3.11.72 \(\int \frac {x^{11/2}}{(a+b x^2+c x^4)^2} \, dx\) [1072]

Optimal. Leaf size=520 \[ -\frac {b \sqrt {x}}{2 c \left (b^2-4 a c\right )}+\frac {x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (b^2-10 a c+\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (b^2-10 a c-\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (b^2-10 a c+\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (b^2-10 a c-\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \]

[Out]

1/2*x^(5/2)*(b*x^2+2*a)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)-1/8*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2)
)^(1/4))*(b^2-10*a*c+b*(-12*a*c+b^2)/(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(5/4)/(-4*a*c+b^2)/(-b-(-4*a*c+b^2)^(1/2))^
(3/4)-1/8*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*(b^2-10*a*c+b*(-12*a*c+b^2)/(-4*a*c+b
^2)^(1/2))*2^(3/4)/c^(5/4)/(-4*a*c+b^2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-1/8*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(
-4*a*c+b^2)^(1/2))^(1/4))*(b^2-10*a*c-b*(-12*a*c+b^2)/(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(5/4)/(-4*a*c+b^2)/(-b+(-4
*a*c+b^2)^(1/2))^(3/4)-1/8*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(b^2-10*a*c-b*(-12*a
*c+b^2)/(-4*a*c+b^2)^(1/2))*2^(3/4)/c^(5/4)/(-4*a*c+b^2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)-1/2*b*x^(1/2)/c/(-4*a*c
+b^2)

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Rubi [A]
time = 1.07, antiderivative size = 520, 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, 1516, 1436, 218, 214, 211} \begin {gather*} -\frac {\left (\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a c+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 )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\left (-\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a c+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 )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\left (\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a c+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 )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\left (-\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}-10 a c+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 )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}+\frac {x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b \sqrt {x}}{2 c \left (b^2-4 a c\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(11/2)/(a + b*x^2 + c*x^4)^2,x]

[Out]

-1/2*(b*Sqrt[x])/(c*(b^2 - 4*a*c)) + (x^(5/2)*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((b^2 - 1
0*a*c + (b*(b^2 - 12*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)]
)/(4*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - ((b^2 - 10*a*c - (b*(b^2 - 12*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)])/(4*2^(1/4)*c^(5/4)*(b^2 - 4*a*
c)*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - ((b^2 - 10*a*c + (b*(b^2 - 12*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)])/(4*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)*(-b - Sqrt[b^2 - 4*a*c])^(3/
4)) - ((b^2 - 10*a*c - (b*(b^2 - 12*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)])/(4*2^(1/4)*c^(5/4)*(b^2 - 4*a*c)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[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 1436

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 1516

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

Rubi steps

\begin {align*} \int \frac {x^{11/2}}{\left (a+b x^2+c x^4\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^{12}}{\left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {x^4 \left (10 a+b x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {b \sqrt {x}}{2 c \left (b^2-4 a c\right )}+\frac {x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {a b+\left (b^2-10 a c\right ) x^4}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{2 c \left (b^2-4 a c\right )}\\ &=-\frac {b \sqrt {x}}{2 c \left (b^2-4 a c\right )}+\frac {x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (b^2-10 a c-\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{4 c \left (b^2-4 a c\right )}+\frac {\left (b^2-10 a c+\frac {b \left (b^2-12 a c\right )}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^4} \, dx,x,\sqrt {x}\right )}{4 c \left (b^2-4 a c\right )}\\ &=-\frac {b \sqrt {x}}{2 c \left (b^2-4 a c\right )}+\frac {x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (b^2-10 a c+\frac {b \left (b^2-12 a c\right )}{\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 )}{4 c \left (b^2-4 a c\right ) \sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\left (b^2-10 a c+\frac {b \left (b^2-12 a c\right )}{\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 )}{4 c \left (b^2-4 a c\right ) \sqrt {-b-\sqrt {b^2-4 a c}}}-\frac {\left (b^2-10 a c-\frac {b \left (b^2-12 a c\right )}{\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 )}{4 c \left (b^2-4 a c\right ) \sqrt {-b+\sqrt {b^2-4 a c}}}-\frac {\left (b^2-10 a c-\frac {b \left (b^2-12 a c\right )}{\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 )}{4 c \left (b^2-4 a c\right ) \sqrt {-b+\sqrt {b^2-4 a c}}}\\ &=-\frac {b \sqrt {x}}{2 c \left (b^2-4 a c\right )}+\frac {x^{5/2} \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (b^2-10 a c+\frac {b \left (b^2-12 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 )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (b^2-10 a c-\frac {b \left (b^2-12 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 )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (b^2-10 a c+\frac {b \left (b^2-12 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 )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {\left (b^2-10 a c-\frac {b \left (b^2-12 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 )}{4 \sqrt [4]{2} c^{5/4} \left (b^2-4 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.39, size = 237, normalized size = 0.46 \begin {gather*} -\frac {4 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b \log \left (\sqrt {x}-\text {$\#$1}\right )-c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]+\frac {\frac {4 c \sqrt {x} \left (a b+b^2 x^2-2 a c x^2\right )}{a+b x^2+c x^4}+\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-4 b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+15 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+3 b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-6 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{b^2-4 a c}}{8 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(11/2)/(a + b*x^2 + c*x^4)^2,x]

[Out]

-1/8*(4*RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[Sqrt[x] - #1] - c*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) &
] + ((4*c*Sqrt[x]*(a*b + b^2*x^2 - 2*a*c*x^2))/(a + b*x^2 + c*x^4) + RootSum[a + b*#1^4 + c*#1^8 & , (-4*b^3*
Log[Sqrt[x] - #1] + 15*a*b*c*Log[Sqrt[x] - #1] + 3*b^2*c*Log[Sqrt[x] - #1]*#1^4 - 6*a*c^2*Log[Sqrt[x] - #1]*#1
^4)/(b*#1^3 + 2*c*#1^7) & ])/(b^2 - 4*a*c))/c^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.06, size = 146, normalized size = 0.28

method	result	size
derivativedivides	\(\frac {-\frac {\left (2 a c -b^{2}\right ) x^{\frac {5}{2}}}{2 c \left (4 a c -b^{2}\right )}+\frac {a b \sqrt {x}}{2 c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (10 a c -b^{2}\right ) \textit {\_R}^{4}-a b \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{8 c \left (4 a c -b^{2}\right )}\)	\(146\)
default	\(\frac {-\frac {\left (2 a c -b^{2}\right ) x^{\frac {5}{2}}}{2 c \left (4 a c -b^{2}\right )}+\frac {a b \sqrt {x}}{2 c \left (4 a c -b^{2}\right )}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (10 a c -b^{2}\right ) \textit {\_R}^{4}-a b \right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{8 c \left (4 a c -b^{2}\right )}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(11/2)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

