3.886 \(\int \frac{1}{(a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=355 \[ \frac{3 \sqrt{c} \left (56 a^2 c^2-10 a b^2 c+b \left (b^2-8 a c\right ) \sqrt{b^2-4 a c}+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left (-\frac{56 a^2 c^2-10 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-8 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

(x*(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*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*
b*c*(b^2 - 8*a*c)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^
2 + b*(b^2 - 8*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2
*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(b^3 - 8*a*b*c - (b^4 - 10*a*b^2*c + 56*a^2*c^2
)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^2*S
qrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 1.83832, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1092, 1178, 1166, 205} \[ \frac{3 \sqrt{c} \left (56 a^2 c^2-10 a b^2 c+b \left (b^2-8 a c\right ) \sqrt{b^2-4 a c}+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} \left (-\frac{56 a^2 c^2-10 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-8 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^(-3),x]

[Out]

(x*(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*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*
b*c*(b^2 - 8*a*c)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^
2 + b*(b^2 - 8*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2
*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(b^3 - 8*a*b*c - (b^4 - 10*a*b^2*c + 56*a^2*c^2
)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^2*S
qrt[b + Sqrt[b^2 - 4*a*c]])

Rule 1092

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^
4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(b^2 - 2*a*c + 2*(p + 1)
*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^2+c x^4\right )^3} \, dx &=\frac{x \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{\int \frac{b^2-2 a c-4 \left (b^2-4 a c\right )-5 b c x^2}{\left (a+b x^2+c x^4\right )^2} \, dx}{4 a \left (b^2-4 a c\right )}\\ &=\frac{x \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 \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\int \frac{3 \left (b^4-9 a b^2 c+28 a^2 c^2\right )+3 b c \left (b^2-8 a c\right ) x^2}{a+b x^2+c x^4} \, dx}{8 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac{x \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 \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{\left (3 c \left (b^4-10 a b^2 c+56 a^2 c^2-b \left (b^2-8 a c\right ) \sqrt{b^2-4 a c}\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{16 a^2 \left (b^2-4 a c\right )^{5/2}}+\frac{\left (3 c \left (b^4-10 a b^2 c+56 a^2 c^2+b \left (b^2-8 a c\right ) \sqrt{b^2-4 a c}\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{16 a^2 \left (b^2-4 a c\right )^{5/2}}\\ &=\frac{x \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 \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{c} \left (b^4-10 a b^2 c+56 a^2 c^2+b \left (b^2-8 a c\right ) \sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{c} \left (b^4-10 a b^2 c+56 a^2 c^2-b \left (b^2-8 a c\right ) \sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 1.05164, size = 372, normalized size = 1.05 \[ \frac{\frac{2 x \left (28 a^2 c^2-25 a b^2 c-24 a b c^2 x^2+3 b^3 c x^2+3 b^4\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \sqrt{c} \left (56 a^2 c^2+b^3 \sqrt{b^2-4 a c}-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} \left (56 a^2 c^2-b^3 \sqrt{b^2-4 a c}-10 a b^2 c+8 a b c \sqrt{b^2-4 a c}+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{4 a x \left (-2 a c+b^2+b c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}}{16 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^(-3),x]

[Out]

((4*a*x*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*x*(3*b^4 - 25*a*b^2*c + 28*a^2*c^2
+ 3*b^3*c*x^2 - 24*a*b*c^2*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c
+ 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^
2 - 4*a*c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^
2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*
c]]])/((b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*a^2)

