860, 270, 1706, 385, 211} \begin {gather*} \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 ) \text {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 ) \text {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 )} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 ) \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 16.16, size = 2218, normalized size = 5.29 \begin {gather*} \text {Result too large to show} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 431, normalized size = 1.03

method	result	size
default	\(\frac {16 \sqrt {e}\, \left (-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \left (a c e -b^{2} e +b c d \right ) \textit {\_R}^{2}+2 \left (4 a b c \,e^{2}-3 a \,c^{2} d e -2 b^{3} e^{2}+3 b^{2} c d e -b \,c^{2} d^{2}\right ) \textit {\_R} +a \,c^{2} d^{2} e -b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}}{8 \left (4 a \,e^{2}-4 d e b +4 c \,d^{2}\right )}-\frac {a c e -b^{2} e +b c d}{2 \left (4 a \,e^{2}-4 d e b +4 c \,d^{2}\right ) \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}+d \right )}\right )}{a^{2}}-\frac {b \left (-\frac {1}{d x \sqrt {e \,x^{2}+d}}-\frac {2 e x}{d^{2} \sqrt {e \,x^{2}+d}}\right )}{a^{2}}+\frac {-\frac {1}{3 d \,x^{3} \sqrt {e \,x^{2}+d}}-\frac {4 e \left (-\frac {1}{d x \sqrt {e \,x^{2}+d}}-\frac {2 e x}{d^{2} \sqrt {e \,x^{2}+d}}\right )}{3 d}}{a}\)	\(431\)
risch	\(-\frac {\sqrt {e \,x^{2}+d}\, \left (-5 a e \,x^{2}-3 b d \,x^{2}+a d \right )}{3 d^{3} a^{2} x^{3}}+\frac {e^{3} \sqrt {e \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}-2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{2 d^{3} \left (a \,e^{2}-d e b +c \,d^{2}\right ) \left (x +\frac {\sqrt {-d e}}{e}\right )}+\frac {e^{3} \sqrt {e \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}+2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{2 d^{3} \left (a \,e^{2}-d e b +c \,d^{2}\right ) \left (x -\frac {\sqrt {-d e}}{e}\right )}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \left (a c e -b^{2} e +b c d \right ) \textit {\_R}^{2}+2 \left (4 a b c \,e^{2}-3 a \,c^{2} d e -2 b^{3} e^{2}+3 b^{2} c d e -b \,c^{2} d^{2}\right ) \textit {\_R} +a \,c^{2} d^{2} e -b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 a^{2} \left (a \,e^{2}-d e b +c \,d^{2}\right )}\)	\(464\)

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,method=_RETURNVERBOSE)

[Out]

16/a^2*e^(1/2)*(-1/8/(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=RootOf(c*_Z^4+(4*b*e-4*c*d)*_Z^3+(1
6*a*e^2-8*b*d*e+6*c*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z+d^4*c))-1/2*(a*c*e-b^2*e+b*c*d)/(4*a*e^2-4*b*d*e+4*c*d^2)
/(((e*x^2+d)^(1/2)-e^(1/2)*x)^2+d))-b/a^2*(-1/d/x/(e*x^2+d)^(1/2)-2*e/d^2*x/(e*x^2+d)^(1/2))+1/a*(-1/3/d/x^3/(
e*x^2+d)^(1/2)-4/3*e/d*(-1/d/x/(e*x^2+d)^(1/2)-2*e/d^2*x/(e*x^2+d)^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)*(x^2*e + d)^(3/2)*x^4), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}

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]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^4\,{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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