2*(-1/4*(2*a*c-b^2)/c/(4*a*c-b^2)*x^(5/2)+1/4*a*b/c/(4*a*c-b^2)*x^(1/2))/(c*x^4+b*x^2+a)+1/8/c/(4*a*c-b^2)*sum
(((10*a*c-b^2)*_R^4-a*b)/(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^(11/2)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*(b*x^(9/2) + 2*a*x^(5/2))/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2) + integrate(-1/4
*(b*x^(7/2) + 10*a*x^(3/2))/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11906 vs. \(2 (424) = 848\).
time = 17.99, size = 11906, normalized size = 22.90 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(11/2)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/8*(4*((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^2)*sqrt(sqrt(1/2)*sqrt(-(b^9 -
45*a*b^7*c + 765*a^2*b^5*c^2 - 5880*a^3*b^3*c^3 + 18000*a^4*b*c^4 + (b^12*c^5 - 24*a*b^10*c^6 + 240*a^2*b^8*c^
7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10 + 4096*a^6*c^11)*sqrt((b^12 - 78*a*b^10*c + 2571*a
^2*b^8*c^2 - 45950*a^3*b^6*c^3 + 470625*a^4*b^4*c^4 - 2625000*a^5*b^2*c^5 + 6250000*a^6*c^6)/(b^18*c^10 - 36*a
*b^16*c^11 + 576*a^2*b^14*c^12 - 5376*a^3*b^12*c^13 + 32256*a^4*b^10*c^14 - 129024*a^5*b^8*c^15 + 344064*a^6*b
^6*c^16 - 589824*a^7*b^4*c^17 + 589824*a^8*b^2*c^18 - 262144*a^9*c^19)))/(b^12*c^5 - 24*a*b^10*c^6 + 240*a^2*b
^8*c^7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10 + 4096*a^6*c^11)))*arctan(-1/2*(sqrt(1/2)*(b^
22 - 91*a*b^20*c + 3683*a^2*b^18*c^2 - 87230*a^3*b^16*c^3 + 1338850*a^4*b^14*c^4 - 13940024*a^5*b^12*c^5 + 100
253344*a^6*b^10*c^6 - 497651072*a^7*b^8*c^7 + 1672046080*a^8*b^6*c^8 - 3627264000*a^9*b^4*c^9 + 4582400000*a^1
0*b^2*c^10 - 2560000000*a^11*c^11 - (b^25*c^5 - 70*a*b^23*c^6 + 2192*a^2*b^21*c^7 - 40672*a^3*b^19*c^8 + 49843
2*a^4*b^17*c^9 - 4254720*a^5*b^15*c^10 + 25976832*a^6*b^13*c^11 - 114475008*a^7*b^11*c^12 + 361955328*a^8*b^9*
c^13 - 802029568*a^9*b^7*c^14 + 1183842304*a^10*b^5*c^15 - 1046478848*a^11*b^3*c^16 + 419430400*a^12*b*c^17)*s
qrt((b^12 - 78*a*b^10*c + 2571*a^2*b^8*c^2 - 45950*a^3*b^6*c^3 + 470625*a^4*b^4*c^4 - 2625000*a^5*b^2*c^5 + 62
50000*a^6*c^6)/(b^18*c^10 - 36*a*b^16*c^11 + 576*a^2*b^14*c^12 - 5376*a^3*b^12*c^13 + 32256*a^4*b^10*c^14 - 12
9024*a^5*b^8*c^15 + 344064*a^6*b^6*c^16 - 589824*a^7*b^4*c^17 + 589824*a^8*b^2*c^18 - 262144*a^9*c^19)))*sqrt(
(81*a^2*b^16 - 8118*a^3*b^14*c + 358651*a^4*b^12*c^2 - 9129750*a^5*b^10*c^3 + 146540625*a^6*b^8*c^4 - 15192500
00*a^7*b^6*c^5 + 9937500000*a^8*b^4*c^6 - 37500000000*a^9*b^2*c^7 + 62500000000*a^10*c^8)*x + 1/2*sqrt(1/2)*(b
^22 - 112*a*b^20*c + 5735*a^2*b^18*c^2 - 176820*a^3*b^16*c^3 + 3634845*a^4*b^14*c^4 - 52073994*a^5*b^12*c^5 +
527503968*a^6*b^10*c^6 - 3751826400*a^7*b^8*c^7 + 18208800000*a^8*b^6*c^8 - 56920000000*a^9*b^4*c^9 + 10240000
0000*a^10*b^2*c^10 - 80000000000*a^11*c^11 - (b^25*c^5 - 91*a*b^23*c^6 + 3641*a^2*b^21*c^7 - 84776*a^3*b^19*c^
8 + 1280016*a^4*b^17*c^9 - 13215744*a^5*b^15*c^10 + 95875584*a^6*b^13*c^11 - 493891584*a^7*b^11*c^12 + 1798938
624*a^8*b^9*c^13 - 4533059584*a^9*b^7*c^14 + 7523860480*a^10*b^5*c^15 - 7405568000*a^11*b^3*c^16 + 3276800000*
a^12*b*c^17)*sqrt((b^12 - 78*a*b^10*c + 2571*a^2*b^8*c^2 - 45950*a^3*b^6*c^3 + 470625*a^4*b^4*c^4 - 2625000*a^
5*b^2*c^5 + 6250000*a^6*c^6)/(b^18*c^10 - 36*a*b^16*c^11 + 576*a^2*b^14*c^12 - 5376*a^3*b^12*c^13 + 32256*a^4*
b^10*c^14 - 129024*a^5*b^8*c^15 + 344064*a^6*b^6*c^16 - 589824*a^7*b^4*c^17 + 589824*a^8*b^2*c^18 - 262144*a^9
*c^19)))*sqrt(-(b^9 - 45*a*b^7*c + 765*a^2*b^5*c^2 - 5880*a^3*b^3*c^3 + 18000*a^4*b*c^4 + (b^12*c^5 - 24*a*b^1
0*c^6 + 240*a^2*b^8*c^7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10 + 4096*a^6*c^11)*sqrt((b^12
- 78*a*b^10*c + 2571*a^2*b^8*c^2 - 45950*a^3*b^6*c^3 + 470625*a^4*b^4*c^4 - 2625000*a^5*b^2*c^5 + 6250000*a^6*
c^6)/(b^18*c^10 - 36*a*b^16*c^11 + 576*a^2*b^14*c^12 - 5376*a^3*b^12*c^13 + 32256*a^4*b^10*c^14 - 129024*a^5*b
^8*c^15 + 344064*a^6*b^6*c^16 - 589824*a^7*b^4*c^17 + 589824*a^8*b^2*c^18 - 262144*a^9*c^19)))/(b^12*c^5 - 24*
a*b^10*c^6 + 240*a^2*b^8*c^7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10 + 4096*a^6*c^11)))*sqrt
(-(b^9 - 45*a*b^7*c + 765*a^2*b^5*c^2 - 5880*a^3*b^3*c^3 + 18000*a^4*b*c^4 + (b^12*c^5 - 24*a*b^10*c^6 + 240*a
^2*b^8*c^7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10 + 4096*a^6*c^11)*sqrt((b^12 - 78*a*b^10*c
+ 2571*a^2*b^8*c^2 - 45950*a^3*b^6*c^3 + 470625*a^4*b^4*c^4 - 2625000*a^5*b^2*c^5 + 6250000*a^6*c^6)/(b^18*c^
10 - 36*a*b^16*c^11 + 576*a^2*b^14*c^12 - 5376*a^3*b^12*c^13 + 32256*a^4*b^10*c^14 - 129024*a^5*b^8*c^15 + 344
064*a^6*b^6*c^16 - 589824*a^7*b^4*c^17 + 589824*a^8*b^2*c^18 - 262144*a^9*c^19)))/(b^12*c^5 - 24*a*b^10*c^6 +
240*a^2*b^8*c^7 - 1280*a^3*b^6*c^8 + 3840*a^4*b^4*c^9 - 6144*a^5*b^2*c^10 + 4096*a^6*c^11)) - sqrt(1/2)*(9*a*b
^30 - 1270*a^2*b^28*c + 82813*a^3*b^26*c^2 - 3305978*a^4*b^24*c^3 + 90231255*a^5*b^22*c^4 - 1780615316*a^6*b^2
0*c^5 + 26199812170*a^7*b^18*c^6 - 292147074792*a^8*b^16*c^7 + 2484388440192*a^9*b^14*c^8 - 16082985454080*a^1
0*b^12*c^9 + 78485701504000*a^11*b^10*c^10 - 283191078400000*a^12*b^8*c^11 + 730734080000000*a^13*b^6*c^12 - 1
272576000000000*a^14*b^4*c^13 + 1337600000000000*a^15*b^2*c^14 - 640000000000000*a^16*c^15 - (9*a*b^33*c^5 - 1
081*a^2*b^31*c^6 + 59923*a^3*b^29*c^7 - 2033390*a^4*b^27*c^8 + 47234960*a^5*b^25*c^9 - 795781312*a^6*b^23*c^10
+ 10050046208*a^7*b^21*c^11 - 96993186304*a^8*b^19*c^12 + 722648002560*a^9*b^17*c^13 - 4169749463040*a^10*b^1
5*c^14 + 18574068219904*a^11*b^13*c^15 - 63226237812736*a^12*b^11*c^16 + 161327426306048*a^13*b^9*c^17 - 29851
0607974400*a^14*b^7*c^18 + 378064076800000*a^15*b^5*c^19 - 293076992000000*a^16*b^3*c^20 + 104857600000000*a^1
7*b*c^21)*sqrt((b^12 - 78*a*b^10*c + 2571*a^2*b...

