3.102 \(\int \frac{1}{x^2 (a x+b x^3+c x^5)^2} \, dx\)

Optimal. Leaf size=361 \[ \frac{\sqrt{c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{5 b^2-14 a c}{6 a^2 x^3 \left (b^2-4 a c\right )}+\frac{b \left (5 b^2-19 a c\right )}{2 a^3 x \left (b^2-4 a c\right )}+\frac{-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

-(5*b^2 - 14*a*c)/(6*a^2*(b^2 - 4*a*c)*x^3) + (b*(5*b^2 - 19*a*c))/(2*a^3*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*
c*x^2)/(2*a*(b^2 - 4*a*c)*x^3*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19
*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)
^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)*Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b +
Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 3.07713, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1585, 1121, 1281, 1166, 205} \[ \frac{\sqrt{c} \left (28 a^2 c^2-29 a b^2 c+b \left (5 b^2-19 a c\right ) \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (28 a^2 c^2-29 a b^2 c-b \left (5 b^2-19 a c\right ) \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{5 b^2-14 a c}{6 a^2 x^3 \left (b^2-4 a c\right )}+\frac{b \left (5 b^2-19 a c\right )}{2 a^3 x \left (b^2-4 a c\right )}+\frac{-2 a c+b^2+b c x^2}{2 a x^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a*x + b*x^3 + c*x^5)^2),x]

[Out]

-(5*b^2 - 14*a*c)/(6*a^2*(b^2 - 4*a*c)*x^3) + (b*(5*b^2 - 19*a*c))/(2*a^3*(b^2 - 4*a*c)*x) + (b^2 - 2*a*c + b*
c*x^2)/(2*a*(b^2 - 4*a*c)*x^3*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + b*(5*b^2 - 19
*a*c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)
^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 - b*(5*b^2 - 19*a*c)*Sqrt[b^2
- 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a^3*(b^2 - 4*a*c)^(3/2)*Sqrt[b +
Sqrt[b^2 - 4*a*c]])

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1121

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

Rule 1281

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

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}{x^2 \left (a x+b x^3+c x^5\right )^2} \, dx &=\int \frac{1}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^3 \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-5 b^2+14 a c-5 b c x^2}{x^4 \left (a+b x^2+c x^4\right )} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) x^3}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^3 \left (a+b x^2+c x^4\right )}+\frac{\int \frac{-3 b \left (5 b^2-19 a c\right )-3 c \left (5 b^2-14 a c\right ) x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx}{6 a^2 \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) x^3}+\frac{b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^3 \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-3 \left (5 b^4-24 a b^2 c+14 a^2 c^2\right )-3 b c \left (5 b^2-19 a c\right ) x^2}{a+b x^2+c x^4} \, dx}{6 a^3 \left (b^2-4 a c\right )}\\ &=-\frac{5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) x^3}+\frac{b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^3 \left (a+b x^2+c x^4\right )}-\frac{\left (c \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 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}{4 a^3 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (c \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 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}{4 a^3 \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac{5 b^2-14 a c}{6 a^2 \left (b^2-4 a c\right ) x^3}+\frac{b \left (5 b^2-19 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^3 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (5 b^4-29 a b^2 c+28 a^2 c^2+b \left (5 b^2-19 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 )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (5 b^4-29 a b^2 c+28 a^2 c^2-b \left (5 b^2-19 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 )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 0.788114, size = 344, normalized size = 0.95 \[ \frac{\frac{6 x \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x^2+b^3 c x^2+b^4\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \sqrt{c} \left (28 a^2 c^2+5 b^3 \sqrt{b^2-4 a c}-29 a b^2 c-19 a b c \sqrt{b^2-4 a c}+5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left (-28 a^2 c^2+5 b^3 \sqrt{b^2-4 a c}+29 a b^2 c-19 a b c \sqrt{b^2-4 a c}-5 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a}{x^3}+\frac{24 b}{x}}{12 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a*x + b*x^3 + c*x^5)^2),x]

