3.11.79 \(\int \frac {1}{x^{3/2} (a+b x^2+c x^4)^2} \, dx\) [1079]

Optimal. Leaf size=573 \[ -\frac {5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x^2+c x^4\right )}+\frac {\sqrt [4]{c} \left (5 b^3-28 a b c-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^3-28 a b c+\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^3-28 a b c-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^3-28 a b c+\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \]

[Out]

1/8*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^3-28*a*b*c-(-18*a*c+5*b^2)*(-4*
a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)-1/8*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^3-28*a*b*c-(-18*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*
a*c+b^2)^(3/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)-1/8*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^3-28*a*b*c+(-18*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b^2)^(3/2)/(-b+(-4*a*c+b^2)^(
1/2))^(1/4)+1/8*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^3-28*a*b*c+(-18*a*
c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b^2)^(3/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)+1/2*(18*a*c-5*b^2)/a
^2/(-4*a*c+b^2)/x^(1/2)+1/2*(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)/x^(1/2)

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Rubi [A]
time = 1.67, antiderivative size = 573, 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, 1518, 1524, 304, 211, 214} \begin {gather*} \frac {\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {5 b^2-18 a c}{2 a^2 \sqrt {x} \left (b^2-4 a c\right )}-\frac {\sqrt [4]{c} \left (-\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}+\frac {\sqrt [4]{c} \left (\left (5 b^2-18 a c\right ) \sqrt {b^2-4 a c}-28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt [4]{\sqrt {b^2-4 a c}-b}}+\frac {-2 a c+b^2+b c x^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + b*x^2 + c*x^4)^2),x]

[Out]

-1/2*(5*b^2 - 18*a*c)/(a^2*(b^2 - 4*a*c)*Sqrt[x]) + (b^2 - 2*a*c + b*c*x^2)/(2*a*(b^2 - 4*a*c)*Sqrt[x]*(a + b*
x^2 + c*x^4)) + (c^(1/4)*(5*b^3 - 28*a*b*c - (5*b^2 - 18*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^(3/4)*a^2*(b^2 - 4*a*c)^(3/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - (c^(
1/4)*(5*b^3 - 28*a*b*c + (5*b^2 - 18*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^(3/4)*a^2*(b^2 - 4*a*c)^(3/2)*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) - (c^(1/4)*(5*b^3 - 28*a*b
*c - (5*b^2 - 18*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^(3/4)*a^2*(b^2 - 4*a*c)^(3/2)*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(1/4)*(5*b^3 - 28*a*b*c + (5*b^2 - 18*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^(3/4)*a^2*(b^2 -
4*a*c)^(3/2)*(-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 1518

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^
(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(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] && LtQ[m, -
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 {1}{x^{3/2} \left (a+b x^2+c x^4\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^4+c x^8\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {-5 b^2+18 a c-5 b c x^4}{x^2 \left (a+b x^4+c x^8\right )} \, dx,x,\sqrt {x}\right )}{2 a \left (b^2-4 a c\right )}\\ &=-\frac {5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (-b \left (5 b^2-23 a c\right )-c \left (5 b^2-18 a c\right ) x^4\right )}{a+b x^4+c x^8} \, dx,x,\sqrt {x}\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x^2+c x^4\right )}-\frac {\left (c \left (5 b^2-18 a c+\frac {5 b^3}{\sqrt {b^2-4 a c}}-\frac {28 a b c}{\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 )}{4 a^2 \left (b^2-4 a c\right )}-\frac {\left (c \left (5 b^2-18 a c-\frac {5 b^3}{\sqrt {b^2-4 a c}}+\frac {28 a b c}{\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 )}{4 a^2 \left (b^2-4 a c\right )}\\ &=-\frac {5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x^2+c x^4\right )}+\frac {\left (\sqrt {c} \left (5 b^2-18 a c+\frac {5 b^3}{\sqrt {b^2-4 a c}}-\frac {28 a b c}{\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 )}{4 \sqrt {2} a^2 \left (b^2-4 a c\right )}-\frac {\left (\sqrt {c} \left (5 b^2-18 a c+\frac {5 b^3}{\sqrt {b^2-4 a c}}-\frac {28 a b c}{\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 )}{4 \sqrt {2} a^2 \left (b^2-4 a c\right )}+\frac {\left (\sqrt {c} \left (5 b^2-18 a c-\frac {5 b^3}{\sqrt {b^2-4 a c}}+\frac {28 a b c}{\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 )}{4 \sqrt {2} a^2 \left (b^2-4 a c\right )}-\frac {\left (\sqrt {c} \left (5 b^2-18 a c-\frac {5 b^3}{\sqrt {b^2-4 a c}}+\frac {28 a b c}{\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 )}{4 \sqrt {2} a^2 \left (b^2-4 a c\right )}\\ &=-\frac {5 b^2-18 a c}{2 a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x^2+c x^4\right )}-\frac {\sqrt [4]{c} \left (5 b^2-18 a c-\frac {5 b^3}{\sqrt {b^2-4 a c}}+\frac {28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^2-18 a c+\frac {5 b^3}{\sqrt {b^2-4 a c}}-\frac {28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^2-18 a c-\frac {5 b^3}{\sqrt {b^2-4 a c}}+\frac {28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^2-18 a c+\frac {5 b^3}{\sqrt {b^2-4 a c}}-\frac {28 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 )}{4\ 2^{3/4} a^2 \left (b^2-4 a c\right ) \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 = 0.44, size = 265, normalized size = 0.46 \begin {gather*} -\frac {-\frac {4 \left (16 a^2 c-5 b^2 x^2 \left (b+c x^2\right )+a \left (-4 b^2+19 b c x^2+18 c^2 x^4\right )\right )}{\left (b^2-4 a c\right ) \sqrt {x} \left (a+b x^2+c x^4\right )}+4 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b \log \left (\sqrt {x}-\text {$\#$1}\right )+c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]+\frac {\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^3 \log \left (\sqrt {x}-\text {$\#$1}\right )-7 a b c \log \left (\sqrt {x}-\text {$\#$1}\right )+b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-2 a c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{b^2-4 a c}}{8 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(a + b*x^2 + c*x^4)^2),x]

