3.11.87 \(\int \frac {\sqrt {x}}{(a+b x^2+c x^4)^3} \, dx\) [1087]

Optimal. Leaf size=658 \[ \frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (5 b^4-45 a b^2 c+52 a^2 c^2+b c \left (5 b^2-44 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \]

[Out]

1/4*x^(3/2)*(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/16*x^(3/2)*(5*b^4-45*a*b^2*c+52*a^2*c^2+b*c
*(-44*a*c+5*b^2)*x^2)/a^2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)-1/64*c^(1/4)*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a
*c+b^2)^(1/2))^(1/4))*(5*b^4-54*a*b^2*c+520*a^2*c^2-b*(-44*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+
b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)+1/64*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^
(1/4))*(5*b^4-54*a*b^2*c+520*a^2*c^2-b*(-44*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b-
(-4*a*c+b^2)^(1/2))^(1/4)+1/64*c^(1/4)*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^4-54
*a*b^2*c+520*a^2*c^2+b*(-44*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b+(-4*a*c+b^2)^(1/
2))^(1/4)-1/64*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^4-54*a*b^2*c+520*a^
2*c^2+b*(-44*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)

________________________________________________________________________________________

Rubi [A]
time = 3.62, antiderivative size = 658, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1129, 1380, 1514, 1524, 304, 211, 214} \begin {gather*} -\frac {\sqrt [4]{c} \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}+\frac {\sqrt [4]{c} \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt [4]{c} \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\sqrt [4]{c} \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b*x^2 + c*x^4)^3,x]

[Out]

(x^(3/2)*(b^2 - 2*a*c + b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x^(3/2)*(5*b^4 - 45*a*b^2*c + 5
2*a^2*c^2 + b*c*(5*b^2 - 44*a*c)*x^2))/(16*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (c^(1/4)*(5*b^4 - 54*a*b
^2*c + 520*a^2*c^2 - b*(5*b^2 - 44*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4
*a*c])^(1/4)])/(32*2^(3/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(1/4)*(5*b^4 - 54*a*b^
2*c + 520*a^2*c^2 + b*(5*b^2 - 44*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*
a*c])^(1/4)])/(32*2^(3/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(1/4)*(5*b^4 - 54*a*b^2
*c + 520*a^2*c^2 - b*(5*b^2 - 44*a*c)*Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*
a*c])^(1/4)])/(32*2^(3/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - (c^(1/4)*(5*b^4 - 54*a*b^2
*c + 520*a^2*c^2 + b*(5*b^2 - 44*a*c)*Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*
a*c])^(1/4)])/(32*2^(3/4)*a^2*(b^2 - 4*a*c)^(5/2)*(-b + Sqrt[b^2 - 4*a*c])^(1/4))

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 214

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

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 1129

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[
k/d, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/d^2) + c*(x^(4*k)/d^4))^p, x], x, (d*x)^(1/k)], x]] /; FreeQ[
{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]

Rule 1380

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(d*x)^(m + 1))*(
b^2 - 2*a*c + b*c*x^n)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*d*n*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(a*n*(p +
1)*(b^2 - 4*a*c)), Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(
p + 1) + 1) + b*c*(m + n*(2*p + 3) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^
2 - 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1]

Rule 1514

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[(-(f*x)^(m + 1))*(a + b*x^n + c*x^(2*n))^(p + 1)*((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n)/(a*f*n
*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(a*n*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)
*Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + n*(2*p + 3) + 1)*(b*d
- 2*a*e)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && LtQ[p, -1] && IntegerQ[p]

