3.11.81 \(\int \frac {x^{13/2}}{(a+b x^2+c x^4)^3} \, dx\) [1081]

Optimal. Leaf size=569 \[ \frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (24 a b+\left (5 b^2+28 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 a c\right )\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^2 \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 a c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} c^{3/4} \left (b^2-4 a c\right )^2 \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \]

[Out]

1/4*x^(7/2)*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/16*x^(3/2)*(24*a*b+(28*a*c+5*b^2)*x^2)/(-4*a*c+b^2)^2
/(c*x^4+b*x^2+a)+1/64*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^2+28*a*c+(-172*a*b*c-
5*b^3)/(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^2/(-b+(-4*a*c+b^2)^(1/2))^(1/4)-1/64*arctanh(2^(1/4)*c
^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^2+28*a*c+(-172*a*b*c-5*b^3)/(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(
3/4)/(-4*a*c+b^2)^2/(-b+(-4*a*c+b^2)^(1/2))^(1/4)+1/64*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^
(1/4))*(5*b^3+172*a*b*c+(28*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)
^(1/2))^(1/4)-1/64*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^3+172*a*b*c+(28*a*c+5*b
^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/c^(3/4)/(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)

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Rubi [A]
time = 1.27, antiderivative size = 569, 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, 1379, 1512, 1524, 304, 211, 214} \begin {gather*} \frac {\left (\sqrt {b^2-4 a c} \left (28 a c+5 b^2\right )+172 a b c+5 b^3\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} c^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}+\frac {\left (-\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\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} c^{3/4} \left (b^2-4 a c\right )^2 \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (x^2 \left (28 a c+5 b^2\right )+24 a b\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (\sqrt {b^2-4 a c} \left (28 a c+5 b^2\right )+172 a b c+5 b^3\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} c^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\left (-\frac {172 a b c+5 b^3}{\sqrt {b^2-4 a c}}+28 a c+5 b^2\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} c^{3/4} \left (b^2-4 a c\right )^2 \sqrt [4]{\sqrt {b^2-4 a c}-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(13/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

(x^(7/2)*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x^(3/2)*(24*a*b + (5*b^2 + 28*a*c)*x^2))/(1
6*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + ((5*b^3 + 172*a*b*c + Sqrt[b^2 - 4*a*c]*(5*b^2 + 28*a*c))*ArcTan[(2^(
1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(32*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b - Sqrt[b^2
- 4*a*c])^(1/4)) + ((5*b^2 + 28*a*c - (5*b^3 + 172*a*b*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)*c^(3/4)*(b^2 - 4*a*c)^2*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) - ((5*b^3
+ 172*a*b*c + Sqrt[b^2 - 4*a*c]*(5*b^2 + 28*a*c))*ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^
(1/4)])/(32*2^(3/4)*c^(3/4)*(b^2 - 4*a*c)^(5/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - ((5*b^2 + 28*a*c - (5*b^3 +
172*a*b*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)*c
^(3/4)*(b^2 - 4*a*c)^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 1379