[Out]

((-4*a)/x^3 + (24*b)/x + (6*x*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x^2 - 3*a*b*c^2*x^2))/((b^2 - 4*a*c)*(a + b
*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(5*b^4 - 29*a*b^2*c + 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*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)^(3/2)*Sqrt[b - Sqrt[b^2
- 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(-5*b^4 + 29*a*b^2*c - 28*a^2*c^2 + 5*b^3*Sqrt[b^2 - 4*a*c] - 19*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)^(3/2)*Sqrt[b + Sqrt[b^2 -
4*a*c]]))/(12*a^3)

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Maple [B]  time = 0.036, size = 913, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(c*x^5+b*x^3+a*x)^2,x)

[Out]

3/2/a^2/(c*x^4+b*x^2+a)*b*c^2/(4*a*c-b^2)*x^3-1/2/a^3/(c*x^4+b*x^2+a)*b^3*c/(4*a*c-b^2)*x^3-1/a/(c*x^4+b*x^2+a
)/(4*a*c-b^2)*x*c^2+2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b^2*c-1/2/a^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x*b^4+19/4/a
^2*c^2/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2
))*b-5/4/a^3*c/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))
*c)^(1/2))*b^3+7/a*c^3/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1
/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))-29/4/a^2*c^2/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/
2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2+5/4/a^3*c/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)
*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4-19/4/a^2*c^
2/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))
*b+5/4/a^3*c/(4*a*c-b^2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2))*b^3+7/a*c^3/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2
^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))-29/4/a^2*c^2/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2
)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2+5/4/a^3*c/(4*a*c-b^2)/(-4*a*c+b^2
)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^4-1
/3/a^2/x^3+2/a^3*b/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 \,{\left (5 \, b^{3} c - 19 \, a b c^{2}\right )} x^{6} +{\left (15 \, b^{4} - 62 \, a b^{2} c + 14 \, a^{2} c^{2}\right )} x^{4} - 2 \, a^{2} b^{2} + 8 \, a^{3} c + 10 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}}{6 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{7} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{5} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{3}\right )}} - \frac{-\int \frac{5 \, b^{4} - 24 \, a b^{2} c + 14 \, a^{2} c^{2} +{\left (5 \, b^{3} c - 19 \, a b c^{2}\right )} x^{2}}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^5+b*x^3+a*x)^2,x, algorithm="maxima")

[Out]

1/6*(3*(5*b^3*c - 19*a*b*c^2)*x^6 + (15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*x^4 - 2*a^2*b^2 + 8*a^3*c + 10*(a*b^3 -
4*a^2*b*c)*x^2)/((a^3*b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*c)*x^3) - 1/2*int
egrate(-(5*b^4 - 24*a*b^2*c + 14*a^2*c^2 + (5*b^3*c - 19*a*b*c^2)*x^2)/(c*x^4 + b*x^2 + a), x)/(a^3*b^2 - 4*a^
4*c)

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Fricas [B]  time = 2.94168, size = 7960, normalized size = 22.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^5+b*x^3+a*x)^2,x, algorithm="fricas")

[Out]

1/12*(6*(5*b^3*c - 19*a*b*c^2)*x^6 + 2*(15*b^4 - 62*a*b^2*c + 14*a^2*c^2)*x^4 - 4*a^2*b^2 + 16*a^3*c + 20*(a*b
^3 - 4*a^2*b*c)*x^2 + 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*
c)*x^3)*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8
*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3
+ 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*
c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log((1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^
2*b^4*c^6 - 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*x + 1/2*sqrt(1/2)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2
- 75579*a^3*b^8*c^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 - (5*a^7*b^
11 - 94*a^8*b^9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12
- 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6
)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2
- 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 -
8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)
/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^
10*c^3))) - 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*c)*x^3)*sq
rt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 4
8*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^
4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^
7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log((1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6
- 50421*a^3*b^2*c^7 + 9604*a^4*c^8)*x - 1/2*sqrt(1/2)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a
^3*b^8*c^3 + 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 - (5*a^7*b^11 - 94*a^
8*b^9*c + 700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12 - 8250*a*b
^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^
6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3
*b^3*c^3 + 1260*a^4*b*c^4 + (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^
10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6
- 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)))
+ 3*sqrt(1/2)*((a^3*b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*c)*x^3)*sqrt(-(25*b^
9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*
c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4
- 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12
*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))*log((1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^
3*b^2*c^7 + 9604*a^4*c^8)*x + 1/2*sqrt(1/2)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^3
+ 160932*a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 + (5*a^7*b^11 - 94*a^8*b^9*c +
700*a^9*b^7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12 - 8250*a*b^10*c + 39
525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^1
5*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 +
1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 395
25*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15
*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))) - 3*sqrt(1
/2)*((a^3*b^2*c - 4*a^4*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*c)*x^3)*sqrt(-(25*b^9 - 315*a*
b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a
^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^
5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 48*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c
+ 48*a^9*b^2*c^2 - 64*a^10*c^3))*log((1125*b^8*c^4 - 12325*a*b^6*c^5 + 43410*a^2*b^4*c^6 - 50421*a^3*b^2*c^7
+ 9604*a^4*c^8)*x - 1/2*sqrt(1/2)*(125*b^14 - 2425*a*b^12*c + 18940*a^2*b^10*c^2 - 75579*a^3*b^8*c^3 + 160932*
a^4*b^6*c^4 - 172990*a^5*b^4*c^5 + 79408*a^6*b^2*c^6 - 10976*a^7*c^7 + (5*a^7*b^11 - 94*a^8*b^9*c + 700*a^9*b^
7*c^2 - 2576*a^10*b^5*c^3 + 4672*a^11*b^3*c^4 - 3328*a^12*b*c^5)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^
8*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c +
48*a^16*b^2*c^2 - 64*a^17*c^3)))*sqrt(-(25*b^9 - 315*a*b^7*c + 1386*a^2*b^5*c^2 - 2415*a^3*b^3*c^3 + 1260*a^4*
b*c^4 - (a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*sqrt((625*b^12 - 8250*a*b^10*c + 39525*a^2*b^8
*c^2 - 83630*a^3*b^6*c^3 + 76686*a^4*b^4*c^4 - 24108*a^5*b^2*c^5 + 2401*a^6*c^6)/(a^14*b^6 - 12*a^15*b^4*c + 4
8*a^16*b^2*c^2 - 64*a^17*c^3)))/(a^7*b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3))))/((a^3*b^2*c - 4*a^4
*c^2)*x^7 + (a^3*b^3 - 4*a^4*b*c)*x^5 + (a^4*b^2 - 4*a^5*c)*x^3)

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Sympy [A]  time = 12.0217, size = 566, normalized size = 1.57 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (1048576 a^{13} c^{6} - 1572864 a^{12} b^{2} c^{5} + 983040 a^{11} b^{4} c^{4} - 327680 a^{10} b^{6} c^{3} + 61440 a^{9} b^{8} c^{2} - 6144 a^{8} b^{10} c + 256 a^{7} b^{12}\right ) + t^{2} \left (- 1290240 a^{7} b c^{7} + 3440640 a^{6} b^{3} c^{6} - 3515904 a^{5} b^{5} c^{5} + 1870848 a^{4} b^{7} c^{4} - 572272 a^{3} b^{9} c^{3} + 101856 a^{2} b^{11} c^{2} - 9840 a b^{13} c + 400 b^{15}\right ) + 38416 a^{2} c^{9} - 17640 a b^{2} c^{8} + 2025 b^{4} c^{7}, \left ( t \mapsto t \log{\left (x + \frac{- 212992 t^{3} a^{12} b c^{5} + 299008 t^{3} a^{11} b^{3} c^{4} - 164864 t^{3} a^{10} b^{5} c^{3} + 44800 t^{3} a^{9} b^{7} c^{2} - 6016 t^{3} a^{8} b^{9} c + 320 t^{3} a^{7} b^{11} - 21952 t a^{7} c^{7} + 289856 t a^{6} b^{2} c^{6} - 682820 t a^{5} b^{4} c^{5} + 642828 t a^{4} b^{6} c^{4} - 302316 t a^{3} b^{8} c^{3} + 75760 t a^{2} b^{10} c^{2} - 9700 t a b^{12} c + 500 t b^{14}}{9604 a^{4} c^{8} - 50421 a^{3} b^{2} c^{7} + 43410 a^{2} b^{4} c^{6} - 12325 a b^{6} c^{5} + 1125 b^{8} c^{4}} \right )} \right )\right )} + \frac{- 8 a^{3} c + 2 a^{2} b^{2} + x^{6} \left (57 a b c^{2} - 15 b^{3} c\right ) + x^{4} \left (- 14 a^{2} c^{2} + 62 a b^{2} c - 15 b^{4}\right ) + x^{2} \left (40 a^{2} b c - 10 a b^{3}\right )}{x^{7} \left (24 a^{4} c^{2} - 6 a^{3} b^{2} c\right ) + x^{5} \left (24 a^{4} b c - 6 a^{3} b^{3}\right ) + x^{3} \left (24 a^{5} c - 6 a^{4} b^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(c*x**5+b*x**3+a*x)**2,x)

[Out]

RootSum(_t**4*(1048576*a**13*c**6 - 1572864*a**12*b**2*c**5 + 983040*a**11*b**4*c**4 - 327680*a**10*b**6*c**3
+ 61440*a**9*b**8*c**2 - 6144*a**8*b**10*c + 256*a**7*b**12) + _t**2*(-1290240*a**7*b*c**7 + 3440640*a**6*b**3
*c**6 - 3515904*a**5*b**5*c**5 + 1870848*a**4*b**7*c**4 - 572272*a**3*b**9*c**3 + 101856*a**2*b**11*c**2 - 984
0*a*b**13*c + 400*b**15) + 38416*a**2*c**9 - 17640*a*b**2*c**8 + 2025*b**4*c**7, Lambda(_t, _t*log(x + (-21299
2*_t**3*a**12*b*c**5 + 299008*_t**3*a**11*b**3*c**4 - 164864*_t**3*a**10*b**5*c**3 + 44800*_t**3*a**9*b**7*c**
2 - 6016*_t**3*a**8*b**9*c + 320*_t**3*a**7*b**11 - 21952*_t*a**7*c**7 + 289856*_t*a**6*b**2*c**6 - 682820*_t*
a**5*b**4*c**5 + 642828*_t*a**4*b**6*c**4 - 302316*_t*a**3*b**8*c**3 + 75760*_t*a**2*b**10*c**2 - 9700*_t*a*b*
*12*c + 500*_t*b**14)/(9604*a**4*c**8 - 50421*a**3*b**2*c**7 + 43410*a**2*b**4*c**6 - 12325*a*b**6*c**5 + 1125
*b**8*c**4)))) + (-8*a**3*c + 2*a**2*b**2 + x**6*(57*a*b*c**2 - 15*b**3*c) + x**4*(-14*a**2*c**2 + 62*a*b**2*c
- 15*b**4) + x**2*(40*a**2*b*c - 10*a*b**3))/(x**7*(24*a**4*c**2 - 6*a**3*b**2*c) + x**5*(24*a**4*b*c - 6*a**
3*b**3) + x**3*(24*a**5*c - 6*a**4*b**2))

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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/x^2/(c*x^5+b*x^3+a*x)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError