∫ √ _x ____(a+bx2 +cx4 )3 dx

3.1087 $\int \frac {\sqrt {x}}{(a+b x^2+c x^4)^3} \, dx$

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

[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 = 5.48, 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 = {1115, 1366, 1500, 1510, 298, 205, 208} \[ \frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\sqrt [4]{c} \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}+\frac {\sqrt [4]{c} \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {\sqrt [4]{c} \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\sqrt [4]{c} \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

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 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]

Rule 208

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

b}, x] && NegQ[a/b]

Rule 298

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 1115

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 1366

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 1500

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)*S

imp[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 1510

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 \operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.49, size = 254, normalized size = 0.39 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8 c+\text {$\#$1}^4 b+a\& ,\frac {-44 \text {$\#$1}^4 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+5 \text {$\#$1}^4 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+260 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )-49 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+5 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )}{2 \text {$\#$1}^5 c+\text {$\#$1} b}\& \right ]+\frac {4 x^{3/2} \left (52 a^2 c^2-45 a b^2 c-44 a b c^2 x^2+5 b^4+5 b^3 c x^2\right )}{a+b x^2+c x^4}-\frac {16 a x^{3/2} \left (4 a c-b^2\right ) \left (-2 a c+b^2+b c x^2\right )}{\left (a+b x^2+c x^4\right )^2}}{64 a^2 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-16*a*(-b^2 + 4*a*c)*x^(3/2)*(b^2 - 2*a*c + b*c*x^2))/(a + b*x^2 + c*x^4)^2 + (4*x^(3/2)*(5*b^4 - 45*a*b^2*c

+ 52*a^2*c^2 + 5*b^3*c*x^2 - 44*a*b*c^2*x^2))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (5*b^4*Lo

g[Sqrt[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)

________________________________________________________________________________________

fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________

giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval

uation time: 191.53Unable to convert to real 1/4 Error: Bad Argument Value

________________________________________________________________________________________

maple [C] time = 0.05, size = 321, normalized size = 0.49 \[ -\frac {\left (\left (44 a c -5 b^{2}\right ) \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{6} b c +\left (-260 a^{2} c^{2}+49 a \,b^{2} c -5 b^{4}\right ) \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{2}\right ) \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )+\sqrt {x}\right )}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2} \left (2 \RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{7} c +\RootOf \left (c \,\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+a \right )^{3} b \right )}+\frac {-\frac {\left (44 a c -5 b^{2}\right ) b \,c^{2} x^{\frac {15}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}+\frac {\left (52 a^{2} c^{2}-89 a \,b^{2} c +10 b^{4}\right ) c \,x^{\frac {11}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {\left (8 a^{2} c^{2}+36 a \,b^{2} c -5 b^{4}\right ) b \,x^{\frac {7}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}+\frac {3 \left (28 a^{2} c^{2}-23 a \,b^{2} c +3 b^{4}\right ) x^{\frac {3}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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*c^2*b*(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(-_R

+x^(1/2)),_R=RootOf(_Z^8*c+_Z^4*b+a))

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (5 \, b^{3} c^{2} - 44 \, a b c^{3}\right )} x^{\frac {15}{2}} + {\left (10 \, b^{4} c - 89 \, a b^{2} c^{2} + 52 \, a^{2} c^{3}\right )} x^{\frac {11}{2}} + {\left (5 \, b^{5} - 36 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} x^{\frac {7}{2}} + 3 \, {\left (3 \, a b^{4} - 23 \, a^{2} b^{2} c + 28 \, a^{3} c^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left ({\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} x^{8} + a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2} + 2 \, {\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} x^{6} + {\left (a^{2} b^{6} - 6 \, a^{3} b^{4} c + 32 \, a^{5} c^{3}\right )} x^{4} + 2 \, {\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2}\right )}} - \int -\frac {{\left (5 \, b^{3} c - 44 \, a b c^{2}\right )} x^{\frac {5}{2}} + {\left (5 \, b^{4} - 49 \, a b^{2} c + 260 \, a^{2} c^{2}\right )} \sqrt {x}}{32 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2} + {\left (a^{2} b^{4} c - 8 \, a^{3} b^{2} c^{2} + 16 \, a^{4} c^{3}\right )} x^{4} + {\left (a^{2} b^{5} - 8 \, a^{3} b^{3} c + 16 \, a^{4} b c^{2}\right )} x^{2}\right )}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [B] time = 8.75, size = 46948, normalized size = 71.35 \[ \text {result too large to display} \]

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^8*c^16 - 19585050869760*a^26*b^6*c^17 + 13056700579840*a^27*b^

4*c^18 - 5497558138880*a^28*b^2*c^19)))^(3/4) + (x^(1/2)*(30525625*b^15*c^10 - 1297573875*a*b^13*c^11 + 998035

58400000*a^7*b*c^17 + 27786809400*a^2*b^11*c^12 - 311511417680*a^3*b^9*c^13 + 1975414457856*a^4*b^7*c^14 - 475

3980591360*a^5*b^5*c^15 - 10990483712000*a^6*b^3*c^16))/(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 - 5033164

8*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 + 2168275*

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 + 41970153273

3440*a^10*b^17*c^10 - 1737502295326720*a^11*b^15*c^11 + 5807000541921280*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)^2

5)^(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^3

4*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 - 16647293239

296*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^1

8 - 5497558138880*a^28*b^2*c^19)))^(1/4)*1i - (((2097152000*a*b^33*c^4 + 466178856428188467200*a^17*b*c^20 - 1

51833804800*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^13*b^9*c^16 - 1771946621413479153664*a^1

4*b^7*c^17 + 2014068018680264916992*a^15*b^5*c^18 - 1418770116510434197504*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 - 20500

48*a^11*b^18*c^5 + 12300288*a^12*b^16*c^6 - 56229888*a^13*b^14*c^7 + 196804608*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 + 1526726656*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) + 11279020326912000*a^18*b*c^18 + 2168275*