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Maple [B]  time = 0.261, size = 3360, normalized size = 9.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*c^2/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/a*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((b+(-4*a*c+
b^2)^(1/2))*c)^(1/2))*b^5+3/16*c/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/a^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc
tan(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^7-3/16*c/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/a^2*2^(1/2)/((b+
(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^8+114*c^4/(-4*a*c+b^2)^(5/
2)/(4*a*c-b^2)^2*a*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2
))*b^2+114*c^4/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x*c*2^(1/2
)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2-3/16*c/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/a^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/
2))*c)^(1/2)*arctanh(x*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^7+27/8*c^2/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2
)^2/a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^6-3/1
6*c/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/a^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x*c*2^(1/2)/((-b+(-
4*a*c+b^2)^(1/2))*c)^(1/2))*b^8+24*c^4/(-4*a*c+b^2)^2/(4*a*c-b^2)^2*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2
)*arctanh(x*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b-24*c^4/(-4*a*c+b^2)^2/(4*a*c-b^2)^2*a*2^(1/2)/((b+(
-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b+3*c^2/(-4*a*c+b^2)^2/(4*a*c
-b^2)^2/a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^5
+27/8*c^2/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/a*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((b+(
-4*a*c+b^2)^(1/2))*c)^(1/2))*b^6-24*c^3/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*
a*x^3*b-15*c^3/(-4*a*c+b^2)^2/(4*a*c-b^2)^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x*c*2^(1/2)/((-b
+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3+27*c^2/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^
2*a*x*b^2-3*c/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2/a*x^3*b^5-168*c^5/(-4*a*c+
b^2)^(5/2)/(4*a*c-b^2)^2*a^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2
))*c)^(1/2))+27/8*c/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2/a*x^3*b^6-15*c^2
/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*a*x*b^3-168*c^5/(-4*a*c+b^2)^(5/2)/
(4*a*c-b^2)^2*a^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1
/2))-27/8*c/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2/a*x^3*b^6+15*c^2/(-4*a*c
+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*a*x*b^3-3*c/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x
^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2/a*x^3*b^5+27*c^2/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*
c+b^2)^(1/2))^2*a*x*b^2-57/2*c^3/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*ar
ctanh(x*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4-57/2*c^3/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2*2^(1/2)/((b
+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4+15*c^3/(-4*a*c+b^2)^2/(
4*a*c-b^2)^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3
-24*c^3/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*a*x^3*b+20*c^3/(-4*a*c+b^2)^(5/2
)/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*a^2*x*b+66*c^3/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+
1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*a*x^3*b^2-15/4*c/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a
*c+b^2)^(1/2))^2*x*b^5+5/16/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2/a*x*b^6+3/16
/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2/a^2*x^3*b^7-3/16/(-4*a*c+b^2)^(5/2)/(4*
a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2/a^2*x^3*b^8-5/16/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2
/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2/a*x*b^7+3/16/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)
^(1/2))^2/a^2*x^3*b^8+5/16/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2/a*x*b^7+3
/16/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2/a^2*x^3*b^7-44*c^3/(-4*a*c+b^2)^2/(4
*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*a^2*x-44*c^3/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*
a*c+b^2)^(1/2)+1/2*b/c)^2*a^2*x-21/4*c/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*x
*b^4+15*c^2/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*x^3*b^3+15*c^2/(-4*a*c+b^2)^
2/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*x^3*b^3-21/4*c/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2*
b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*x*b^4-72*c^4/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/
2*b/c)^2*a^2*x^3-45/2*c^2/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*x^3*b^4+15
/4*c/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*x*b^5+72*c^4/(-4*a*c+b^2)^(5/2)
/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*a^2*x^3+45/2*c^2/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2
+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*x^3*b^4+5/16/(-4*a*c+b^2)^2/(4*a*c-b^2)^2/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1
/2*b/c)^2/a*x*b^6-66*c^3/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*a*x^3*b^2-2
0*c^3/(-4*a*c+b^2)^(5/2)/(4*a*c-b^2)^2/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*a^2*x*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} x^{7} +{\left (6 \, b^{4} c - 49 \, a b^{2} c^{2} + 28 \, a^{2} c^{3}\right )} x^{5} +{\left (3 \, b^{5} - 20 \, a b^{3} c - 4 \, a^{2} b c^{2}\right )} x^{3} +{\left (5 \, a b^{4} - 37 \, a^{2} b^{2} c + 44 \, a^{3} c^{2}\right )} x}{8 \,{\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 )}} - \frac{-3 \, \int \frac{b^{4} - 9 \, a b^{2} c + 28 \, a^{2} c^{2} +{\left (b^{3} c - 8 \, a b c^{2}\right )} x^{2}}{c x^{4} + b x^{2} + a}\,{d x}}{8 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*(b^3*c^2 - 8*a*b*c^3)*x^7 + (6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*x^5 + (3*b^5 - 20*a*b^3*c - 4*a^2*b*c
^2)*x^3 + (5*a*b^4 - 37*a^2*b^2*c + 44*a^3*c^2)*x)/((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) - 3/8*integrate(-(b^4 - 9*a*b^2*c + 28*a^2*c^2 + (b^
3*c - 8*a*b*c^2)*x^2)/(c*x^4 + b*x^2 + a), x)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)

