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

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Rubi [A]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.01, size = 49, normalized size = 1.00 \[ -\frac {b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac {b^2 \log (x)}{a^3}+\frac {b}{2 a^2 x^2}-\frac {1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^2)),x]

[Out]

-1/4*1/(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)

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fricas [A]  time = 0.65, size = 45, normalized size = 0.92 \[ -\frac {2 \, b^{2} x^{4} \log \left (b x^{2} + a\right ) - 4 \, b^{2} x^{4} \log \relax (x) - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \]

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)

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giac [A]  time = 1.19, size = 57, normalized size = 1.16 \[ \frac {b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} - \frac {3 \, b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \]

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)

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maple [A]  time = 0.01, size = 44, normalized size = 0.90 \[ \frac {b^{2} \ln \relax (x )}{a^{3}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}+\frac {b}{2 a^{2} x^{2}}-\frac {1}{4 a \,x^{4}} \]

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

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maxima [A]  time = 1.36, size = 47, normalized size = 0.96 \[ -\frac {b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {2 \, b x^{2} - a}{4 \, a^{2} x^{4}} \]

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)

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mupad [B]  time = 0.08, size = 46, normalized size = 0.94 \[ \frac {b^2\,\ln \relax (x)}{a^3}-\frac {b^2\,\ln \left (b\,x^2+a\right )}{2\,a^3}-\frac {\frac {1}{4\,a}-\frac {b\,x^2}{2\,a^2}}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x^2)),x)

[Out]

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

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sympy [A]  time = 0.30, size = 42, normalized size = 0.86 \[ \frac {- a + 2 b x^{2}}{4 a^{2} x^{4}} + \frac {b^{2} \log {\relax (x )}}{a^{3}} - \frac {b^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3}} \]

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)

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