[Out]

-1/8*((-4*(16*a^2*c - 5*b^2*x^2*(b + c*x^2) + a*(-4*b^2 + 19*b*c*x^2 + 18*c^2*x^4)))/((b^2 - 4*a*c)*Sqrt[x]*(a
+ b*x^2 + c*x^4)) + 4*RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[Sqrt[x] - #1] + c*Log[Sqrt[x] - #1]*#1^4)/(b*#1
+ 2*c*#1^5) & ] + RootSum[a + b*#1^4 + c*#1^8 & , (b^3*Log[Sqrt[x] - #1] - 7*a*b*c*Log[Sqrt[x] - #1] + b^2*c*L
og[Sqrt[x] - #1]*#1^4 - 2*a*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*#1 + 2*c*#1^5) & ]/(b^2 - 4*a*c))/a^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.10, size = 172, normalized size = 0.30

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\frac {c \left (2 a c -b^{2}\right ) x^{\frac {7}{2}}}{16 a c -4 b^{2}}+\frac {b \left (3 a c -b^{2}\right ) x^{\frac {3}{2}}}{16 a c -4 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (18 a c -5 b^{2}\right ) \textit {\_R}^{6}+b \left (23 a c -5 b^{2}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a c -16 b^{2}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(172\)
default	\(-\frac {2 \left (\frac {\frac {c \left (2 a c -b^{2}\right ) x^{\frac {7}{2}}}{16 a c -4 b^{2}}+\frac {b \left (3 a c -b^{2}\right ) x^{\frac {3}{2}}}{16 a c -4 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (18 a c -5 b^{2}\right ) \textit {\_R}^{6}+b \left (23 a c -5 b^{2}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a c -16 b^{2}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(172\)
risch	\(-\frac {2}{a^{2} \sqrt {x}}-\frac {c^{2} x^{\frac {7}{2}}}{a \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}+\frac {c \,x^{\frac {7}{2}} b^{2}}{2 a^{2} \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}-\frac {3 b \,x^{\frac {3}{2}} c}{2 a \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}+\frac {b^{3} x^{\frac {3}{2}}}{2 a^{2} \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (c \left (18 a c -5 b^{2}\right ) \textit {\_R}^{6}+b \left (23 a c -5 b^{2}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{8 a^{2} \left (4 a c -b^{2}\right )}\)	\(245\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a^2*((1/4*c*(2*a*c-b^2)/(4*a*c-b^2)*x^(7/2)+1/4*b*(3*a*c-b^2)/(4*a*c-b^2)*x^(3/2))/(c*x^4+b*x^2+a)+1/16/(4*
a*c-b^2)*sum((c*(18*a*c-5*b^2)*_R^6+b*(23*a*c-5*b^2)*_R^2)/(2*_R^7*c+_R^3*b)*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+_
Z^4*b+a)))-2/a^2/x^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((5*b^2*c - 18*a*c^2)*x^(7/2) + (5*b^3 - 19*a*b*c)*x^(3/2) + 4*(a*b^2 - 4*a^2*c)/sqrt(x))/(a^3*b^2 - 4*a^
4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2) - integrate(1/4*((5*b^2*c - 18*a*c^2)*x^(5/2) +
(5*b^3 - 23*a*b*c)*sqrt(x))/(a^3*b^2 - 4*a^4*c + (a^2*b^2*c - 4*a^3*c^2)*x^4 + (a^2*b^3 - 4*a^3*b*c)*x^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 15175 vs. \(2 (469) = 938\).
time = 203.77, size = 15175, normalized size = 26.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*((a^2*b^2*c - 4*a^3*c^2)*x^5 + (a^2*b^3 - 4*a^3*b*c)*x^3 + (a^3*b^2 - 4*a^4*c)*x)*sqrt(sqrt(1/2)*sqrt(
-(625*b^13 - 14625*a*b^11*c + 137475*a^2*b^9*c^2 - 655590*a^3*b^7*c^3 + 1634841*a^4*b^5*c^4 - 1932840*a^5*b^3*
c^5 + 758160*a^6*b*c^6 - (a^9*b^12 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4
- 6144*a^14*b^2*c^5 + 4096*a^15*c^6)*sqrt((390625*b^20 - 13593750*a*b^18*c + 203859375*a^2*b^16*c^2 - 1716818
750*a^3*b^14*c^3 + 8861859375*a^4*b^12*c^4 - 28693007250*a^5*b^10*c^5 + 57219314050*a^6*b^8*c^6 - 66126243780*
a^7*b^6*c^7 + 39067874721*a^8*b^4*c^8 - 9017490888*a^9*b^2*c^9 + 688747536*a^10*c^10)/(a^18*b^18 - 36*a^19*b^1
6*c + 576*a^20*b^14*c^2 - 5376*a^21*b^12*c^3 + 32256*a^22*b^10*c^4 - 129024*a^23*b^8*c^5 + 344064*a^24*b^6*c^6
- 589824*a^25*b^4*c^7 + 589824*a^26*b^2*c^8 - 262144*a^27*c^9)))/(a^9*b^12 - 24*a^10*b^10*c + 240*a^11*b^8*c^
2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5 + 4096*a^15*c^6)))*arctan(-1/2*((3125*b^16 - 906
25*a*b^14*c + 1113125*a^2*b^12*c^2 - 7497500*a^3*b^10*c^3 + 29884825*a^4*b^8*c^4 - 70582238*a^5*b^6*c^5 + 9241
8696*a^6*b^4*c^6 - 55357344*a^7*b^2*c^7 + 7558272*a^8*c^8 + (5*a^9*b^15 - 148*a^10*b^13*c + 1872*a^11*b^11*c^2
- 13120*a^12*b^9*c^3 + 55040*a^13*b^7*c^4 - 138240*a^14*b^5*c^5 + 192512*a^15*b^3*c^6 - 114688*a^16*b*c^7)*sq
rt((390625*b^20 - 13593750*a*b^18*c + 203859375*a^2*b^16*c^2 - 1716818750*a^3*b^14*c^3 + 8861859375*a^4*b^12*c
^4 - 28693007250*a^5*b^10*c^5 + 57219314050*a^6*b^8*c^6 - 66126243780*a^7*b^6*c^7 + 39067874721*a^8*b^4*c^8 -
9017490888*a^9*b^2*c^9 + 688747536*a^10*c^10)/(a^18*b^18 - 36*a^19*b^16*c + 576*a^20*b^14*c^2 - 5376*a^21*b^12
*c^3 + 32256*a^22*b^10*c^4 - 129024*a^23*b^8*c^5 + 344064*a^24*b^6*c^6 - 589824*a^25*b^4*c^7 + 589824*a^26*b^2
*c^8 - 262144*a^27*c^9)))*sqrt((29460504150390625*b^32*c^14 - 1906321237792968750*a*b^30*c^15 + 57016785906005
859375*a^2*b^28*c^16 - 1044377592206152343750*a^3*b^26*c^17 + 13083244775189443359375*a^4*b^24*c^18 - 11853080
2949736785156250*a^5*b^22*c^19 + 800604258039631850781250*a^6*b^20*c^20 - 4094671424700247652812500*a^7*b^18*c
^21 + 15936145834452127418390625*a^8*b^16*c^22 - 47001721928239386407700000*a^9*b^14*c^23 + 103639285070964365
022360000*a^10*b^12*c^24 - 166605129319347087666624000*a^11*b^10*c^25 + 187261266135093054493017600*a^12*b^8*c
^26 - 137324626675735662427299840*a^13*b^6*c^27 + 58232653591556101463064576*a^14*b^4*c^28 - 11390785137798347
628085248*a^15*b^2*c^29 + 796764763524754885902336*a^16*c^30)*x - 1/2*sqrt(1/2)*(174322509765625*b^41*c^9 - 13
760750732421875*a*b^39*c^10 + 510877840332031250*a^2*b^37*c^11 - 11849949052880859375*a^3*b^35*c^12 + 19239592
9770468750000*a^4*b^33*c^13 - 2321414560345529296875*a^5*b^31*c^14 + 21567618386883643359375*a^6*b^29*c^15 - 1
57732451339060345625000*a^7*b^27*c^16 + 920408933613565924734375*a^8*b^25*c^17 - 4317209296832579116815625*a^9
*b^23*c^18 + 16318538811415112602143125*a^10*b^21*c^19 - 49611749311377582465071000*a^11*b^19*c^20 + 120537558
422009905693280400*a^12*b^17*c^21 - 231348963491962085612943360*a^13*b^15*c^22 + 344476360343672339627579904*a
^14*b^13*c^23 - 387248525625301291868147712*a^15*b^11*c^24 + 315446330908747793410007040*a^16*b^9*c^25 - 17459
6571921689117767434240*a^17*b^7*c^26 + 58984300529125601196441600*a^18*b^5*c^27 - 10043642926062594638217216*a
^19*b^3*c^28 + 647029821682654173462528*a^20*b*c^29 + (278916015625*a^9*b^40*c^9 - 22184550781250*a^10*b^38*c^
10 + 832288361328125*a^11*b^36*c^11 - 19570939608593750*a^12*b^34*c^12 + 323271617376718750*a^13*b^32*c^13 - 3
983893429538503125*a^14*b^30*c^14 + 37970074015626890625*a^15*b^28*c^15 - 286266780074640405000*a^16*b^26*c^16
+ 1731540717669235278000*a^17*b^24*c^17 - 8471573265172367334400*a^18*b^22*c^18 + 33638878345847517227264*a^1
9*b^20*c^19 - 108320215826758770219008*a^20*b^18*c^20 + 281439039542942343016448*a^21*b^16*c^21 - 584308534716
933471797248*a^22*b^14*c^22 + 954415526802929103863808*a^23*b^12*c^23 - 1198125396055113290219520*a^24*b^10*c^
24 + 1116075874846199793057792*a^25*b^8*c^25 - 730808896711525612388352*a^26*b^6*c^26 + 3077927915494084989419
52*a^27*b^4*c^27 - 70848277770584815828992*a^28*b^2*c^28 + 6140942214464815497216*a^29*c^29)*sqrt((390625*b^20
- 13593750*a*b^18*c + 203859375*a^2*b^16*c^2 - 1716818750*a^3*b^14*c^3 + 8861859375*a^4*b^12*c^4 - 2869300725
0*a^5*b^10*c^5 + 57219314050*a^6*b^8*c^6 - 66126243780*a^7*b^6*c^7 + 39067874721*a^8*b^4*c^8 - 9017490888*a^9*
b^2*c^9 + 688747536*a^10*c^10)/(a^18*b^18 - 36*a^19*b^16*c + 576*a^20*b^14*c^2 - 5376*a^21*b^12*c^3 + 32256*a^
22*b^10*c^4 - 129024*a^23*b^8*c^5 + 344064*a^24*b^6*c^6 - 589824*a^25*b^4*c^7 + 589824*a^26*b^2*c^8 - 262144*a
^27*c^9)))*sqrt(-(625*b^13 - 14625*a*b^11*c + 137475*a^2*b^9*c^2 - 655590*a^3*b^7*c^3 + 1634841*a^4*b^5*c^4 -
1932840*a^5*b^3*c^5 + 758160*a^6*b*c^6 - (a^9*b^12 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3
840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5 + 4096*a^15*c^6)*sqrt((390625*b^20 - 13593750*a*b^18*c + 203859375*a^2*b^
16*c^2 - 1716818750*a^3*b^14*c^3 + 8861859375*a...

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 11.42, size = 2500, normalized size = 4.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((x^(1/2)*(602332119171072*a^31*b*c^21 - 54080000*a^20*b^23*c^10 + 2604992000*a^21*b^21*c^11 - 5703444480
0*a^22*b^19*c^12 + 749118545920*a^23*b^17*c^13 - 6557747642368*a^24*b^15*c^14 + 40169229778944*a^25*b^13*c^15
- 175670703423488*a^26*b^11*c^16 + 548447002296320*a^27*b^9*c^17 - 1197821248143360*a^28*b^7*c^18 + 1742819580
444672*a^29*b^5*c^19 - 1520311317037056*a^30*b^3*c^20) + (-(625*b^25 - 625*b^10*(-(4*a*c - b^2)^15)^(1/2) + 31
05423360*a^12*b*c^12 + 638475*a^2*b^21*c^2 - 8264990*a^3*b^19*c^3 + 71483001*a^4*b^17*c^4 - 434478624*a^5*b^15
*c^5 + 1898983360*a^6*b^13*c^6 - 5996689920*a^7*b^11*c^7 + 13524825600*a^8*b^9*c^8 - 21122310144*a^9*b^7*c^9 +
21483012096*a^10*b^5*c^10 - 12575047680*a^11*b^3*c^11 + 26244*a^5*c^5*(-(4*a*c - b^2)^15)^(1/2) - 29625*a*b^2
3*c - 68475*a^2*b^6*c^2*(-(4*a*c - b^2)^15)^(1/2) + 181990*a^3*b^4*c^3*(-(4*a*c - b^2)^15)^(1/2) - 171801*a^4*
b^2*c^4*(-(4*a*c - b^2)^15)^(1/2) + 10875*a*b^8*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(a^9*b^24 + 16777216*a^21*c
^12 - 48*a^10*b^22*c + 1056*a^11*b^20*c^2 - 14080*a^12*b^18*c^3 + 126720*a^13*b^16*c^4 - 811008*a^14*b^14*c^5
+ 3784704*a^15*b^12*c^6 - 12976128*a^16*b^10*c^7 + 32440320*a^17*b^8*c^8 - 57671680*a^18*b^6*c^9 + 69206016*a^
19*b^4*c^10 - 50331648*a^20*b^2*c^11)))^(3/4)*(32768000*a^21*b^34*c^4 - 25649407252758528*a^38*c^21 - 21233664
00*a^22*b^32*c^5 + 64398295040*a^23*b^30*c^6 - 1213399564288*a^24*b^28*c^7 + 15898363035648*a^25*b^26*c^8 - 15
3599583715328*a^26*b^24*c^9 + 1132021560639488*a^27*b^22*c^10 - 6492917279490048*a^28*b^20*c^11 + 292983989851
91424*a^29*b^18*c^12 - 104398826088955904*a^30*b^16*c^13 + 293000581579014144*a^31*b^14*c^14 - 641705669216436
224*a^32*b^12*c^15 + 1077743462209552384*a^33*b^10*c^16 - 1348355710714380288*a^34*b^8*c^17 + 1198053158392168
448*a^35*b^6*c^18 - 695801744382230528*a^36*b^4*c^19 + 223957324438437888*a^37*b^2*c^20 + x^(1/2)*(-(625*b^25
- 625*b^10*(-(4*a*c - b^2)^15)^(1/2) + 3105423360*a^12*b*c^12 + 638475*a^2*b^21*c^2 - 8264990*a^3*b^19*c^3 + 7
1483001*a^4*b^17*c^4 - 434478624*a^5*b^15*c^5 + 1898983360*a^6*b^13*c^6 - 5996689920*a^7*b^11*c^7 + 1352482560
0*a^8*b^9*c^8 - 21122310144*a^9*b^7*c^9 + 21483012096*a^10*b^5*c^10 - 12575047680*a^11*b^3*c^11 + 26244*a^5*c^
5*(-(4*a*c - b^2)^15)^(1/2) - 29625*a*b^23*c - 68475*a^2*b^6*c^2*(-(4*a*c - b^2)^15)^(1/2) + 181990*a^3*b^4*c^
3*(-(4*a*c - b^2)^15)^(1/2) - 171801*a^4*b^2*c^4*(-(4*a*c - b^2)^15)^(1/2) + 10875*a*b^8*c*(-(4*a*c - b^2)^15)