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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**(11/2)/(c*x**4+b*x**2+a)**2,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^(11/2)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate(x^(11/2)/(c*x^4 + b*x^2 + a)^2, x)

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Mupad [B]
time = 11.85, size = 2500, normalized size = 4.81 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(11/2)/(a + b*x^2 + c*x^4)^2,x)

[Out]

2*atan(((((9*a^3*b^9 - 397*a^4*b^7*c + 130000*a^7*b*c^4 + 6549*a^5*b^5*c^2 - 47800*a^6*b^3*c^3)/(2*(b^8*c + 25
6*a^4*c^5 - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4)) + ((x^(1/2)*(1006632960*a^10*b*c^11 + 4096*a^3*b
^15*c^4 + 147456*a^4*b^13*c^5 - 4915200*a^5*b^11*c^6 + 53739520*a^6*b^9*c^7 - 298844160*a^7*b^7*c^8 + 91855257
6*a^8*b^5*c^9 - 1493172224*a^9*b^3*c^10))/(16*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*
a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)) - ((-(b^21 + b^6*(-(4*a*c - b^2)^15)^(1/2) + 73728000*a^10
*b*c^10 + 2085*a^2*b^17*c^2 - 36320*a^3*b^15*c^3 + 404160*a^4*b^13*c^4 - 3001344*a^5*b^11*c^5 + 15064576*a^6*b
^9*c^6 - 50503680*a^7*b^7*c^7 + 108380160*a^8*b^5*c^8 - 134676480*a^9*b^3*c^9 - 2500*a^3*c^3*(-(4*a*c - b^2)^1
5)^(1/2) - 69*a*b^19*c + 525*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 39*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2))/(81
92*(16777216*a^12*c^17 + b^24*c^5 - 48*a*b^22*c^6 + 1056*a^2*b^20*c^7 - 14080*a^3*b^18*c^8 + 126720*a^4*b^16*c
^9 - 811008*a^5*b^14*c^10 + 3784704*a^6*b^12*c^11 - 12976128*a^7*b^10*c^12 + 32440320*a^8*b^8*c^13 - 57671680*
a^9*b^6*c^14 + 69206016*a^10*b^4*c^15 - 50331648*a^11*b^2*c^16)))^(1/4)*(167772160*a^9*c^11 + 40960*a^3*b^12*c
^5 - 983040*a^4*b^10*c^6 + 9830400*a^5*b^8*c^7 - 52428800*a^6*b^6*c^8 + 157286400*a^7*b^4*c^9 - 251658240*a^8*
b^2*c^10)*1i)/(2*(b^8*c + 256*a^4*c^5 - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4)))*(-(b^21 + b^6*(-(4*
a*c - b^2)^15)^(1/2) + 73728000*a^10*b*c^10 + 2085*a^2*b^17*c^2 - 36320*a^3*b^15*c^3 + 404160*a^4*b^13*c^4 - 3
001344*a^5*b^11*c^5 + 15064576*a^6*b^9*c^6 - 50503680*a^7*b^7*c^7 + 108380160*a^8*b^5*c^8 - 134676480*a^9*b^3*
c^9 - 2500*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 69*a*b^19*c + 525*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 39*a*
b^4*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^17 + b^24*c^5 - 48*a*b^22*c^6 + 1056*a^2*b^20*c^7 - 14
080*a^3*b^18*c^8 + 126720*a^4*b^16*c^9 - 811008*a^5*b^14*c^10 + 3784704*a^6*b^12*c^11 - 12976128*a^7*b^10*c^12
+ 32440320*a^8*b^8*c^13 - 57671680*a^9*b^6*c^14 + 69206016*a^10*b^4*c^15 - 50331648*a^11*b^2*c^16)))^(3/4)*1i
)*(-(b^21 + b^6*(-(4*a*c - b^2)^15)^(1/2) + 73728000*a^10*b*c^10 + 2085*a^2*b^17*c^2 - 36320*a^3*b^15*c^3 + 40
4160*a^4*b^13*c^4 - 3001344*a^5*b^11*c^5 + 15064576*a^6*b^9*c^6 - 50503680*a^7*b^7*c^7 + 108380160*a^8*b^5*c^8
- 134676480*a^9*b^3*c^9 - 2500*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 69*a*b^19*c + 525*a^2*b^2*c^2*(-(4*a*c - b
^2)^15)^(1/2) - 39*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^17 + b^24*c^5 - 48*a*b^22*c^6 + 1
056*a^2*b^20*c^7 - 14080*a^3*b^18*c^8 + 126720*a^4*b^16*c^9 - 811008*a^5*b^14*c^10 + 3784704*a^6*b^12*c^11 - 1
2976128*a^7*b^10*c^12 + 32440320*a^8*b^8*c^13 - 57671680*a^9*b^6*c^14 + 69206016*a^10*b^4*c^15 - 50331648*a^11
*b^2*c^16)))^(1/4)*1i - (x^(1/2)*(81*a^4*b^10 - 2000000*a^9*c^5 - 3744*a^5*b^8*c + 66322*a^6*b^6*c^2 - 547800*
a^7*b^4*c^3 + 1980000*a^8*b^2*c^4))/(16*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^
6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))*(-(b^21 + b^6*(-(4*a*c - b^2)^15)^(1/2) + 73728000*a^10*b*c^10
+ 2085*a^2*b^17*c^2 - 36320*a^3*b^15*c^3 + 404160*a^4*b^13*c^4 - 3001344*a^5*b^11*c^5 + 15064576*a^6*b^9*c^6 -
50503680*a^7*b^7*c^7 + 108380160*a^8*b^5*c^8 - 134676480*a^9*b^3*c^9 - 2500*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2)
- 69*a*b^19*c + 525*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 39*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(1677
7216*a^12*c^17 + b^24*c^5 - 48*a*b^22*c^6 + 1056*a^2*b^20*c^7 - 14080*a^3*b^18*c^8 + 126720*a^4*b^16*c^9 - 811
008*a^5*b^14*c^10 + 3784704*a^6*b^12*c^11 - 12976128*a^7*b^10*c^12 + 32440320*a^8*b^8*c^13 - 57671680*a^9*b^6*
c^14 + 69206016*a^10*b^4*c^15 - 50331648*a^11*b^2*c^16)))^(1/4) - (((9*a^3*b^9 - 397*a^4*b^7*c + 130000*a^7*b*
c^4 + 6549*a^5*b^5*c^2 - 47800*a^6*b^3*c^3)/(2*(b^8*c + 256*a^4*c^5 - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*
b^2*c^4)) - ((x^(1/2)*(1006632960*a^10*b*c^11 + 4096*a^3*b^15*c^4 + 147456*a^4*b^13*c^5 - 4915200*a^5*b^11*c^6
+ 53739520*a^6*b^9*c^7 - 298844160*a^7*b^7*c^8 + 918552576*a^8*b^5*c^9 - 1493172224*a^9*b^3*c^10))/(16*(b^12*
c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6))
+ ((-(b^21 + b^6*(-(4*a*c - b^2)^15)^(1/2) + 73728000*a^10*b*c^10 + 2085*a^2*b^17*c^2 - 36320*a^3*b^15*c^3 + 4
04160*a^4*b^13*c^4 - 3001344*a^5*b^11*c^5 + 15064576*a^6*b^9*c^6 - 50503680*a^7*b^7*c^7 + 108380160*a^8*b^5*c^
8 - 134676480*a^9*b^3*c^9 - 2500*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 69*a*b^19*c + 525*a^2*b^2*c^2*(-(4*a*c -
b^2)^15)^(1/2) - 39*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(16777216*a^12*c^17 + b^24*c^5 - 48*a*b^22*c^6 +
1056*a^2*b^20*c^7 - 14080*a^3*b^18*c^8 + 126720*a^4*b^16*c^9 - 811008*a^5*b^14*c^10 + 3784704*a^6*b^12*c^11 -
12976128*a^7*b^10*c^12 + 32440320*a^8*b^8*c^13 - 57671680*a^9*b^6*c^14 + 69206016*a^10*b^4*c^15 - 50331648*a^1
1*b^2*c^16)))^(1/4)*(167772160*a^9*c^11 + 40960*a^3*b^12*c^5 - 983040*a^4*b^10*c^6 + 9830400*a^5*b^8*c^7 - 524
28800*a^6*b^6*c^8 + 157286400*a^7*b^4*c^9 - 251...

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