Rule 1524

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/
2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2
, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {x^2}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (-5 b^2+26 a c-9 b c x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )}{4 a \left (b^2-4 a c\right )}\\ &=\frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (5 b^4-45 a b^2 c+52 a^2 c^2+b c \left (5 b^2-44 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (5 b^4-49 a b^2 c+260 a^2 c^2+b c \left (5 b^2-44 a c\right ) x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{16 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (5 b^4-45 a b^2 c+52 a^2 c^2+b c \left (5 b^2-44 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (c \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^2 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (c \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{32 a^2 \left (b^2-4 a c\right )^{5/2}}\\ &=\frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (5 b^4-45 a b^2 c+52 a^2 c^2+b c \left (5 b^2-44 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (\sqrt {c} \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2}}-\frac {\left (\sqrt {c} \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2}}-\frac {\left (\sqrt {c} \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (\sqrt {c} \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+\sqrt {b^2-4 a c}}+\sqrt {2} \sqrt {c} x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2}}\\ &=\frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (5 b^4-45 a b^2 c+52 a^2 c^2+b c \left (5 b^2-44 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.48, size = 255, normalized size = 0.39 \begin {gather*} \frac {\frac {4 x^{3/2} \left (84 a^3 c^2+5 b^3 x^2 \left (b+c x^2\right )^2+a^2 c \left (-69 b^2-8 b c x^2+52 c^2 x^4\right )+a b \left (9 b^3-36 b^2 c x^2-89 b c^2 x^4-44 c^3 x^6\right )\right )}{\left (a+b x^2+c x^4\right )^2}+\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {5 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )-49 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+260 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+5 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-44 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{64 a^2 \left (b^2-4 a c\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*x^(3/2)*(84*a^3*c^2 + 5*b^3*x^2*(b + c*x^2)^2 + a^2*c*(-69*b^2 - 8*b*c*x^2 + 52*c^2*x^4) + a*b*(9*b^3 - 36
*b^2*c*x^2 - 89*b*c^2*x^4 - 44*c^3*x^6)))/(a + b*x^2 + c*x^4)^2 + RootSum[a + b*#1^4 + c*#1^8 & , (5*b^4*Log[S
qrt[x] - #1] - 49*a*b^2*c*Log[Sqrt[x] - #1] + 260*a^2*c^2*Log[Sqrt[x] - #1] + 5*b^3*c*Log[Sqrt[x] - #1]*#1^4 -
44*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(64*a^2*(b^2 - 4*a*c)^2)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.10, size = 321, normalized size = 0.49

method	result	size
derivativedivides	\(\frac {\frac {3 \left (28 a^{2} c^{2}-23 a \,b^{2} c +3 b^{4}\right ) x^{\frac {3}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {b \left (8 a^{2} c^{2}+36 a \,b^{2} c -5 b^{4}\right ) x^{\frac {7}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (52 a^{2} c^{2}-89 a \,b^{2} c +10 b^{4}\right ) x^{\frac {11}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \,c^{2} \left (44 a c -5 b^{2}\right ) x^{\frac {15}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c \left (-44 a c +5 b^{2}\right ) \textit {\_R}^{6}+\left (260 a^{2} c^{2}-49 a \,b^{2} c +5 b^{4}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\)	\(321\)
default	\(\frac {\frac {3 \left (28 a^{2} c^{2}-23 a \,b^{2} c +3 b^{4}\right ) x^{\frac {3}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {b \left (8 a^{2} c^{2}+36 a \,b^{2} c -5 b^{4}\right ) x^{\frac {7}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (52 a^{2} c^{2}-89 a \,b^{2} c +10 b^{4}\right ) x^{\frac {11}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \,c^{2} \left (44 a c -5 b^{2}\right ) x^{\frac {15}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c \left (-44 a c +5 b^{2}\right ) \textit {\_R}^{6}+\left (260 a^{2} c^{2}-49 a \,b^{2} c +5 b^{4}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\)	\(321\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(3/32*(28*a^2*c^2-23*a*b^2*c+3*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^(3/2)-1/32*b*(8*a^2*c^2+36*a*b^2*c-5*b^4)
/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)+1/32/a^2*c*(52*a^2*c^2-89*a*b^2*c+10*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x
^(11/2)-1/32*b*c^2*(44*a*c-5*b^2)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(15/2))/(c*x^4+b*x^2+a)^2+1/64/a^2/(16*a^2*
c^2-8*a*b^2*c+b^4)*sum((b*c*(-44*a*c+5*b^2)*_R^6+(260*a^2*c^2-49*a*b^2*c+5*b^4)*_R^2)/(2*_R^7*c+_R^3*b)*ln(x^(
1/2)-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))

________________________________________________________________________________________

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

[Out]

1/16*((5*b^3*c^2 - 44*a*b*c^3)*x^(15/2) + (10*b^4*c - 89*a*b^2*c^2 + 52*a^2*c^3)*x^(11/2) + (5*b^5 - 36*a*b^3*
c - 8*a^2*b*c^2)*x^(7/2) + 3*(3*a*b^4 - 23*a^2*b^2*c + 28*a^3*c^2)*x^(3/2))/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16
*a^4*c^4)*x^8 + a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x^6 + (a^2*b
^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2) - integrate(-1/32*((5*b^3*c
- 44*a*b*c^2)*x^(5/2) + (5*b^4 - 49*a*b^2*c + 260*a^2*c^2)*sqrt(x))/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^
2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [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(x**(1/2)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x)/(c*x^4 + b*x^2 + a)^3, x)

