3.854 \(\int \frac{1}{x^5 (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=114 \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c}}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

[Out]

-1/(4*a*x^4) + b/(2*a^2*x^2) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*Sqrt[b^2 - 4*
a*c]) + ((b^2 - a*c)*Log[x])/a^3 - ((b^2 - a*c)*Log[a + b*x^2 + c*x^4])/(4*a^3)

________________________________________________________________________________________

Rubi [A]  time = 0.196236, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {1114, 709, 800, 634, 618, 206, 628} \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c}}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

-1/(4*a*x^4) + b/(2*a^2*x^2) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*Sqrt[b^2 - 4*
a*c]) + ((b^2 - a*c)*Log[x])/a^3 - ((b^2 - a*c)*Log[a + b*x^2 + c*x^4])/(4*a^3)

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 709

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c
*e*x, x])/(a + b*x + c*x^2), 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[m, -1]

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 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 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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.222882, size = 188, normalized size = 1.65 \[ \frac{-\frac{a^2}{x^4}-\frac{\left (b^2 \sqrt{b^2-4 a c}-a c \sqrt{b^2-4 a c}-3 a b c+b^3\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\sqrt{b^2-4 a c}}+\frac{\left (-b^2 \sqrt{b^2-4 a c}+a c \sqrt{b^2-4 a c}-3 a b c+b^3\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\sqrt{b^2-4 a c}}+4 \log (x) \left (b^2-a c\right )+\frac{2 a b}{x^2}}{4 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^2 + c*x^4)),x]

[Out]

(-(a^2/x^4) + (2*a*b)/x^2 + 4*(b^2 - a*c)*Log[x] - ((b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*
a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c] + ((b^3 - 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*
Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*a^3)

________________________________________________________________________________________

Maple [A]  time = 0.171, size = 159, normalized size = 1.4 \begin{align*} -{\frac{1}{4\,a{x}^{4}}}-{\frac{c\ln \left ( x \right ) }{{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}}{4\,{a}^{3}}}+{\frac{3\,bc}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}}{2\,{a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(c*x^4+b*x^2+a),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.7616, size = 819, normalized size = 7.18 \begin{align*} \left [-\frac{{\left (b^{3} - 3 \, a b c\right )} \sqrt{b^{2} - 4 \, a c} x^{4} \log \left (\frac{2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c -{\left (2 \, c x^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \,{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} \log \left (x\right ) + a^{2} b^{2} - 4 \, a^{3} c - 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}}{4 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}, \frac{2 \,{\left (b^{3} - 3 \, a b c\right )} \sqrt{-b^{2} + 4 \, a c} x^{4} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \,{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{4} \log \left (x\right ) - a^{2} b^{2} + 4 \, a^{3} c + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}}{4 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[-1/4*((b^3 - 3*a*b*c)*sqrt(b^2 - 4*a*c)*x^4*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^4 - 5*a*b^2*c + 4*a^2*c^2)*x^4*log(c*x^4 + b*x^2 + a) - 4*(b^4 - 5*a*b^2*
c + 4*a^2*c^2)*x^4*log(x) + a^2*b^2 - 4*a^3*c - 2*(a*b^3 - 4*a^2*b*c)*x^2)/((a^3*b^2 - 4*a^4*c)*x^4), 1/4*(2*(
b^3 - 3*a*b*c)*sqrt(-b^2 + 4*a*c)*x^4*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (b^4 - 5*a*b^2
*c + 4*a^2*c^2)*x^4*log(c*x^4 + b*x^2 + a) + 4*(b^4 - 5*a*b^2*c + 4*a^2*c^2)*x^4*log(x) - a^2*b^2 + 4*a^3*c +
2*(a*b^3 - 4*a^2*b*c)*x^2)/((a^3*b^2 - 4*a^4*c)*x^4)]

________________________________________________________________________________________

Sympy [B]  time = 9.74828, size = 423, normalized size = 3.71 \begin{align*} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) \log{\left (x^{2} + \frac{8 a^{4} c \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) - 2 a^{3} b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) - 2 a^{2} c^{2} + 4 a b^{2} c - b^{4}}{3 a b c^{2} - b^{3} c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) \log{\left (x^{2} + \frac{8 a^{4} c \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) - 2 a^{3} b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 a^{3} \left (4 a c - b^{2}\right )} + \frac{a c - b^{2}}{4 a^{3}}\right ) - 2 a^{2} c^{2} + 4 a b^{2} c - b^{4}}{3 a b c^{2} - b^{3} c} \right )} + \frac{- a + 2 b x^{2}}{4 a^{2} x^{4}} - \frac{\left (a c - b^{2}\right ) \log{\left (x \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(c*x**4+b*x**2+a),x)

[Out]

(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**3))*log(x**2 + (8*a**4*c*(
-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**3)) - 2*a**3*b**2*(-b*sqrt(
-4*a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**3)) - 2*a**2*c**2 + 4*a*b**2*c - b*
*4)/(3*a*b*c**2 - b**3*c)) + (b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a
**3))*log(x**2 + (8*a**4*c*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**
3)) - 2*a**3*b**2*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*a**3*(4*a*c - b**2)) + (a*c - b**2)/(4*a**3)) - 2*a
**2*c**2 + 4*a*b**2*c - b**4)/(3*a*b*c**2 - b**3*c)) + (-a + 2*b*x**2)/(4*a**2*x**4) - (a*c - b**2)*log(x)/a**
3

________________________________________________________________________________________

Giac [A]  time = 1.17367, size = 170, normalized size = 1.49 \begin{align*} -\frac{{\left (b^{2} - a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac{{\left (b^{2} - a c\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac{{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{3}} - \frac{3 \, b^{2} x^{4} - 3 \, a c x^{4} - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/4*(b^2 - a*c)*log(c*x^4 + b*x^2 + a)/a^3 + 1/2*(b^2 - a*c)*log(x^2)/a^3 - 1/2*(b^3 - 3*a*b*c)*arctan((2*c*x
^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^3) - 1/4*(3*b^2*x^4 - 3*a*c*x^4 - 2*a*b*x^2 + a^2)/(a^3*x^4)