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

Optimal. Leaf size=49 \[ -\frac{c^2 \log \left (b+c x^2\right )}{2 b^3}+\frac{c^2 \log (x)}{b^3}+\frac{c}{2 b^2 x^2}-\frac{1}{4 b x^4} \]

[Out]

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

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Rubi [A]  time = 0.0350715, antiderivative size = 49, 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} \[ -\frac{c^2 \log \left (b+c x^2\right )}{2 b^3}+\frac{c^2 \log (x)}{b^3}+\frac{c}{2 b^2 x^2}-\frac{1}{4 b x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(b*x^2 + c*x^4)),x]

[Out]

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

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

Mathematica [A]  time = 0.0070187, size = 49, normalized size = 1. \[ -\frac{c^2 \log \left (b+c x^2\right )}{2 b^3}+\frac{c^2 \log (x)}{b^3}+\frac{c}{2 b^2 x^2}-\frac{1}{4 b x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(b*x^2 + c*x^4)),x]

[Out]

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

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Maple [A]  time = 0.049, size = 44, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,b{x}^{4}}}+{\frac{c}{2\,{b}^{2}{x}^{2}}}+{\frac{{c}^{2}\ln \left ( x \right ) }{{b}^{3}}}-{\frac{{c}^{2}\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.97574, size = 63, normalized size = 1.29 \begin{align*} -\frac{c^{2} \log \left (c x^{2} + b\right )}{2 \, b^{3}} + \frac{c^{2} \log \left (x^{2}\right )}{2 \, b^{3}} + \frac{2 \, c x^{2} - b}{4 \, b^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.49403, size = 108, normalized size = 2.2 \begin{align*} -\frac{2 \, c^{2} x^{4} \log \left (c x^{2} + b\right ) - 4 \, c^{2} x^{4} \log \left (x\right ) - 2 \, b c x^{2} + b^{2}}{4 \, b^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.530072, size = 42, normalized size = 0.86 \begin{align*} \frac{- b + 2 c x^{2}}{4 b^{2} x^{4}} + \frac{c^{2} \log{\left (x \right )}}{b^{3}} - \frac{c^{2} \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2567, size = 77, normalized size = 1.57 \begin{align*} \frac{c^{2} \log \left (x^{2}\right )}{2 \, b^{3}} - \frac{c^{2} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{3}} - \frac{3 \, c^{2} x^{4} - 2 \, b c x^{2} + b^{2}}{4 \, b^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c^2*log(x^2)/b^3 - 1/2*c^2*log(abs(c*x^2 + b))/b^3 - 1/4*(3*c^2*x^4 - 2*b*c*x^2 + b^2)/(b^3*x^4)