________________________________________________________________________________________

Mupad [B]
time = 8.75, size = 2500, normalized size = 3.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a + b*x^2 + c*x^4)^3,x)

[Out]

((x^(11/2)*(10*b^4*c + 52*a^2*c^3 - 89*a*b^2*c^2))/(16*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) - (x^(7/2)*(8*a^2
*b*c^2 - 5*b^5 + 36*a*b^3*c))/(16*a*(a*b^4 + 16*a^3*c^2 - 8*a^2*b^2*c)) + (3*x^(3/2)*(3*b^4 + 28*a^2*c^2 - 23*
a*b^2*c))/(16*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (b*c^2*x^(15/2)*(44*a*c - 5*b^2))/(16*(a^2*b^4 + 16*a^4*c^2
- 8*a^3*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c*x^6) + atan(((((2097152000*a*b^33*c^4
+ 466178856428188467200*a^17*b*c^20 - 151833804800*a^2*b^31*c^5 + 5340020080640*a^3*b^29*c^6 - 120300087803904
*a^4*b^27*c^7 + 1933149881761792*a^5*b^25*c^8 - 23398590986584064*a^6*b^23*c^9 + 219878252263505920*a^7*b^21*c
^10 - 1631099300505190400*a^8*b^19*c^11 + 9625014804028588032*a^9*b^17*c^12 - 45207702606568226816*a^10*b^15*c
^13 + 168027072287612076032*a^11*b^13*c^14 - 487882094458626375680*a^12*b^11*c^15 + 1082673222923122114560*a^1
3*b^9*c^16 - 1771946621413479153664*a^14*b^7*c^17 + 2014068018680264916992*a^15*b^5*c^18 - 1418770116510434197
504*a^16*b^3*c^19)/(268435456*(a^6*b^28 + 268435456*a^20*c^14 - 56*a^7*b^26*c + 1456*a^8*b^24*c^2 - 23296*a^9*
b^22*c^3 + 256256*a^10*b^20*c^4 - 2050048*a^11*b^18*c^5 + 12300288*a^12*b^16*c^6 - 56229888*a^13*b^14*c^7 + 19
6804608*a^14*b^12*c^8 - 524812288*a^15*b^10*c^9 + 1049624576*a^16*b^8*c^10 - 1526726656*a^17*b^6*c^11 + 152672
6656*a^18*b^4*c^12 - 939524096*a^19*b^2*c^13)) - (x^(1/2)*(-(625*b^37 - 625*b^12*(-(4*a*c - b^2)^25)^(1/2) + 1
1279020326912000*a^18*b*c^18 + 2168275*a^2*b^33*c^2 - 57758230*a^3*b^31*c^3 + 1109954201*a^4*b^29*c^4 - 162857
49400*a^5*b^27*c^5 + 188531780400*a^6*b^25*c^6 - 1756313913600*a^7*b^23*c^7 + 13317068448000*a^8*b^21*c^8 - 82
629338933248*a^9*b^19*c^9 + 419701532733440*a^10*b^17*c^10 - 1737502295326720*a^11*b^15*c^11 + 580700054192128
0*a^12*b^13*c^12 - 15422593991966720*a^13*b^11*c^13 + 31764369743282176*a^14*b^9*c^14 - 48851227886223360*a^15
*b^7*c^15 + 52725360025927680*a^16*b^5*c^16 - 35577189126635520*a^17*b^3*c^17 - 285610000*a^6*c^6*(-(4*a*c - b
^2)^25)^(1/2) - 52625*a*b^35*c - 380775*a^2*b^8*c^2*(-(4*a*c - b^2)^25)^(1/2) + 4075730*a^3*b^6*c^3*(-(4*a*c -
b^2)^25)^(1/2) - 28545201*a^4*b^4*c^4*(-(4*a*c - b^2)^25)^(1/2) + 121578600*a^5*b^2*c^5*(-(4*a*c - b^2)^25)^(
1/2) + 21375*a*b^10*c*(-(4*a*c - b^2)^25)^(1/2))/(33554432*(a^9*b^40 + 1099511627776*a^29*c^20 - 80*a^10*b^38*
c + 3040*a^11*b^36*c^2 - 72960*a^12*b^34*c^3 + 1240320*a^13*b^32*c^4 - 15876096*a^14*b^30*c^5 + 158760960*a^15
*b^28*c^6 - 1270087680*a^16*b^26*c^7 + 8255569920*a^17*b^24*c^8 - 44029706240*a^18*b^22*c^9 + 193730707456*a^1
9*b^20*c^10 - 704475299840*a^20*b^18*c^11 + 2113425899520*a^21*b^16*c^12 - 5202279137280*a^22*b^14*c^13 + 1040
4558274560*a^23*b^12*c^14 - 16647293239296*a^24*b^10*c^15 + 20809116549120*a^25*b^8*c^16 - 19585050869760*a^26
*b^6*c^17 + 13056700579840*a^27*b^4*c^18 - 5497558138880*a^28*b^2*c^19)))^(1/4)*(2378463553205043200*a^18*c^19
- 419430400*a^3*b^30*c^4 + 26675773440*a^4*b^28*c^5 - 814718386176*a^5*b^26*c^6 + 15745652097024*a^6*b^24*c^7
- 214134184476672*a^7*b^22*c^8 + 2159815572848640*a^8*b^20*c^9 - 16615360157450240*a^9*b^18*c^10 + 9886257942
1544448*a^10*b^16*c^11 - 456983970538586112*a^11*b^14*c^12 + 1635439433677275136*a^12*b^12*c^13 - 448054836609
4172160*a^13*b^10*c^14 + 9201889778671288320*a^14*b^8*c^15 - 13675039531022155776*a^15*b^6*c^16 + 138416023484
90686464*a^16*b^4*c^17 - 8502514621498785792*a^17*b^2*c^18))/(4194304*(a^6*b^24 + 16777216*a^18*c^12 - 48*a^7*
b^22*c + 1056*a^8*b^20*c^2 - 14080*a^9*b^18*c^3 + 126720*a^10*b^16*c^4 - 811008*a^11*b^14*c^5 + 3784704*a^12*b
^12*c^6 - 12976128*a^13*b^10*c^7 + 32440320*a^14*b^8*c^8 - 57671680*a^15*b^6*c^9 + 69206016*a^16*b^4*c^10 - 50
331648*a^17*b^2*c^11)))*(-(625*b^37 - 625*b^12*(-(4*a*c - b^2)^25)^(1/2) + 11279020326912000*a^18*b*c^18 + 216
8275*a^2*b^33*c^2 - 57758230*a^3*b^31*c^3 + 1109954201*a^4*b^29*c^4 - 16285749400*a^5*b^27*c^5 + 188531780400*
a^6*b^25*c^6 - 1756313913600*a^7*b^23*c^7 + 13317068448000*a^8*b^21*c^8 - 82629338933248*a^9*b^19*c^9 + 419701
532733440*a^10*b^17*c^10 - 1737502295326720*a^11*b^15*c^11 + 5807000541921280*a^12*b^13*c^12 - 154225939919667
20*a^13*b^11*c^13 + 31764369743282176*a^14*b^9*c^14 - 48851227886223360*a^15*b^7*c^15 + 52725360025927680*a^16
*b^5*c^16 - 35577189126635520*a^17*b^3*c^17 - 285610000*a^6*c^6*(-(4*a*c - b^2)^25)^(1/2) - 52625*a*b^35*c - 3
80775*a^2*b^8*c^2*(-(4*a*c - b^2)^25)^(1/2) + 4075730*a^3*b^6*c^3*(-(4*a*c - b^2)^25)^(1/2) - 28545201*a^4*b^4
*c^4*(-(4*a*c - b^2)^25)^(1/2) + 121578600*a^5*b^2*c^5*(-(4*a*c - b^2)^25)^(1/2) + 21375*a*b^10*c*(-(4*a*c - b
^2)^25)^(1/2))/(33554432*(a^9*b^40 + 1099511627776*a^29*c^20 - 80*a^10*b^38*c + 3040*a^11*b^36*c^2 - 72960*a^1
2*b^34*c^3 + 1240320*a^13*b^32*c^4 - 15876096*a^14*b^30*c^5 + 158760960*a^15*b^28*c^6 - 1270087680*a^16*b^26*c
^7 + 8255569920*a^17*b^24*c^8 - 44029706240*a^18*b^22*c^9 + 193730707456*a^19*b^20*c^10 - 704475299840*a^20*b^
18*c^11 + 2113425899520*a^21*b^16*c^12 - 5202279137280*a^22*b^14*c^13 + 10404558274560*a^23*b^12*c^14 - 166472
93239296*a^24*b^10*c^15 + 20809116549120*a^25*b...

________________________________________________________________________________________