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

Optimal. Leaf size=219 \[ -\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac {\log (x) \left (3 b^2-2 a c\right )}{a^4}+\frac {b \left (3 b^2-11 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )}-\frac {3 b^2-8 a c}{4 a^2 x^4 \left (b^2-4 a c\right )}+\frac {b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {-2 a c+b^2+b c x^2}{2 a x^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

1/4*(8*a*c-3*b^2)/a^2/(-4*a*c+b^2)/x^4+1/2*b*(-11*a*c+3*b^2)/a^3/(-4*a*c+b^2)/x^2+1/2*(b*c*x^2-2*a*c+b^2)/a/(-
4*a*c+b^2)/x^4/(c*x^4+b*x^2+a)+1/2*b*(30*a^2*c^2-20*a*b^2*c+3*b^4)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/a^4
/(-4*a*c+b^2)^(3/2)+(-2*a*c+3*b^2)*ln(x)/a^4-1/4*(-2*a*c+3*b^2)*ln(c*x^4+b*x^2+a)/a^4

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Rubi [A]  time = 0.31, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1585, 1114, 740, 800, 634, 618, 206, 628} \[ \frac {b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {b \left (3 b^2-11 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )}-\frac {3 b^2-8 a c}{4 a^2 x^4 \left (b^2-4 a c\right )}-\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac {\log (x) \left (3 b^2-2 a c\right )}{a^4}+\frac {-2 a c+b^2+b c x^2}{2 a x^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*b^2 - 8*a*c)/(4*a^2*(b^2 - 4*a*c)*x^4) + (b*(3*b^2 - 11*a*c))/(2*a^3*(b^2 - 4*a*c)*x^2) + (b^2 - 2*a*c + b
*c*x^2)/(2*a*(b^2 - 4*a*c)*x^4*(a + b*x^2 + c*x^4)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*x^
2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*Log[x])/a^4 - ((3*b^2 - 2*a*c)*Log[a + b
*x^2 + c*x^4])/(4*a^4)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 740

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1114

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

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]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (a x+b x^3+c x^5\right )^2} \, dx &=\int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \frac {-3 b^2+8 a c-3 b c x}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}-\frac {\operatorname {Subst}\left (\int \left (\frac {-3 b^2+8 a c}{a x^3}+\frac {3 b^3-11 a b c}{a^2 x^2}+\frac {\left (b^2-4 a c\right ) \left (-3 b^2+2 a c\right )}{a^3 x}+\frac {b \left (3 b^4-17 a b^2 c+19 a^2 c^2\right )+c \left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}\\ &=-\frac {3 b^2-8 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (3 b^2-11 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac {\operatorname {Subst}\left (\int \frac {b \left (3 b^4-17 a b^2 c+19 a^2 c^2\right )+c \left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac {3 b^2-8 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (3 b^2-11 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac {\left (3 b^2-2 a c\right ) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}-\frac {\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4 \left (b^2-4 a c\right )}\\ &=-\frac {3 b^2-8 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (3 b^2-11 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}+\frac {\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac {\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac {3 b^2-8 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (3 b^2-11 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}+\frac {b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 328, normalized size = 1.50 \[ \frac {\frac {2 a \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x^2+b^4+b^3 c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (8 a^2 c^2 \sqrt {b^2-4 a c}+30 a^2 b c^2-20 a b^3 c-14 a b^2 c \sqrt {b^2-4 a c}+3 b^4 \sqrt {b^2-4 a c}+3 b^5\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (-8 a^2 c^2 \sqrt {b^2-4 a c}+30 a^2 b c^2-20 a b^3 c+14 a b^2 c \sqrt {b^2-4 a c}-3 b^4 \sqrt {b^2-4 a c}+3 b^5\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {a^2}{x^4}+4 \log (x) \left (3 b^2-2 a c\right )+\frac {4 a b}{x^2}}{4 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2/x^4) + (4*a*b)/x^2 + (2*a*(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)) + 4*(3*b^2 - 2*a*c)*Log[x] - ((3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2 + 3*b^4*Sqrt[b^2 - 4*a*c] -
14*a*b^2*c*Sqrt[b^2 - 4*a*c] + 8*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*
c)^(3/2) + ((3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2 - 3*b^4*Sqrt[b^2 - 4*a*c] + 14*a*b^2*c*Sqrt[b^2 - 4*a*c] - 8*a^
2*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^4)

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fricas [B]  time = 1.41, size = 1242, normalized size = 5.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 - 2*(3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*x^6 - (6*a*b^6 - 49*
a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*x^4 - 3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2 + ((3*b^5*c - 20*
a*b^3*c^2 + 30*a^2*b*c^3)*x^8 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*x^6 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b
*c^2)*x^4)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c - (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^
4 + b*x^2 + a)) + ((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^8 + (3*b^7 - 26*a*b^5*c + 64*a^2*b
^3*c^2 - 32*a^3*b*c^3)*x^6 + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*x^4)*log(c*x^4 + b*x^2 + a
) - 4*((3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^8 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*
a^3*b*c^3)*x^6 + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*x^4)*log(x))/((a^4*b^4*c - 8*a^5*b^2*c
^2 + 16*a^6*c^3)*x^8 + (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^6 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x^4),
-1/4*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 - 2*(3*a*b^5*c - 23*a^2*b^3*c^2 + 44*a^3*b*c^3)*x^6 - (6*a*b^6 - 49*
a^2*b^4*c + 108*a^3*b^2*c^2 - 32*a^4*c^3)*x^4 - 3*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2 - 2*((3*b^5*c - 2
0*a*b^3*c^2 + 30*a^2*b*c^3)*x^8 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*x^6 + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3
*b*c^2)*x^4)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + ((3*b^6*c - 26*a*b^4
*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^8 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*x^6 + (3*a*b^6
- 26*a^2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*x^4)*log(c*x^4 + b*x^2 + a) - 4*((3*b^6*c - 26*a*b^4*c^2 + 64*a^
2*b^2*c^3 - 32*a^3*c^4)*x^8 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*x^6 + (3*a*b^6 - 26*a^2*b^4
*c + 64*a^3*b^2*c^2 - 32*a^4*c^3)*x^4)*log(x))/((a^4*b^4*c - 8*a^5*b^2*c^2 + 16*a^6*c^3)*x^8 + (a^4*b^5 - 8*a^
5*b^3*c + 16*a^6*b*c^2)*x^6 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x^4)]

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giac [A]  time = 2.04, size = 274, normalized size = 1.25 \[ -\frac {{\left (3 \, b^{5} - 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b^{4} c x^{4} - 14 \, a b^{2} c^{2} x^{4} + 8 \, a^{2} c^{3} x^{4} + 3 \, b^{5} x^{2} - 12 \, a b^{3} c x^{2} + 2 \, a^{2} b c^{2} x^{2} + 5 \, a b^{4} - 22 \, a^{2} b^{2} c + 12 \, a^{3} c^{2}}{4 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} {\left (c x^{4} + b x^{2} + a\right )}} - \frac {{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} + \frac {{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {9 \, b^{2} x^{4} - 6 \, a c x^{4} - 4 \, a b x^{2} + a^{2}}{4 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^2 - 4*a^5*c)*sqrt(-b
^2 + 4*a*c)) + 1/4*(3*b^4*c*x^4 - 14*a*b^2*c^2*x^4 + 8*a^2*c^3*x^4 + 3*b^5*x^2 - 12*a*b^3*c*x^2 + 2*a^2*b*c^2*
x^2 + 5*a*b^4 - 22*a^2*b^2*c + 12*a^3*c^2)/((a^4*b^2 - 4*a^5*c)*(c*x^4 + b*x^2 + a)) - 1/4*(3*b^2 - 2*a*c)*log
(c*x^4 + b*x^2 + a)/a^4 + 1/2*(3*b^2 - 2*a*c)*log(x^2)/a^4 - 1/4*(9*b^2*x^4 - 6*a*c*x^4 - 4*a*b*x^2 + a^2)/(a^
4*x^4)

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maple [B]  time = 0.02, size = 443, normalized size = 2.02 \[ \frac {3 b \,c^{2} x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a^{2}}-\frac {b^{3} c \,x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a^{3}}+\frac {15 b \,c^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {10 b^{3} c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{3}}+\frac {3 b^{5} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{4}}-\frac {c^{2}}{\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a}+\frac {2 b^{2} c}{\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {2 c^{2} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (4 a c -b^{2}\right ) a^{2}}-\frac {b^{4}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a^{3}}-\frac {7 b^{2} c \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 \left (4 a c -b^{2}\right ) a^{3}}+\frac {3 b^{4} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (4 a c -b^{2}\right ) a^{4}}-\frac {2 c \ln \relax (x )}{a^{3}}+\frac {3 b^{2} \ln \relax (x )}{a^{4}}+\frac {b}{a^{3} x^{2}}-\frac {1}{4 a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/a^2/x^4-2/a^3*c*ln(x)+3/a^4*b^2*ln(x)+1/a^3*b/x^2+3/2/a^2/(c*x^4+b*x^2+a)*b*c^2/(4*a*c-b^2)*x^2-1/2/a^3/(
c*x^4+b*x^2+a)*b^3*c/(4*a*c-b^2)*x^2-1/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*c^2+2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^2
*c-1/2/a^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*b^4+2/a^2/(4*a*c-b^2)*c^2*ln(c*x^4+b*x^2+a)-7/2/a^3/(4*a*c-b^2)*c*ln(c*
x^4+b*x^2+a)*b^2+3/4/a^4/(4*a*c-b^2)*ln(c*x^4+b*x^2+a)*b^4+15/a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-
b^2)^(1/2))*b*c^2-10/a^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^3*c+3/2/a^4/(4*a*c-b^2)^(3/
2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (3 \, b^{3} c - 11 \, a b c^{2}\right )} x^{6} + {\left (6 \, b^{4} - 25 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x^{4} - a^{2} b^{2} + 4 \, a^{3} c + 3 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}}{4 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{8} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{6} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{4}\right )}} - \frac {\frac {1}{4} \, {\left (3 \, b^{4} - 14 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right ) + \frac {{\left (3 \, b^{5} - 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c}}}{a^{4} b^{2} - 4 \, a^{5} c} + \frac {{\left (3 \, b^{2} - 2 \, a c\right )} \log \relax (x)}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(3*b^3*c - 11*a*b*c^2)*x^6 + (6*b^4 - 25*a*b^2*c + 8*a^2*c^2)*x^4 - a^2*b^2 + 4*a^3*c + 3*(a*b^3 - 4*a^
2*b*c)*x^2)/((a^3*b^2*c - 4*a^4*c^2)*x^8 + (a^3*b^3 - 4*a^4*b*c)*x^6 + (a^4*b^2 - 4*a^5*c)*x^4) - integrate(((
3*b^4*c - 14*a*b^2*c^2 + 8*a^2*c^3)*x^3 + (3*b^5 - 17*a*b^3*c + 19*a^2*b*c^2)*x)/(c*x^4 + b*x^2 + a), x)/(a^4*
b^2 - 4*a^5*c) + (3*b^2 - 2*a*c)*log(x)/a^4

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mupad [B]  time = 7.47, size = 5999, normalized size = 27.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*atan((x^2*((((((b*((2240*a^10*b*c^7 - 6*a^6*b^9*c^3 + 40*a^7*b^7*c^4 + 108*a^8*b^5*c^5 - 1248*a^9*b^3*c^6)/
(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2) - ((2560*a^13*b*c^6 + 12*a^9*b^9*c^2 - 184*a^10*b^7*
c^3 + 1056*a^11*b^5*c^4 - 2688*a^12*b^3*c^5)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b
^6*c))/(2*(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c +
192*a^6*b^2*c^2)))*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c))/(4*a^4*(4*a*c - b^2)^(3/2)) - (b*(3*b^4 + 30*a^2*c^2 - 2
0*a*b^2*c)*(2560*a^13*b*c^6 + 12*a^9*b^9*c^2 - 184*a^10*b^7*c^3 + 1056*a^11*b^5*c^4 - 2688*a^12*b^3*c^5)*(6*b^
8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(8*a^4*(4*a*c - b^2)^(3/2)*(a^9*b^6 - 64*a^
12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(6*b^8
+ 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c +
192*a^6*b^2*c^2)) + (b*((1760*a^7*b*c^8 + 54*a^3*b^9*c^4 - 657*a^4*b^7*c^5 + 2775*a^5*b^5*c^6 - 4484*a^6*b^3*c
^7)/(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2) + (((2240*a^10*b*c^7 - 6*a^6*b^9*c^3 + 40*a^7*b^
7*c^4 + 108*a^8*b^5*c^5 - 1248*a^9*b^3*c^6)/(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2) - ((2560
*a^13*b*c^6 + 12*a^9*b^9*c^2 - 184*a^10*b^7*c^3 + 1056*a^11*b^5*c^4 - 2688*a^12*b^3*c^5)*(6*b^8 + 256*a^4*c^4
+ 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2
)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^
3*b^2*c^3 - 76*a*b^6*c))/(2*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(3*b^4 + 30*a^2*c^2 -
20*a*b^2*c))/(4*a^4*(4*a*c - b^2)^(3/2)) + (b^3*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c)^3*(2560*a^13*b*c^6 + 12*a^9
*b^9*c^2 - 184*a^10*b^7*c^3 + 1056*a^11*b^5*c^4 - 2688*a^12*b^3*c^5))/(64*a^12*(4*a*c - b^2)^(9/2)*(a^9*b^6 -
64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2)))*(9*b^8 + 80*a^4*c^4 + 270*a^2*b^4*c^2 - 285*a^3*b^2*c^3 - 87*
a*b^6*c))/(8*a^3*c^2*(4*a*c - b^2)^(7/2)*(54*b^10 - 1600*a^5*c^5 + 3480*a^2*b^6*c^2 - 7200*a^3*b^4*c^3 + 5775*
a^4*b^2*c^4 - 720*a*b^8*c)) + (3*b*((27*b^9*c^5 - 297*a*b^7*c^6 + 1089*a^2*b^5*c^7 - 1331*a^3*b^3*c^8)/(a^9*b^
6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2) - (((1760*a^7*b*c^8 + 54*a^3*b^9*c^4 - 657*a^4*b^7*c^5 + 27
75*a^5*b^5*c^6 - 4484*a^6*b^3*c^7)/(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2) + (((2240*a^10*b*
c^7 - 6*a^6*b^9*c^3 + 40*a^7*b^7*c^4 + 108*a^8*b^5*c^5 - 1248*a^9*b^3*c^6)/(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^
4*c + 48*a^11*b^2*c^2) - ((2560*a^13*b*c^6 + 12*a^9*b^9*c^2 - 184*a^10*b^7*c^3 + 1056*a^11*b^5*c^4 - 2688*a^12
*b^3*c^5)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(a^9*b^6 - 64*a^12*c^3 -
12*a^10*b^4*c + 48*a^11*b^2*c^2)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(6*b^8 + 256*a^4
*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b
^2*c^2)))*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(4*a^4*b^6 - 256*a^7*c^3
- 48*a^5*b^4*c + 192*a^6*b^2*c^2)) + (b*((b*((2240*a^10*b*c^7 - 6*a^6*b^9*c^3 + 40*a^7*b^7*c^4 + 108*a^8*b^5*c
^5 - 1248*a^9*b^3*c^6)/(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2) - ((2560*a^13*b*c^6 + 12*a^9*
b^9*c^2 - 184*a^10*b^7*c^3 + 1056*a^11*b^5*c^4 - 2688*a^12*b^3*c^5)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 5
76*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2)*(4*a^4*b^6 - 256*a^
7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c))/(4*a^4*(4*a*c - b^2)^(3/2)) - (b*
(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c)*(2560*a^13*b*c^6 + 12*a^9*b^9*c^2 - 184*a^10*b^7*c^3 + 1056*a^11*b^5*c^4 - 2
688*a^12*b^3*c^5)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(8*a^4*(4*a*c - b^2)
^(3/2)*(a^9*b^6 - 64*a^12*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192
*a^6*b^2*c^2)))*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c))/(4*a^4*(4*a*c - b^2)^(3/2)) - (b^2*(3*b^4 + 30*a^2*c^2 - 20
*a*b^2*c)^2*(2560*a^13*b*c^6 + 12*a^9*b^9*c^2 - 184*a^10*b^7*c^3 + 1056*a^11*b^5*c^4 - 2688*a^12*b^3*c^5)*(6*b
^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(32*a^8*(4*a*c - b^2)^3*(a^9*b^6 - 64*a^12
*c^3 - 12*a^10*b^4*c + 48*a^11*b^2*c^2)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(3*b^6 -
25*a^3*c^3 + 50*a^2*b^2*c^2 - 23*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^3*(54*b^10 - 1600*a^5*c^5 + 3480*a^2*b^6*c
^2 - 7200*a^3*b^4*c^3 + 5775*a^4*b^2*c^4 - 720*a*b^8*c)))*(16*a^12*b^6*(4*a*c - b^2)^(9/2) - 1024*a^15*c^3*(4*
a*c - b^2)^(9/2) - 192*a^13*b^4*c*(4*a*c - b^2)^(9/2) + 768*a^14*b^2*c^2*(4*a*c - b^2)^(9/2)))/(9*b^10*c^2 - 1
20*a*b^8*c^3 + 580*a^2*b^6*c^4 - 1200*a^3*b^4*c^5 + 900*a^4*b^2*c^6) + (((b*((36*a^3*b^8*c^3 - 309*a^4*b^6*c^4
+ 778*a^5*b^4*c^5 - 473*a^6*b^2*c^6)/(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c) - (((12*a^6*b^8*c^2 - 116*a^7*b^6
*c^3 + 348*a^8*b^4*c^4 - 304*a^9*b^2*c^5)/(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c) + ((4*a^10*b^6*c^2 - 32*a^11*
b^4*c^3 + 64*a^12*b^2*c^4)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(a^9*b^4
+ 16*a^11*c^2 - 8*a^10*b^2*c)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(6*b^8 + 256*a^4*c
^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2
*c^2)))*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c))/(4*a^4*(4*a*c - b^2)^(3/2)) - (((b*((12*a^6*b^8*c^2 - 116*a^7*b^6*c
^3 + 348*a^8*b^4*c^4 - 304*a^9*b^2*c^5)/(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c) + ((4*a^10*b^6*c^2 - 32*a^11*b^
4*c^3 + 64*a^12*b^2*c^4)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(a^9*b^4 +
16*a^11*c^2 - 8*a^10*b^2*c)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(3*b^4 + 30*a^2*c^2
- 20*a*b^2*c))/(4*a^4*(4*a*c - b^2)^(3/2)) + (b*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c)*(4*a^10*b^6*c^2 - 32*a^11*b^
4*c^3 + 64*a^12*b^2*c^4)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(8*a^4*(4*a*c
- b^2)^(3/2)*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2
)))*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(4*a^4*b^6 - 256*a^7*c^3 - 48*a
^5*b^4*c + 192*a^6*b^2*c^2)) + (b^3*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c)^3*(4*a^10*b^6*c^2 - 32*a^11*b^4*c^3 + 64
*a^12*b^2*c^4))/(64*a^12*(4*a*c - b^2)^(9/2)*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))*(16*a^12*b^6*(4*a*c - b^
2)^(9/2) - 1024*a^15*c^3*(4*a*c - b^2)^(9/2) - 192*a^13*b^4*c*(4*a*c - b^2)^(9/2) + 768*a^14*b^2*c^2*(4*a*c -
b^2)^(9/2))*(9*b^8 + 80*a^4*c^4 + 270*a^2*b^4*c^2 - 285*a^3*b^2*c^3 - 87*a*b^6*c))/(8*a^3*c^2*(4*a*c - b^2)^(7
/2)*(9*b^10*c^2 - 120*a*b^8*c^3 + 580*a^2*b^6*c^4 - 1200*a^3*b^4*c^5 + 900*a^4*b^2*c^6)*(54*b^10 - 1600*a^5*c^
5 + 3480*a^2*b^6*c^2 - 7200*a^3*b^4*c^3 + 5775*a^4*b^2*c^4 - 720*a*b^8*c)) - (3*b*(16*a^12*b^6*(4*a*c - b^2)^(
9/2) - 1024*a^15*c^3*(4*a*c - b^2)^(9/2) - 192*a^13*b^4*c*(4*a*c - b^2)^(9/2) + 768*a^14*b^2*c^2*(4*a*c - b^2)
^(9/2))*((((36*a^3*b^8*c^3 - 309*a^4*b^6*c^4 + 778*a^5*b^4*c^5 - 473*a^6*b^2*c^6)/(a^9*b^4 + 16*a^11*c^2 - 8*a
^10*b^2*c) - (((12*a^6*b^8*c^2 - 116*a^7*b^6*c^3 + 348*a^8*b^4*c^4 - 304*a^9*b^2*c^5)/(a^9*b^4 + 16*a^11*c^2 -
8*a^10*b^2*c) + ((4*a^10*b^6*c^2 - 32*a^11*b^4*c^3 + 64*a^12*b^2*c^4)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2
- 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b
^4*c + 192*a^6*b^2*c^2)))*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(4*a^4*b^
6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 -
76*a*b^6*c))/(2*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)) - (27*b^8*c^4 - 216*a*b^6*c^5 + 4
95*a^2*b^4*c^6 - 242*a^3*b^2*c^7)/(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c) + (b*((b*((12*a^6*b^8*c^2 - 116*a^7*b
^6*c^3 + 348*a^8*b^4*c^4 - 304*a^9*b^2*c^5)/(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c) + ((4*a^10*b^6*c^2 - 32*a^1
1*b^4*c^3 + 64*a^12*b^2*c^4)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(a^9*b
^4 + 16*a^11*c^2 - 8*a^10*b^2*c)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)))*(3*b^4 + 30*a^2*
c^2 - 20*a*b^2*c))/(4*a^4*(4*a*c - b^2)^(3/2)) + (b*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c)*(4*a^10*b^6*c^2 - 32*a^1
1*b^4*c^3 + 64*a^12*b^2*c^4)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(8*a^4*(4
*a*c - b^2)^(3/2)*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2
*c^2)))*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c))/(4*a^4*(4*a*c - b^2)^(3/2)) + (b^2*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c
)^2*(4*a^10*b^6*c^2 - 32*a^11*b^4*c^3 + 64*a^12*b^2*c^4)*(6*b^8 + 256*a^4*c^4 + 336*a^2*b^4*c^2 - 576*a^3*b^2*
c^3 - 76*a*b^6*c))/(32*a^8*(4*a*c - b^2)^3*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)*(4*a^4*b^6 - 256*a^7*c^3 - 4
8*a^5*b^4*c + 192*a^6*b^2*c^2)))*(3*b^6 - 25*a^3*c^3 + 50*a^2*b^2*c^2 - 23*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^
3*(9*b^10*c^2 - 120*a*b^8*c^3 + 580*a^2*b^6*c^4 - 1200*a^3*b^4*c^5 + 900*a^4*b^2*c^6)*(54*b^10 - 1600*a^5*c^5
+ 3480*a^2*b^6*c^2 - 7200*a^3*b^4*c^3 + 5775*a^4*b^2*c^4 - 720*a*b^8*c)))*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c))/(
2*a^4*(4*a*c - b^2)^(3/2)) - (log(x)*(2*a*c - 3*b^2))/a^4 - (log(a + b*x^2 + c*x^4)*(6*b^8 + 256*a^4*c^4 + 336
*a^2*b^4*c^2 - 576*a^3*b^2*c^3 - 76*a*b^6*c))/(2*(4*a^4*b^6 - 256*a^7*c^3 - 48*a^5*b^4*c + 192*a^6*b^2*c^2)) -
(1/(4*a) - (3*b*x^2)/(4*a^2) + (x^4*(6*b^4 + 8*a^2*c^2 - 25*a*b^2*c))/(4*a^3*(4*a*c - b^2)) - (b*c*x^6*(11*a*
c - 3*b^2))/(2*a^3*(4*a*c - b^2)))/(a*x^4 + b*x^6 + c*x^8)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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