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

Rule 1512

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :
> Simp[f^(n - 1)*(f*x)^(m - n + 1)*(a + b*x^n + c*x^(2*n))^(p + 1)*((b*d - 2*a*e - (b*e - 2*c*d)*x^n)/(n*(p +
1)*(b^2 - 4*a*c))), x] + Dist[f^n/(n*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^(p + 1)
*Simp[(n - m - 1)*(b*d - 2*a*e) + (2*n*p + 2*n + m + 1)*(b*e - 2*c*d)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m, n - 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 {x^{13/2}}{\left (a+b x^2+c x^4\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {x^{14}}{\left (a+b x^4+c x^8\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\text {Subst}\left (\int \frac {x^6 \left (14 a-5 b x^4\right )}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )}{4 \left (b^2-4 a c\right )}\\ &=\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (24 a b+\left (5 b^2+28 a c\right ) x^2\right )}{16 \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 (-72 a b+\left (5 b^2+28 a c\right ) x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{16 \left (b^2-4 a c\right )^2}\\ &=\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (24 a b+\left (5 b^2+28 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 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 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\sqrt {b^2-4 a c}}\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 \left (b^2-4 a c\right )^2}\\ &=\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (24 a b+\left (5 b^2+28 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 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} \sqrt {c} \left (b^2-4 a c\right )^{5/2}}+\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 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} \sqrt {c} \left (b^2-4 a c\right )^{5/2}}-\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\sqrt {b^2-4 a c}}\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} \sqrt {c} \left (b^2-4 a c\right )^2}+\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\sqrt {b^2-4 a c}}\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} \sqrt {c} \left (b^2-4 a c\right )^2}\\ &=\frac {x^{7/2} \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (24 a b+\left (5 b^2+28 a c\right ) x^2\right )}{16 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 a c\right )\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} c^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\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} c^{3/4} \left (b^2-4 a c\right )^2 \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\left (5 b^3+172 a b c+\sqrt {b^2-4 a c} \left (5 b^2+28 a c\right )\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} c^{3/4} \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\left (5 b^2+28 a c-\frac {5 b^3+172 a b c}{\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} c^{3/4} \left (b^2-4 a c\right )^2 \sqrt [4]{-b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.76, size = 392, normalized size = 0.69 \begin {gather*} \frac {1}{64} \left (\frac {4 x^{3/2} \left (4 a^2 \left (6 b-c x^2\right )+b^2 x^4 \left (9 b+5 c x^2\right )+a \left (37 b^2 x^2+36 b c x^4+28 c^2 x^6\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^2}+\frac {8 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )-13 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4+2 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{a c^2 \left (-b^2+4 a c\right )}+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {8 b^5 \log \left (\sqrt {x}-\text {$\#$1}\right )-136 a b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right )+344 a^2 b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+8 b^4 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-11 a b^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-36 a^2 c^3 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{a c^2 \left (b^2-4 a c\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(13/2)/(a + b*x^2 + c*x^4)^3,x]

[Out]

((4*x^(3/2)*(4*a^2*(6*b - c*x^2) + b^2*x^4*(9*b + 5*c*x^2) + a*(37*b^2*x^2 + 36*b*c*x^4 + 28*c^2*x^6)))/((b^2
- 4*a*c)^2*(a + b*x^2 + c*x^4)^2) + (8*RootSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 13*a*b*c*Log[S
qrt[x] - #1] + b^2*c*Log[Sqrt[x] - #1]*#1^4 + 2*a*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ])/(a*c^2*(-
b^2 + 4*a*c)) + RootSum[a + b*#1^4 + c*#1^8 & , (8*b^5*Log[Sqrt[x] - #1] - 136*a*b^3*c*Log[Sqrt[x] - #1] + 344
*a^2*b*c^2*Log[Sqrt[x] - #1] + 8*b^4*c*Log[Sqrt[x] - #1]*#1^4 - 11*a*b^2*c^2*Log[Sqrt[x] - #1]*#1^4 - 36*a^2*c
^3*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(a*c^2*(b^2 - 4*a*c)^2))/64

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.10, size = 242, normalized size = 0.43

method	result	size
derivativedivides	\(\frac {\frac {3 a^{2} b \,x^{\frac {3}{2}}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (4 a c -37 b^{2}\right ) x^{\frac {7}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {9 b \left (4 a c +b^{2}\right ) x^{\frac {11}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 c \left (28 a c +5 b^{2}\right ) x^{\frac {15}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-28 a c -5 b^{2}\right ) \textit {\_R}^{6}+72 b \,\textit {\_R}^{2} a \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\)	\(242\)
default	\(\frac {\frac {3 a^{2} b \,x^{\frac {3}{2}}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (4 a c -37 b^{2}\right ) x^{\frac {7}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {9 b \left (4 a c +b^{2}\right ) x^{\frac {11}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {2 c \left (28 a c +5 b^{2}\right ) x^{\frac {15}{2}}}{512 a^{2} c^{2}-256 a \,b^{2} c +32 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (-28 a c -5 b^{2}\right ) \textit {\_R}^{6}+72 b \,\textit {\_R}^{2} a \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\)	\(242\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(13/2)/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

2*(3/4*a^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)-1/32*a*(4*a*c-37*b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)+9/32*
b*(4*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(11/2)+1/32*c*(28*a*c+5*b^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/(16*a^2*c^2-8*a*b^2*c+b^4)*sum(((-28*a*c-5*b^2)*_R^6+72*b*_R^2*a)/(2*_R^7*c+_R^3*b)*ln(x
^(1/2)-_R),_R=RootOf(_Z^8*c+_Z^4*b+a))

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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(x^(13/2)/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/16*((5*b^2*c + 28*a*c^2)*x^(15/2) + 9*(b^3 + 4*a*b*c)*x^(11/2) + 24*a^2*b*x^(3/2) + (37*a*b^2 - 4*a^2*c)*x^(
7/2))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3
*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2) + integ
rate(1/32*((5*b^2 + 28*a*c)*x^(5/2) - 72*a*b*sqrt(x))/(a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2 + (b^4*c - 8*a*b^2*c^2
+ 16*a^2*c^3)*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^2), 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(x^(13/2)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

Timed out

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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**(13/2)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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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^(13/2)/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

integrate(x^(13/2)/(c*x^4 + b*x^2 + a)^3, x)

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Mupad [B]
time = 8.01, size = 2500, normalized size = 4.39 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(13/2)/(a + b*x^2 + c*x^4)^3,x)

[Out]

((9*x^(11/2)*(b^3 + 4*a*b*c))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^(7/2)*(37*a*b^2 - 4*a^2*c))/(16*(b^4 +
16*a^2*c^2 - 8*a*b^2*c)) + (c*x^(15/2)*(28*a*c + 5*b^2))/(16*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*a^2*b*x^(3/2
))/(2*(b^4 + 16*a^2*c^2 - 8*a*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(((((
386183668047020032*a^16*c^16 + 2097152000*a^3*b^26*c^3 - 7615312560128*a^4*b^24*c^4 + 295658569334784*a^5*b^22
*c^5 - 5154027327193088*a^6*b^20*c^6 + 52821290217635840*a^7*b^18*c^7 - 350572668266741760*a^8*b^16*c^8 + 1560
295235622273024*a^9*b^14*c^9 - 4628236966960300032*a^10*b^12*c^10 + 8604139182719238144*a^11*b^10*c^11 - 79240
26369753743360*a^12*b^8*c^12 - 1942353261163970560*a^13*b^6*c^13 + 11823215659242749952*a^14*b^4*c^14 - 841919
8028392431616*a^15*b^2*c^15)/(268435456*(b^28 + 268435456*a^14*c^14 + 1456*a^2*b^24*c^2 - 23296*a^3*b^22*c^3 +
256256*a^4*b^20*c^4 - 2050048*a^5*b^18*c^5 + 12300288*a^6*b^16*c^6 - 56229888*a^7*b^14*c^7 + 196804608*a^8*b^
12*c^8 - 524812288*a^9*b^10*c^9 + 1049624576*a^10*b^8*c^10 - 1526726656*a^11*b^6*c^11 + 1526726656*a^12*b^4*c^
12 - 939524096*a^13*b^2*c^13 - 56*a*b^26*c)) - (x^(1/2)*(-(625*b^31 + 625*b^6*(-(4*a*c - b^2)^25)^(1/2) - 1519
2104632320*a^15*b*c^15 - 89000*a^2*b^27*c^2 + 27186416*a^3*b^25*c^3 - 1342297600*a^4*b^23*c^4 + 25492409600*a^
5*b^21*c^5 - 265188833280*a^6*b^19*c^6 + 1688816578560*a^7*b^17*c^7 - 6664504147968*a^8*b^15*c^8 + 14462970429
440*a^9*b^13*c^9 - 4163326443520*a^10*b^11*c^10 - 70455242260480*a^11*b^9*c^11 + 206669464207360*a^12*b^7*c^12
- 267459844112384*a^13*b^5*c^13 + 150009114787840*a^14*b^3*c^14 - 38416*a^3*c^3*(-(4*a*c - b^2)^25)^(1/2) + 2
3125*a*b^29*c + 1911000*a^2*b^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 54375*a*b^4*c*(-(4*a*c - b^2)^25)^(1/2))/(3355
4432*(1099511627776*a^20*c^23 + b^40*c^3 - 80*a*b^38*c^4 + 3040*a^2*b^36*c^5 - 72960*a^3*b^34*c^6 + 1240320*a^
4*b^32*c^7 - 15876096*a^5*b^30*c^8 + 158760960*a^6*b^28*c^9 - 1270087680*a^7*b^26*c^10 + 8255569920*a^8*b^24*c
^11 - 44029706240*a^9*b^22*c^12 + 193730707456*a^10*b^20*c^13 - 704475299840*a^11*b^18*c^14 + 2113425899520*a^
12*b^16*c^15 - 5202279137280*a^13*b^14*c^16 + 10404558274560*a^14*b^12*c^17 - 16647293239296*a^15*b^10*c^18 +
