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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0282816, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^2)),x]

[Out]

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

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

Mathematica [A]  time = 0.006423, size = 49, normalized size = 1. \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 44, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,a{x}^{4}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 2.11841, size = 63, normalized size = 1.29 \begin{align*} -\frac{b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac{b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac{2 \, b x^{2} - a}{4 \, a^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.29664, size = 108, normalized size = 2.2 \begin{align*} -\frac{2 \, b^{2} x^{4} \log \left (b x^{2} + a\right ) - 4 \, b^{2} x^{4} \log \left (x\right ) - 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/(b*x^2+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.452362, size = 42, normalized size = 0.86 \begin{align*} \frac{- a + 2 b x^{2}}{4 a^{2} x^{4}} + \frac{b^{2} \log{\left (x \right )}}{a^{3}} - \frac{b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.63094, size = 77, normalized size = 1.57 \begin{align*} \frac{b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac{b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} - \frac{3 \, b^{2} 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/(b*x^2+a),x, algorithm="giac")

[Out]

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