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Fricas [B]  time = 3.7484, size = 9839, normalized size = 27.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(6*(b^3*c^2 - 8*a*b*c^3)*x^7 + 2*(6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*x^5 + 2*(3*b^5 - 20*a*b^3*c - 4*a^
2*b*c^2)*x^3 - 3*sqrt(1/2)*((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)*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4
+ (a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8
- 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^
2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))/(a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a
^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5))*log(27*(21*b^8*c^3 - 447*a*b^6*c^4 + 4189*a^2*b^4*c^5 - 19208*
a^3*b^2*c^6 + 38416*a^4*c^7)*x + 27/2*sqrt(1/2)*(b^14 - 32*a*b^12*c + 464*a^2*b^10*c^2 - 3885*a^3*b^8*c^3 + 20
088*a^4*b^6*c^4 - 63680*a^5*b^4*c^5 + 113792*a^6*b^2*c^6 - 87808*a^7*c^7 - (a^5*b^15 - 31*a^6*b^13*c + 424*a^7
*b^11*c^2 - 3280*a^8*b^9*c^3 + 15360*a^9*b^7*c^4 - 43264*a^10*b^5*c^5 + 67584*a^11*b^3*c^6 - 45056*a^12*b*c^7)
*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*
a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c
^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 + (a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^
9*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b
^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))/(a^5*b^10 - 2
0*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5))) + 3*sqrt(1/2)*((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)*sqr
t(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 + (a^5*b^10 - 20*a^6*b^8*c + 160*a^7
*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078
*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*
c^4 - 1024*a^15*c^5)))/(a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*
a^10*c^5))*log(27*(21*b^8*c^3 - 447*a*b^6*c^4 + 4189*a^2*b^4*c^5 - 19208*a^3*b^2*c^6 + 38416*a^4*c^7)*x - 27/2
*sqrt(1/2)*(b^14 - 32*a*b^12*c + 464*a^2*b^10*c^2 - 3885*a^3*b^8*c^3 + 20088*a^4*b^6*c^4 - 63680*a^5*b^4*c^5 +
113792*a^6*b^2*c^6 - 87808*a^7*c^7 - (a^5*b^15 - 31*a^6*b^13*c + 424*a^7*b^11*c^2 - 3280*a^8*b^9*c^3 + 15360*
a^9*b^7*c^4 - 43264*a^10*b^5*c^5 + 67584*a^11*b^3*c^6 - 45056*a^12*b*c^7)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4
*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 128
0*a^14*b^2*c^4 - 1024*a^15*c^5)))*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4
+ (a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8
- 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^
2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))/(a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a
^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5))) - 3*sqrt(1/2)*((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)*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^
2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 - (a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9
*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^
10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))/(a^5*b^10 - 20
*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5))*log(27*(21*b^8*c^3 - 447*a
*b^6*c^4 + 4189*a^2*b^4*c^5 - 19208*a^3*b^2*c^6 + 38416*a^4*c^7)*x + 27/2*sqrt(1/2)*(b^14 - 32*a*b^12*c + 464*
a^2*b^10*c^2 - 3885*a^3*b^8*c^3 + 20088*a^4*b^6*c^4 - 63680*a^5*b^4*c^5 + 113792*a^6*b^2*c^6 - 87808*a^7*c^7 +
(a^5*b^15 - 31*a^6*b^13*c + 424*a^7*b^11*c^2 - 3280*a^8*b^9*c^3 + 15360*a^9*b^7*c^4 - 43264*a^10*b^5*c^5 + 67
584*a^11*b^3*c^6 - 45056*a^12*b*c^7)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^
4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))*sqr
t(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 - (a^5*b^10 - 20*a^6*b^8*c + 160*a^7
*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078
*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*
c^4 - 1024*a^15*c^5)))/(a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*
a^10*c^5))) + 3*sqrt(1/2)*((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)*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4
- (a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8 -
22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2
- 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))/(a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^
8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5))*log(27*(21*b^8*c^3 - 447*a*b^6*c^4 + 4189*a^2*b^4*c^5 - 19208*a
^3*b^2*c^6 + 38416*a^4*c^7)*x - 27/2*sqrt(1/2)*(b^14 - 32*a*b^12*c + 464*a^2*b^10*c^2 - 3885*a^3*b^8*c^3 + 200
88*a^4*b^6*c^4 - 63680*a^5*b^4*c^5 + 113792*a^6*b^2*c^6 - 87808*a^7*c^7 + (a^5*b^15 - 31*a^6*b^13*c + 424*a^7*
b^11*c^2 - 3280*a^8*b^9*c^3 + 15360*a^9*b^7*c^4 - 43264*a^10*b^5*c^5 + 67584*a^11*b^3*c^6 - 45056*a^12*b*c^7)*
sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^10 - 20*a^11*b^8*c + 160*a
^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))*sqrt(-(b^9 - 21*a*b^7*c + 189*a^2*b^5*c^
2 - 840*a^3*b^3*c^3 + 1680*a^4*b*c^4 - (a^5*b^10 - 20*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9
*b^2*c^4 - 1024*a^10*c^5)*sqrt((b^8 - 22*a*b^6*c + 219*a^2*b^4*c^2 - 1078*a^3*b^2*c^3 + 2401*a^4*c^4)/(a^10*b^
10 - 20*a^11*b^8*c + 160*a^12*b^6*c^2 - 640*a^13*b^4*c^3 + 1280*a^14*b^2*c^4 - 1024*a^15*c^5)))/(a^5*b^10 - 20
*a^6*b^8*c + 160*a^7*b^6*c^2 - 640*a^8*b^4*c^3 + 1280*a^9*b^2*c^4 - 1024*a^10*c^5))) + 2*(5*a*b^4 - 37*a^2*b^2
*c + 44*a^3*c^2)*x)/((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)

