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

Optimal. Leaf size=95 \[ -\frac{2 c^3}{b^5 \left (b+c x^2\right )}-\frac{c^3}{4 b^4 \left (b+c x^2\right )^2}-\frac{3 c^2}{b^5 x^2}+\frac{5 c^3 \log \left (b+c x^2\right )}{b^6}-\frac{10 c^3 \log (x)}{b^6}+\frac{3 c}{4 b^4 x^4}-\frac{1}{6 b^3 x^6} \]

[Out]

-1/(6*b^3*x^6) + (3*c)/(4*b^4*x^4) - (3*c^2)/(b^5*x^2) - c^3/(4*b^4*(b + c*x^2)^2) - (2*c^3)/(b^5*(b + c*x^2))
- (10*c^3*Log[x])/b^6 + (5*c^3*Log[b + c*x^2])/b^6

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Rubi [A]  time = 0.0847774, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 266, 44} \[ -\frac{2 c^3}{b^5 \left (b+c x^2\right )}-\frac{c^3}{4 b^4 \left (b+c x^2\right )^2}-\frac{3 c^2}{b^5 x^2}+\frac{5 c^3 \log \left (b+c x^2\right )}{b^6}-\frac{10 c^3 \log (x)}{b^6}+\frac{3 c}{4 b^4 x^4}-\frac{1}{6 b^3 x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(b*x^2 + c*x^4)^3),x]

[Out]

-1/(6*b^3*x^6) + (3*c)/(4*b^4*x^4) - (3*c^2)/(b^5*x^2) - c^3/(4*b^4*(b + c*x^2)^2) - (2*c^3)/(b^5*(b + c*x^2))
- (10*c^3*Log[x])/b^6 + (5*c^3*Log[b + c*x^2])/b^6

Rule 1584

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

Rule 266

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \left (b x^2+c x^4\right )^3} \, dx &=\int \frac{1}{x^7 \left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^3 x^4}-\frac{3 c}{b^4 x^3}+\frac{6 c^2}{b^5 x^2}-\frac{10 c^3}{b^6 x}+\frac{c^4}{b^4 (b+c x)^3}+\frac{4 c^4}{b^5 (b+c x)^2}+\frac{10 c^4}{b^6 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{6 b^3 x^6}+\frac{3 c}{4 b^4 x^4}-\frac{3 c^2}{b^5 x^2}-\frac{c^3}{4 b^4 \left (b+c x^2\right )^2}-\frac{2 c^3}{b^5 \left (b+c x^2\right )}-\frac{10 c^3 \log (x)}{b^6}+\frac{5 c^3 \log \left (b+c x^2\right )}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0728489, size = 85, normalized size = 0.89 \[ -\frac{\frac{b \left (20 b^2 c^2 x^4-5 b^3 c x^2+2 b^4+90 b c^3 x^6+60 c^4 x^8\right )}{x^6 \left (b+c x^2\right )^2}-60 c^3 \log \left (b+c x^2\right )+120 c^3 \log (x)}{12 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(b*x^2 + c*x^4)^3),x]

[Out]

-((b*(2*b^4 - 5*b^3*c*x^2 + 20*b^2*c^2*x^4 + 90*b*c^3*x^6 + 60*c^4*x^8))/(x^6*(b + c*x^2)^2) + 120*c^3*Log[x]
- 60*c^3*Log[b + c*x^2])/(12*b^6)

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Maple [A]  time = 0.058, size = 90, normalized size = 1. \begin{align*} -{\frac{1}{6\,{b}^{3}{x}^{6}}}+{\frac{3\,c}{4\,{b}^{4}{x}^{4}}}-3\,{\frac{{c}^{2}}{{b}^{5}{x}^{2}}}-{\frac{{c}^{3}}{4\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}}-2\,{\frac{{c}^{3}}{{b}^{5} \left ( c{x}^{2}+b \right ) }}-10\,{\frac{{c}^{3}\ln \left ( x \right ) }{{b}^{6}}}+5\,{\frac{{c}^{3}\ln \left ( c{x}^{2}+b \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/b^3/x^6+3/4*c/b^4/x^4-3*c^2/b^5/x^2-1/4*c^3/b^4/(c*x^2+b)^2-2*c^3/b^5/(c*x^2+b)-10*c^3*ln(x)/b^6+5*c^3*ln
(c*x^2+b)/b^6

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Maxima [A]  time = 1.03071, size = 139, normalized size = 1.46 \begin{align*} -\frac{60 \, c^{4} x^{8} + 90 \, b c^{3} x^{6} + 20 \, b^{2} c^{2} x^{4} - 5 \, b^{3} c x^{2} + 2 \, b^{4}}{12 \,{\left (b^{5} c^{2} x^{10} + 2 \, b^{6} c x^{8} + b^{7} x^{6}\right )}} + \frac{5 \, c^{3} \log \left (c x^{2} + b\right )}{b^{6}} - \frac{5 \, c^{3} \log \left (x^{2}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(60*c^4*x^8 + 90*b*c^3*x^6 + 20*b^2*c^2*x^4 - 5*b^3*c*x^2 + 2*b^4)/(b^5*c^2*x^10 + 2*b^6*c*x^8 + b^7*x^6
) + 5*c^3*log(c*x^2 + b)/b^6 - 5*c^3*log(x^2)/b^6

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Fricas [A]  time = 1.56263, size = 308, normalized size = 3.24 \begin{align*} -\frac{60 \, b c^{4} x^{8} + 90 \, b^{2} c^{3} x^{6} + 20 \, b^{3} c^{2} x^{4} - 5 \, b^{4} c x^{2} + 2 \, b^{5} - 60 \,{\left (c^{5} x^{10} + 2 \, b c^{4} x^{8} + b^{2} c^{3} x^{6}\right )} \log \left (c x^{2} + b\right ) + 120 \,{\left (c^{5} x^{10} + 2 \, b c^{4} x^{8} + b^{2} c^{3} x^{6}\right )} \log \left (x\right )}{12 \,{\left (b^{6} c^{2} x^{10} + 2 \, b^{7} c x^{8} + b^{8} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(60*b*c^4*x^8 + 90*b^2*c^3*x^6 + 20*b^3*c^2*x^4 - 5*b^4*c*x^2 + 2*b^5 - 60*(c^5*x^10 + 2*b*c^4*x^8 + b^2
*c^3*x^6)*log(c*x^2 + b) + 120*(c^5*x^10 + 2*b*c^4*x^8 + b^2*c^3*x^6)*log(x))/(b^6*c^2*x^10 + 2*b^7*c*x^8 + b^
8*x^6)

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Sympy [A]  time = 2.05117, size = 104, normalized size = 1.09 \begin{align*} - \frac{2 b^{4} - 5 b^{3} c x^{2} + 20 b^{2} c^{2} x^{4} + 90 b c^{3} x^{6} + 60 c^{4} x^{8}}{12 b^{7} x^{6} + 24 b^{6} c x^{8} + 12 b^{5} c^{2} x^{10}} - \frac{10 c^{3} \log{\left (x \right )}}{b^{6}} + \frac{5 c^{3} \log{\left (\frac{b}{c} + x^{2} \right )}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b**4 - 5*b**3*c*x**2 + 20*b**2*c**2*x**4 + 90*b*c**3*x**6 + 60*c**4*x**8)/(12*b**7*x**6 + 24*b**6*c*x**8 +
12*b**5*c**2*x**10) - 10*c**3*log(x)/b**6 + 5*c**3*log(b/c + x**2)/b**6

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Giac [A]  time = 1.30037, size = 149, normalized size = 1.57 \begin{align*} -\frac{5 \, c^{3} \log \left (x^{2}\right )}{b^{6}} + \frac{5 \, c^{3} \log \left ({\left | c x^{2} + b \right |}\right )}{b^{6}} - \frac{30 \, c^{5} x^{4} + 68 \, b c^{4} x^{2} + 39 \, b^{2} c^{3}}{4 \,{\left (c x^{2} + b\right )}^{2} b^{6}} + \frac{110 \, c^{3} x^{6} - 36 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} - 2 \, b^{3}}{12 \, b^{6} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*c^3*log(x^2)/b^6 + 5*c^3*log(abs(c*x^2 + b))/b^6 - 1/4*(30*c^5*x^4 + 68*b*c^4*x^2 + 39*b^2*c^3)/((c*x^2 + b
)^2*b^6) + 1/12*(110*c^3*x^6 - 36*b*c^2*x^4 + 9*b^2*c*x^2 - 2*b^3)/(b^6*x^6)