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

Optimal. Leaf size=200 \[ -\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac {\log (x)}{a^3}+\frac {16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

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Rubi [A]  time = 0.30, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1114, 740, 822, 800, 634, 618, 206, 628} \begin {gather*} \frac {16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}-\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac {\log (x)}{a^3}+\frac {-2 a c+b^2+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^2 + c*x^4)^3),x]

[Out]

(b^2 - 2*a*c + b*c*x^2)/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(
b^2 - 7*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b
+ 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(5/2)) + Log[x]/a^3 - Log[a + b*x^2 + c*x^4]/(4*a^3)

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 822

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

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]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {-2 \left (b^2-4 a c\right )-3 b c x}{x \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {2 \left (b^2-4 a c\right )^2+2 b c \left (b^2-7 a c\right ) x}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {2 \left (-b^2+4 a c\right )^2}{a x}+\frac {2 \left (-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x\right )}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\log (x)}{a^3}+\frac {\operatorname {Subst}\left (\int \frac {-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\log (x)}{a^3}-\frac {\operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3}-\frac {\left (b \left (b^4-10 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^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac {\left (b \left (b^4-10 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^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b \left (b^4-10 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^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.52, size = 342, normalized size = 1.71 \begin {gather*} \frac {\frac {a^2 \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}+\frac {a \left (16 a^2 c^2-15 a b^2 c-14 a b c^2 x^2+2 b^4+2 b^3 c x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (16 a^2 c^2 \sqrt {b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt {b^2-4 a c}+b^4 \sqrt {b^2-4 a c}+b^5\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {\left (-16 a^2 c^2 \sqrt {b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt {b^2-4 a c}-b^4 \sqrt {b^2-4 a c}+b^5\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+4 \log (x)}{4 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2*(b^2 - 2*a*c + b*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (a*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2
*b^3*c*x^2 - 14*a*b*c^2*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + 4*Log[x] - ((b^5 - 10*a*b^3*c + 30*a^2*b
*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqrt[b^2 - 4*a*c] + 16*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)^(5/2) + ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c
*Sqrt[b^2 - 4*a*c] - 16*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)^(5/2))/
(4*a^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.89, size = 2017, normalized size = 10.08

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*c^2 - 96*a^5*c^3 + 2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*
x^6 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3 - 64*a^4*c^4)*x^4 + 2*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c
^2 + 4*a^4*b*c^3)*x^2 + ((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^8 + a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 +
2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*x^6 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*x^4 + 2*(a*
b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*x^2)*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)) - ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^8 +
a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^
4)*x^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48
*a^3*b^3*c^2 - 64*a^4*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^
3*c^5)*x^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 -
64*a^3*b*c^4)*x^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^4 + 2*(a*b^7 - 12*a^
2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x^2)*log(x))/(a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^2 - 64*a^8*c^3 +
(a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*x^8 + 2*(a^3*b^7*c - 12*a^4*b^5*c^2 + 48*a^5*b^3*
c^3 - 64*a^6*b*c^4)*x^6 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128*a^7*c^4)*x^4 + 2*(a^
4*b^7 - 12*a^5*b^5*c + 48*a^6*b^3*c^2 - 64*a^7*b*c^3)*x^2), 1/4*(3*a^2*b^6 - 33*a^3*b^4*c + 108*a^4*b^2*c^2 -
96*a^5*c^3 + 2*(a*b^5*c^2 - 11*a^2*b^3*c^3 + 28*a^3*b*c^4)*x^6 + (4*a*b^6*c - 45*a^2*b^4*c^2 + 132*a^3*b^2*c^3
- 64*a^4*c^4)*x^4 + 2*(a*b^7 - 10*a^2*b^5*c + 23*a^3*b^3*c^2 + 4*a^4*b*c^3)*x^2 + 2*((b^5*c^2 - 10*a*b^3*c^3
+ 30*a^2*b*c^4)*x^8 + a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*x^6 +
(b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*x^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*x^2)*sqrt(-b
^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^
4 - 64*a^3*c^5)*x^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*
b^3*c^3 - 64*a^3*b*c^4)*x^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^4 + 2*(a*b^
7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((b^6*c^2 - 12*a*b^4*c^3 + 4
8*a^2*b^2*c^4 - 64*a^3*c^5)*x^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c
^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x
^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x^2)*log(x))/(a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^
2*c^2 - 64*a^8*c^3 + (a^3*b^6*c^2 - 12*a^4*b^4*c^3 + 48*a^5*b^2*c^4 - 64*a^6*c^5)*x^8 + 2*(a^3*b^7*c - 12*a^4*
b^5*c^2 + 48*a^5*b^3*c^3 - 64*a^6*b*c^4)*x^6 + (a^3*b^8 - 10*a^4*b^6*c + 24*a^5*b^4*c^2 + 32*a^6*b^2*c^3 - 128
*a^7*c^4)*x^4 + 2*(a^4*b^7 - 12*a^5*b^5*c + 48*a^6*b^3*c^2 - 64*a^7*b*c^3)*x^2)]

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giac [A]  time = 1.88, size = 323, normalized size = 1.62 \begin {gather*} -\frac {{\left (b^{5} - 10 \, 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^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b^{4} c^{2} x^{8} - 24 \, a b^{2} c^{3} x^{8} + 48 \, a^{2} c^{4} x^{8} + 6 \, b^{5} c x^{6} - 44 \, a b^{3} c^{2} x^{6} + 68 \, a^{2} b c^{3} x^{6} + 3 \, b^{6} x^{4} - 10 \, a b^{4} c x^{4} - 58 \, a^{2} b^{2} c^{2} x^{4} + 128 \, a^{3} c^{3} x^{4} + 10 \, a b^{5} x^{2} - 72 \, a^{2} b^{3} c x^{2} + 92 \, a^{3} b c^{2} x^{2} + 9 \, a^{2} b^{4} - 66 \, a^{3} b^{2} c + 96 \, a^{4} c^{2}}{8 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac {\log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac {\log \left (x^{2}\right )}{2 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^3*b^4 - 8*a^4*b^2*c + 16*a
^5*c^2)*sqrt(-b^2 + 4*a*c)) + 1/8*(3*b^4*c^2*x^8 - 24*a*b^2*c^3*x^8 + 48*a^2*c^4*x^8 + 6*b^5*c*x^6 - 44*a*b^3*
c^2*x^6 + 68*a^2*b*c^3*x^6 + 3*b^6*x^4 - 10*a*b^4*c*x^4 - 58*a^2*b^2*c^2*x^4 + 128*a^3*c^3*x^4 + 10*a*b^5*x^2
- 72*a^2*b^3*c*x^2 + 92*a^3*b*c^2*x^2 + 9*a^2*b^4 - 66*a^3*b^2*c + 96*a^4*c^2)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^
5*c^2)*(c*x^4 + b*x^2 + a)^2) - 1/4*log(c*x^4 + b*x^2 + a)/a^3 + 1/2*log(x^2)/a^3

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maple [B]  time = 0.03, size = 822, normalized size = 4.11 \begin {gather*} -\frac {7 b \,c^{3} x^{6}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {b^{3} c^{2} x^{6}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {29 b^{2} c^{2} x^{4}}{4 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {b^{4} c \,x^{4}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}+\frac {4 c^{3} x^{4}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 b^{3} c \,x^{2}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {b^{5} x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {b \,c^{2} x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 a \,c^{2}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 b^{4}}{4 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {15 b \,c^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a}+\frac {5 b^{3} c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b^{5} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{3}}-\frac {21 b^{2} c}{4 \left (c \,x^{4}+b \,x^{2}+a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {4 c^{2} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {2 b^{2} c \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {b^{4} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}+\frac {\ln \relax (x )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/2/a/(c*x^4+b*x^2+a)^2*b*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2/a^2/(c*x^4+b*x^2+a)^2*b^3*c^2/(16*a^2*c^2-8*
a*b^2*c+b^4)*x^6+4/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-29/4/a/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c
^2-8*a*b^2*c+b^4)*x^4*b^2+1/a^2/(c*x^4+b*x^2+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*b^4-1/2/(c*x^4+b*x^2+a)^2*b
/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*c^2-3/a/(c*x^4+b*x^2+a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*c+1/2/a^2/(c*x^4+
b*x^2+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+6*a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2-21/4/(c*x^4
+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2*c+3/4/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^4-4/a/(16*a^
2*c^2-8*a*b^2*c+b^4)*c^2*ln(c*x^4+b*x^2+a)+2/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln(c*x^4+b*x^2+a)*b^2-1/4/a^3/(1
6*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*b^4-15/a/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x
^2+b)/(4*a*c-b^2)^(1/2))*b*c^2+5/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^
2)^(1/2))*b^3*c-1/2/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b^5
+ln(x)/a^3

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 10.95, size = 9339, normalized size = 46.70

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^2 + c*x^4)^3),x)

[Out]

log(x)/a^3 + ((3*(b^4 + 8*a^2*c^2 - 7*a*b^2*c))/(4*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^4*(4*b^4*c + 16*a^2*
c^3 - 29*a*b^2*c^2))/(4*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (b*x^2*(a^2*c^2 - b^4 + 6*a*b^2*c))/(2*a^2*(b^4
+ 16*a^2*c^2 - 8*a*b^2*c)) - (b*c^2*x^6*(7*a*c - b^2))/(2*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b
^2) + a^2 + c^2*x^8 + 2*a*b*x^2 + 2*b*c*x^6) - (log((((a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a
*c - b^2)^5))^(1/2) + 1)*((b^2*c^3*(4*b^6 - 497*a^3*c^3 + 302*a^2*b^2*c^2 - 61*a*b^4*c))/(a^4*(4*a*c - b^2)^4)
- ((a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a*c - b^2)^5))^(1/2) + 1)*((4*b^2*c^2*(b^4 + 23*a^2
*c^2 - 9*a*b^2*c))/(a^2*(4*a*c - b^2)^2) + (b*c^2*(a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a*c -
b^2)^5))^(1/2) + 1)*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^3 + (2*b*c^3*x^2*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*(
4*a*c - b^2)^2)))/(4*a^3) + (b*c^4*x^2*(6*b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*(4*a*c - b^2
)^4)))/(4*a^3) - (b^2*c^4*(7*a*c - b^2)^2)/(a^6*(4*a*c - b^2)^4) + (b^3*c^5*x^2*(7*a*c - b^2)^3)/(a^6*(4*a*c -
b^2)^6))*(((a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a*c - b^2)^5))^(1/2) - 1)*(((a^3*(-(b^2*(b^
4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a*c - b^2)^5))^(1/2) - 1)*((4*b^2*c^2*(b^4 + 23*a^2*c^2 - 9*a*b^2*c))/
(a^2*(4*a*c - b^2)^2) - (b*c^2*(a^3*(-(b^2*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^2)/(a^6*(4*a*c - b^2)^5))^(1/2) - 1
)*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^3 + (2*b*c^3*x^2*(b^4 + 10*a^2*c^2 - 2*a*b^2*c))/(a^2*(4*a*c - b^2)^2)))/(
4*a^3) + (b^2*c^3*(4*b^6 - 497*a^3*c^3 + 302*a^2*b^2*c^2 - 61*a*b^4*c))/(a^4*(4*a*c - b^2)^4) + (b*c^4*x^2*(6*
b^6 - 560*a^3*c^3 + 409*a^2*b^2*c^2 - 89*a*b^4*c))/(a^4*(4*a*c - b^2)^4)))/(4*a^3) + (b^2*c^4*(7*a*c - b^2)^2)
/(a^6*(4*a*c - b^2)^4) - (b^3*c^5*x^2*(7*a*c - b^2)^3)/(a^6*(4*a*c - b^2)^6)))*(2*b^10 - 2048*a^5*c^5 + 320*a^
2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c +
640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)) + (b*atan((((b*((4*a^2*b^8*c^3 - 61*a^3*b^6*c^4 + 302*
a^4*b^4*c^5 - 497*a^5*b^2*c^6)/(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3) - ((
(4*a^4*b^10*c^2 - 68*a^5*b^8*c^3 + 444*a^6*b^6*c^4 - 1312*a^7*b^4*c^5 + 1472*a^8*b^2*c^6)/(a^6*b^8 + 256*a^10*
c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3) + ((4*a^7*b^10*c^2 - 64*a^8*b^8*c^3 + 384*a^9*b^6*c^4 -
1024*a^10*b^4*c^5 + 1024*a^11*b^2*c^6)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4
*b^2*c^4 - 40*a*b^8*c))/(2*(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3)*(4*a^3*b
^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)))*(2*b^10 - 2048*a^
5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 8
0*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3
*(4*a*c - b^2)^(5/2)) - (((b*((4*a^4*b^10*c^2 - 68*a^5*b^8*c^3 + 444*a^6*b^6*c^4 - 1312*a^7*b^4*c^5 + 1472*a^8
*b^2*c^6)/(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3) + ((4*a^7*b^10*c^2 - 64*a
^8*b^8*c^3 + 384*a^9*b^6*c^4 - 1024*a^10*b^4*c^5 + 1024*a^11*b^2*c^6)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2
- 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c
^2 - 256*a^9*b^2*c^3)*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^
7*b^2*c^4)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*(4*a*c - b^2)^(5/2)) + (b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)
*(4*a^7*b^10*c^2 - 64*a^8*b^8*c^3 + 384*a^9*b^6*c^4 - 1024*a^10*b^4*c^5 + 1024*a^11*b^2*c^6)*(2*b^10 - 2048*a^
5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(8*a^3*(4*a*c - b^2)^(5/2)*(a^6*b
^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3)*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c
+ 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^
3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2
560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)) + (b^3*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)^3*(4*a^7*b^10*c^2 - 64*a^8*b^8*c^3
+ 384*a^9*b^6*c^4 - 1024*a^10*b^4*c^5 + 1024*a^11*b^2*c^6))/(64*a^9*(4*a*c - b^2)^(15/2)*(a^6*b^8 + 256*a^10*
c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3)))*(3*b^8 + 160*a^4*c^4 + 180*a^2*b^4*c^2 - 325*a^3*b^2*
c^3 - 39*a*b^6*c)*(16*a^9*b^12*(4*a*c - b^2)^(15/2) + 65536*a^15*c^6*(4*a*c - b^2)^(15/2) - 384*a^10*b^10*c*(4
*a*c - b^2)^(15/2) + 3840*a^11*b^8*c^2*(4*a*c - b^2)^(15/2) - 20480*a^12*b^6*c^3*(4*a*c - b^2)^(15/2) + 61440*
a^13*b^4*c^4*(4*a*c - b^2)^(15/2) - 98304*a^14*b^2*c^5*(4*a*c - b^2)^(15/2)))/(8*a^3*c^2*(4*a*c - b^2)^(13/2)*
(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 600*a^3*b^4*c^5 + 900*a^4*b^2*c^6)*(6*b^10 - 6400*a^5*c^5 + 960*a
^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*c^4 - 120*a*b^8*c)) - (x^2*((((((b*((5120*a^10*b*c^9 + 2*a^4*b^13
*c^3 - 36*a^5*b^11*c^4 + 276*a^6*b^9*c^5 - 1216*a^7*b^7*c^6 + 3456*a^8*b^5*c^7 - 6144*a^9*b^3*c^8)/(a^6*b^12 +
4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5) -
((2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c)*(163840*a^13*b*
c^9 - 12*a^6*b^15*c^2 + 328*a^7*b^13*c^3 - 3840*a^8*b^11*c^4 + 24960*a^9*b^9*c^5 - 97280*a^10*b^7*c^6 + 227328
*a^11*b^5*c^7 - 294912*a^12*b^3*c^8))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^
6*b^4*c^3 + 5120*a^7*b^2*c^4)*(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 +
3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*(4*a*c - b^2)^(5/2)) - (b*(b
^4 + 30*a^2*c^2 - 10*a*b^2*c)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 -
40*a*b^8*c)*(163840*a^13*b*c^9 - 12*a^6*b^15*c^2 + 328*a^7*b^13*c^3 - 3840*a^8*b^11*c^4 + 24960*a^9*b^9*c^5 -
97280*a^10*b^7*c^6 + 227328*a^11*b^5*c^7 - 294912*a^12*b^3*c^8))/(8*a^3*(4*a*c - b^2)^(5/2)*(4*a^3*b^10 - 409
6*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)*(a^6*b^12 + 4096*a^12*c^6 -
24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5)))*(2*b^10 - 2048*a
^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4*a^3*b^10 - 4096*a^8*c^5 -
80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)) + (b*((8960*a^7*b*c^9 - 6*a^2*b^11*c^4
+ 137*a^3*b^9*c^5 - 1217*a^4*b^7*c^6 + 5256*a^5*b^5*c^7 - 11024*a^6*b^3*c^8)/(a^6*b^12 + 4096*a^12*c^6 - 24*a^
7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5) + (((5120*a^10*b*c^9 +
2*a^4*b^13*c^3 - 36*a^5*b^11*c^4 + 276*a^6*b^9*c^5 - 1216*a^7*b^7*c^6 + 3456*a^8*b^5*c^7 - 6144*a^9*b^3*c^8)/(
a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*
b^2*c^5) - ((2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c)*(1638
40*a^13*b*c^9 - 12*a^6*b^15*c^2 + 328*a^7*b^13*c^3 - 3840*a^8*b^11*c^4 + 24960*a^9*b^9*c^5 - 97280*a^10*b^7*c^
6 + 227328*a^11*b^5*c^7 - 294912*a^12*b^3*c^8))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2
- 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)*(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9
*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5)))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^
3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*
b^4*c^3 + 5120*a^7*b^2*c^4)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*(4*a*c - b^2)^(5/2)) + (b^3*(b^4 + 30*a^
2*c^2 - 10*a*b^2*c)^3*(163840*a^13*b*c^9 - 12*a^6*b^15*c^2 + 328*a^7*b^13*c^3 - 3840*a^8*b^11*c^4 + 24960*a^9*
b^9*c^5 - 97280*a^10*b^7*c^6 + 227328*a^11*b^5*c^7 - 294912*a^12*b^3*c^8))/(64*a^9*(4*a*c - b^2)^(15/2)*(a^6*b
^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c
^5)))*(3*b^8 + 160*a^4*c^4 + 180*a^2*b^4*c^2 - 325*a^3*b^2*c^3 - 39*a*b^6*c))/(8*a^3*c^2*(4*a*c - b^2)^(13/2)*
(6*b^10 - 6400*a^5*c^5 + 960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*c^4 - 120*a*b^8*c)) + (3*b*((b^9*c^
5 - 21*a*b^7*c^6 + 147*a^2*b^5*c^7 - 343*a^3*b^3*c^8)/(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*
c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5) + (((8960*a^7*b*c^9 - 6*a^2*b^11*c^4 + 137*a^3
*b^9*c^5 - 1217*a^4*b^7*c^6 + 5256*a^5*b^5*c^7 - 11024*a^6*b^3*c^8)/(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c
+ 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5) + (((5120*a^10*b*c^9 + 2*a^4*b^1
3*c^3 - 36*a^5*b^11*c^4 + 276*a^6*b^9*c^5 - 1216*a^7*b^7*c^6 + 3456*a^8*b^5*c^7 - 6144*a^9*b^3*c^8)/(a^6*b^12
+ 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5)
- ((2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c)*(163840*a^13*b
*c^9 - 12*a^6*b^15*c^2 + 328*a^7*b^13*c^3 - 3840*a^8*b^11*c^4 + 24960*a^9*b^9*c^5 - 97280*a^10*b^7*c^6 + 22732
8*a^11*b^5*c^7 - 294912*a^12*b^3*c^8))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a
^6*b^4*c^3 + 5120*a^7*b^2*c^4)*(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3
+ 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5)))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*
a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 +
5120*a^7*b^2*c^4)))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8
*c))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)) -
(b*((b*((5120*a^10*b*c^9 + 2*a^4*b^13*c^3 - 36*a^5*b^11*c^4 + 276*a^6*b^9*c^5 - 1216*a^7*b^7*c^6 + 3456*a^8*b^
5*c^7 - 6144*a^9*b^3*c^8)/(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 384
0*a^10*b^4*c^4 - 6144*a^11*b^2*c^5) - ((2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*
b^2*c^4 - 40*a*b^8*c)*(163840*a^13*b*c^9 - 12*a^6*b^15*c^2 + 328*a^7*b^13*c^3 - 3840*a^8*b^11*c^4 + 24960*a^9*
b^9*c^5 - 97280*a^10*b^7*c^6 + 227328*a^11*b^5*c^7 - 294912*a^12*b^3*c^8))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*
a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)*(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c +
240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)
)/(4*a^3*(4*a*c - b^2)^(5/2)) - (b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 -
1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c)*(163840*a^13*b*c^9 - 12*a^6*b^15*c^2 + 328*a^7*b^13*c^3 - 38
40*a^8*b^11*c^4 + 24960*a^9*b^9*c^5 - 97280*a^10*b^7*c^6 + 227328*a^11*b^5*c^7 - 294912*a^12*b^3*c^8))/(8*a^3*
(4*a*c - b^2)^(5/2)*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*
b^2*c^4)*(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 -
6144*a^11*b^2*c^5)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*(4*a*c - b^2)^(5/2)) + (b^2*(b^4 + 30*a^2*c^2 - 1
0*a*b^2*c)^2*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c)*(163
840*a^13*b*c^9 - 12*a^6*b^15*c^2 + 328*a^7*b^13*c^3 - 3840*a^8*b^11*c^4 + 24960*a^9*b^9*c^5 - 97280*a^10*b^7*c
^6 + 227328*a^11*b^5*c^7 - 294912*a^12*b^3*c^8))/(32*a^6*(4*a*c - b^2)^5*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b
^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)*(a^6*b^12 + 4096*a^12*c^6 - 24*a^7*b^10*c + 240*
a^8*b^8*c^2 - 1280*a^9*b^6*c^3 + 3840*a^10*b^4*c^4 - 6144*a^11*b^2*c^5)))*(b^6 - 45*a^3*c^3 + 40*a^2*b^2*c^2 -
11*a*b^4*c))/(8*a^3*c^2*(4*a*c - b^2)^6*(6*b^10 - 6400*a^5*c^5 + 960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^
4*b^2*c^4 - 120*a*b^8*c)))*(16*a^9*b^12*(4*a*c - b^2)^(15/2) + 65536*a^15*c^6*(4*a*c - b^2)^(15/2) - 384*a^10*
b^10*c*(4*a*c - b^2)^(15/2) + 3840*a^11*b^8*c^2*(4*a*c - b^2)^(15/2) - 20480*a^12*b^6*c^3*(4*a*c - b^2)^(15/2)
+ 61440*a^13*b^4*c^4*(4*a*c - b^2)^(15/2) - 98304*a^14*b^2*c^5*(4*a*c - b^2)^(15/2)))/(b^10*c^2 - 20*a*b^8*c^
3 + 160*a^2*b^6*c^4 - 600*a^3*b^4*c^5 + 900*a^4*b^2*c^6) + (3*b*(b^6 - 45*a^3*c^3 + 40*a^2*b^2*c^2 - 11*a*b^4*
c)*((((4*a^2*b^8*c^3 - 61*a^3*b^6*c^4 + 302*a^4*b^4*c^5 - 497*a^5*b^2*c^6)/(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^
6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3) - (((4*a^4*b^10*c^2 - 68*a^5*b^8*c^3 + 444*a^6*b^6*c^4 - 1312*a^7*b^4*
c^5 + 1472*a^8*b^2*c^6)/(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3) + ((4*a^7*b
^10*c^2 - 64*a^8*b^8*c^3 + 384*a^9*b^6*c^4 - 1024*a^10*b^4*c^5 + 1024*a^11*b^2*c^6)*(2*b^10 - 2048*a^5*c^5 + 3
20*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c
+ 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3)*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4
*c^3 + 5120*a^7*b^2*c^4)))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40
*a*b^8*c))/(2*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^
4)))*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(2*(4*a^3*b
^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)) - (b^6*c^4 - 14*a*
b^4*c^5 + 49*a^2*b^2*c^6)/(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3) + (b*((b*
((4*a^4*b^10*c^2 - 68*a^5*b^8*c^3 + 444*a^6*b^6*c^4 - 1312*a^7*b^4*c^5 + 1472*a^8*b^2*c^6)/(a^6*b^8 + 256*a^10
*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3) + ((4*a^7*b^10*c^2 - 64*a^8*b^8*c^3 + 384*a^9*b^6*c^4
- 1024*a^10*b^4*c^5 + 1024*a^11*b^2*c^6)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^
4*b^2*c^4 - 40*a*b^8*c))/(2*(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3)*(4*a^3*
b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)))*(b^4 + 30*a^2*c^
2 - 10*a*b^2*c))/(4*a^3*(4*a*c - b^2)^(5/2)) + (b*(b^4 + 30*a^2*c^2 - 10*a*b^2*c)*(4*a^7*b^10*c^2 - 64*a^8*b^8
*c^3 + 384*a^9*b^6*c^4 - 1024*a^10*b^4*c^5 + 1024*a^11*b^2*c^6)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 128
0*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(8*a^3*(4*a*c - b^2)^(5/2)*(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^
6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3)*(4*a^3*b^10 - 4096*a^8*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6
*b^4*c^3 + 5120*a^7*b^2*c^4)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(4*a^3*(4*a*c - b^2)^(5/2)) + (b^2*(b^4 + 30*a
^2*c^2 - 10*a*b^2*c)^2*(4*a^7*b^10*c^2 - 64*a^8*b^8*c^3 + 384*a^9*b^6*c^4 - 1024*a^10*b^4*c^5 + 1024*a^11*b^2*
c^6)*(2*b^10 - 2048*a^5*c^5 + 320*a^2*b^6*c^2 - 1280*a^3*b^4*c^3 + 2560*a^4*b^2*c^4 - 40*a*b^8*c))/(32*a^6*(4*
a*c - b^2)^5*(a^6*b^8 + 256*a^10*c^4 - 16*a^7*b^6*c + 96*a^8*b^4*c^2 - 256*a^9*b^2*c^3)*(4*a^3*b^10 - 4096*a^8
*c^5 - 80*a^4*b^8*c + 640*a^5*b^6*c^2 - 2560*a^6*b^4*c^3 + 5120*a^7*b^2*c^4)))*(16*a^9*b^12*(4*a*c - b^2)^(15/
2) + 65536*a^15*c^6*(4*a*c - b^2)^(15/2) - 384*a^10*b^10*c*(4*a*c - b^2)^(15/2) + 3840*a^11*b^8*c^2*(4*a*c - b
^2)^(15/2) - 20480*a^12*b^6*c^3*(4*a*c - b^2)^(15/2) + 61440*a^13*b^4*c^4*(4*a*c - b^2)^(15/2) - 98304*a^14*b^
2*c^5*(4*a*c - b^2)^(15/2)))/(8*a^3*c^2*(4*a*c - b^2)^6*(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 600*a^3*b
^4*c^5 + 900*a^4*b^2*c^6)*(6*b^10 - 6400*a^5*c^5 + 960*a^2*b^6*c^2 - 3850*a^3*b^4*c^3 + 7775*a^4*b^2*c^4 - 120
*a*b^8*c)))*(b^4 + 30*a^2*c^2 - 10*a*b^2*c))/(2*a^3*(4*a*c - b^2)^(5/2))

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sympy [F(-1)]  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/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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