20809116549120*a^16*b^8*c^19 - 19585050869760*a^17*b^6*c^20 + 13056700579840*a^18*b^4*c^21 - 5497558138880*a^1
9*b^2*c^22)))^(1/4)*(27584547717644288*a^15*c^16 + 99891544064*a^3*b^24*c^4 - 4092566962176*a^4*b^22*c^5 + 758
24426385408*a^5*b^20*c^6 - 837991069122560*a^6*b^18*c^7 + 6133342147706880*a^7*b^16*c^8 - 31188471955587072*a^
8*b^14*c^9 + 112343150323826688*a^9*b^12*c^10 - 286537128244936704*a^10*b^10*c^11 + 507743474590679040*a^11*b^
8*c^12 - 599365778533253120*a^12*b^6*c^13 + 436356582645694464*a^13*b^4*c^14 - 170573835886657536*a^14*b^2*c^1
5))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*a^4*b^16*c^4 - 81100
8*a^5*b^14*c^5 + 3784704*a^6*b^12*c^6 - 12976128*a^7*b^10*c^7 + 32440320*a^8*b^8*c^8 - 57671680*a^9*b^6*c^9 +
69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*(-(625*b^31 + 625*b^6*(-(4*a*c - b^2)^25)^(1/
2) - 15192104632320*a^15*b*c^15 - 89000*a^2*b^27*c^2 + 27186416*a^3*b^25*c^3 - 1342297600*a^4*b^23*c^4 + 25492
409600*a^5*b^21*c^5 - 265188833280*a^6*b^19*c^6 + 1688816578560*a^7*b^17*c^7 - 6664504147968*a^8*b^15*c^8 + 14
462970429440*a^9*b^13*c^9 - 4163326443520*a^10*b^11*c^10 - 70455242260480*a^11*b^9*c^11 + 206669464207360*a^12
*b^7*c^12 - 267459844112384*a^13*b^5*c^13 + 150009114787840*a^14*b^3*c^14 - 38416*a^3*c^3*(-(4*a*c - b^2)^25)^
(1/2) + 23125*a*b^29*c + 1911000*a^2*b^2*c^2*(-(4*a*c - b^2)^25)^(1/2) + 54375*a*b^4*c*(-(4*a*c - b^2)^25)^(1/
2))/(33554432*(1099511627776*a^20*c^23 + b^40*c^3 - 80*a*b^38*c^4 + 3040*a^2*b^36*c^5 - 72960*a^3*b^34*c^6 + 1
240320*a^4*b^32*c^7 - 15876096*a^5*b^30*c^8 + 158760960*a^6*b^28*c^9 - 1270087680*a^7*b^26*c^10 + 8255569920*a
^8*b^24*c^11 - 44029706240*a^9*b^22*c^12 + 193730707456*a^10*b^20*c^13 - 704475299840*a^11*b^18*c^14 + 2113425
899520*a^12*b^16*c^15 - 5202279137280*a^13*b^14*c^16 + 10404558274560*a^14*b^12*c^17 - 16647293239296*a^15*b^1
0*c^18 + 20809116549120*a^16*b^8*c^19 - 19585050869760*a^17*b^6*c^20 + 13056700579840*a^18*b^4*c^21 - 54975581
38880*a^19*b^2*c^22)))^(3/4) - (x^(1/2)*(3705625*a^3*b^15*c - 6402256896*a^10*b*c^8 + 281098125*a^4*b^13*c^2 +
7885779000*a^5*b^11*c^3 + 95525940400*a^6*b^9*c^4 + 387469862400*a^7*b^7*c^5 - 497953639680*a^8*b^5*c^6 - 117
420369920*a^9*b^3*c^7))/(4194304*(b^24 + 16777216*a^12*c^12 + 1056*a^2*b^20*c^2 - 14080*a^3*b^18*c^3 + 126720*
a^4*b^16*c^4 - 811008*a^5*b^14*c^5 + 3784704*a^6*b^12*c^6 - 12976128*a^7*b^10*c^7 + 32440320*a^8*b^8*c^8 - 576
71680*a^9*b^6*c^9 + 69206016*a^10*b^4*c^10 - 50331648*a^11*b^2*c^11 - 48*a*b^22*c)))*(-(625*b^31 + 625*b^6*(-(
4*a*c - b^2)^25)^(1/2) - 15192104632320*a^15*b*c^15 - 89000*a^2*b^27*c^2 + 27186416*a^3*b^25*c^3 - 1342297600*
a^4*b^23*c^4 + 25492409600*a^5*b^21*c^5 - 265188833280*a^6*b^19*c^6 + 1688816578560*a^7*b^17*c^7 - 66645041479
68*a^8*b^15*c^8 + 14462970429440*a^9*b^13*c^9 -...

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