^(1/2))/(8192*(a^9*b^24 + 16777216*a^21*c^12 - 48*a^10*b^22*c + 1056*a^11*b^20*c^2 - 14080*a^12*b^18*c^3 + 126
720*a^13*b^16*c^4 - 811008*a^14*b^14*c^5 + 3784704*a^15*b^12*c^6 - 12976128*a^16*b^10*c^7 + 32440320*a^17*b^8*
c^8 - 57671680*a^18*b^6*c^9 + 69206016*a^19*b^4*c^10 - 50331648*a^20*b^2*c^11)))^(1/4)*(91197892454252544*a^40
*c^21 - 52428800*a^23*b^34*c^4 + 3418357760*a^24*b^32*c^5 - 104457043968*a^25*b^30*c^6 + 1986074247168*a^26*b^
28*c^7 - 26302715265024*a^27*b^26*c^8 + 257340683059200*a^28*b^24*c^9 - 1924694567550976*a^29*b^22*c^10 + 1123
0133666971648*a^30*b^20*c^11 - 51694329453871104*a^31*b^18*c^12 + 188531248770056192*a^32*b^16*c^13 - 54372155
6635811840*a^33*b^14*c^14 + 1229750704231415808*a^34*b^12*c^15 - 2146620531372195840*a^35*b^10*c^16 + 28158800
65059913728*a^36*b^8*c^17 - 2657721914474102784*a^37*b^6*c^18 + 1675831642591068160*a^38*b^4*c^19 - 6124895493
22387456*a^39*b^2*c^20)))*(-(625*b^25 - 625*b^10*(-(4*a*c - b^2)^15)^(1/2) + 3105423360*a^12*b*c^12 + 638475*a
^2*b^21*c^2 - 8264990*a^3*b^19*c^3 + 71483001*a^4*b^17*c^4 - 434478624*a^5*b^15*c^5 + 1898983360*a^6*b^13*c^6
- 5996689920*a^7*b^11*c^7 + 13524825600*a^8*b^9*c^8 - 21122310144*a^9*b^7*c^9 + 21483012096*a^10*b^5*c^10 - 12
575047680*a^11*b^3*c^11 + 26244*a^5*c^5*(-(4*a*c - b^2)^15)^(1/2) - 29625*a*b^23*c - 68475*a^2*b^6*c^2*(-(4*a*
c - b^2)^15)^(1/2) + 181990*a^3*b^4*c^3*(-(4*a*c - b^2)^15)^(1/2) - 171801*a^4*b^2*c^4*(-(4*a*c - b^2)^15)^(1/
2) + 10875*a*b^8*c*(-(4*a*c - b^2)^15)^(1/2))/(8192*(a^9*b^24 + 16777216*a^21*c^12 - 48*a^10*b^22*c + 1056*a^1
1*b^20*c^2 - 14080*a^12*b^18*c^3 + 126720*a^13*b^16*c^4 - 811008*a^14*b^14*c^5 + 3784704*a^15*b^12*c^6 - 12976
128*a^16*b^10*c^7 + 32440320*a^17*b^8*c^8 - 57671680*a^18*b^6*c^9 + 69206016*a^19*b^4*c^10 - 50331648*a^20*b^2
*c^11)))^(1/4)*1i + (x^(1/2)*(602332119171072*a^31*b*c^21 - 54080000*a^20*b^23*c^10 + 2604992000*a^21*b^21*c^1
1 - 57034444800*a^22*b^19*c^12 + 749118545920*a^23*b^17*c^13 - 6557747642368*a^24*b^15*c^14 + 40169229778944*a
^25*b^13*c^15 - 175670703423488*a^26*b^11*c^16 + 548447002296320*a^27*b^9*c^17 - 1197821248143360*a^28*b^7*c^1
8 + 1742819580444672*a^29*b^5*c^19 - 1520311317037056*a^30*b^3*c^20) + (-(625*b^25 - 625*b^10*(-(4*a*c - b^2)^
15)^(1/2) + 3105423360*a^12*b*c^12 + 638475*a^2*b^21*c^2 - 8264990*a^3*b^19*c^3 + 71483001*a^4*b^17*c^4 - 4344
78624*a^5*b^15*c^5 + 1898983360*a^6*b^13*c^6 - 5996689920*a^7*b^11*c^7 + 13524825600*a^8*b^9*c^8 - 21122310144
*a^9*b^7*c^9 + 21483012096*a^10*b^5*c^10 - 12575047680*a^11*b^3*c^11 + 26244*a^5*c^5*(-(4*a*c - b^2)^15)^(1/2)
- 29625*a*b^23*c - 68475*a^2*b^6*c^2*(-(4*a*c ...

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