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Sympy [B]  time = 21.0109, size = 818, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x**7*(24*a*b*c**3 - 3*b**3*c**2) + x**5*(-28*a**2*c**3 + 49*a*b**2*c**2 - 6*b**4*c) + x**3*(4*a**2*b*c**2 +
20*a*b**3*c - 3*b**5) + x*(-44*a**3*c**2 + 37*a**2*b**2*c - 5*a*b**4))/(128*a**6*c**2 - 64*a**5*b**2*c + 8*a**
4*b**4 + x**8*(128*a**4*c**4 - 64*a**3*b**2*c**3 + 8*a**2*b**4*c**2) + x**6*(256*a**4*b*c**3 - 128*a**3*b**3*c
**2 + 16*a**2*b**5*c) + x**4*(256*a**5*c**3 - 48*a**3*b**4*c + 8*a**2*b**6) + x**2*(256*a**5*b*c**2 - 128*a**4
*b**3*c + 16*a**3*b**5)) + RootSum(_t**4*(68719476736*a**15*c**10 - 171798691840*a**14*b**2*c**9 + 19327352832
0*a**13*b**4*c**8 - 128849018880*a**12*b**6*c**7 + 56371445760*a**11*b**8*c**6 - 16911433728*a**10*b**10*c**5
+ 3523215360*a**9*b**12*c**4 - 503316480*a**8*b**14*c**3 + 47185920*a**7*b**16*c**2 - 2621440*a**6*b**18*c + 6
5536*a**5*b**20) + _t**2*(-3963617280*a**9*b*c**9 + 6936330240*a**8*b**3*c**8 - 5400428544*a**7*b**5*c**7 + 24
64874496*a**6*b**7*c**6 - 730054656*a**5*b**9*c**5 + 146165760*a**4*b**11*c**4 - 19860480*a**3*b**13*c**3 + 17
71776*a**2*b**15*c**2 - 94464*a*b**17*c + 2304*b**19) + 49787136*a**4*c**9 - 27433728*a**3*b**2*c**8 + 6446304
*a**2*b**4*c**7 - 734832*a*b**6*c**6 + 35721*b**8*c**5, Lambda(_t, _t*log(x + (184549376*_t**3*a**12*b*c**7 -
276824064*_t**3*a**11*b**3*c**6 + 177209344*_t**3*a**10*b**5*c**5 - 62914560*_t**3*a**9*b**7*c**4 + 13434880*_
t**3*a**8*b**9*c**3 - 1736704*_t**3*a**7*b**11*c**2 + 126976*_t**3*a**6*b**13*c - 4096*_t**3*a**5*b**15 + 6322
176*_t*a**7*c**7 - 13515264*_t*a**6*b**2*c**6 + 8576640*_t*a**5*b**4*c**5 - 2831328*_t*a**4*b**6*c**4 + 556416
*_t*a**3*b**8*c**3 - 66816*_t*a**2*b**10*c**2 + 4608*_t*a*b**12*c - 144*_t*b**14)/(1037232*a**4*c**7 - 518616*
a**3*b**2*c**6 + 113103*a**2*b**4*c**5 - 12069*a*b**6*c**4 + 567*b**8*c**3))))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError