3.19.52 \(\int \frac {1}{(a+\frac {b}{x^2}) x^3} \, dx\) [1852]

Optimal. Leaf size=15 \[ -\frac {\log \left (a+\frac {b}{x^2}\right )}{2 b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {266} \begin {gather*} -\frac {\log \left (a+\frac {b}{x^2}\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)*x^3),x]

[Out]

-1/2*Log[a + b/x^2]/b

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.47 \begin {gather*} \frac {\log (x)}{b}-\frac {\log \left (b+a x^2\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)*x^3),x]

[Out]

Log[x]/b - Log[b + a*x^2]/(2*b)

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Maple [A]
time = 0.03, size = 21, normalized size = 1.40

method	result	size
derivativedivides	\(-\frac {\ln \left (\frac {b}{x^{2}}+a \right )}{2 b}\)	\(14\)
default	\(-\frac {\ln \left (a \,x^{2}+b \right )}{2 b}+\frac {\ln \left (x \right )}{b}\)	\(21\)
norman	\(-\frac {\ln \left (a \,x^{2}+b \right )}{2 b}+\frac {\ln \left (x \right )}{b}\)	\(21\)
risch	\(-\frac {\ln \left (a \,x^{2}+b \right )}{2 b}+\frac {\ln \left (x \right )}{b}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x^2+a)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} -\frac {\log \left (a + \frac {b}{x^{2}}\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(a + b/x^2)/b

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Fricas [A]
time = 0.37, size = 18, normalized size = 1.20 \begin {gather*} -\frac {\log \left (a x^{2} + b\right ) - 2 \, \log \left (x\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(a*x^2 + b) - 2*log(x))/b

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Sympy [A]
time = 0.08, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log {\left (x \right )}}{b} - \frac {\log {\left (x^{2} + \frac {b}{a} \right )}}{2 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/b - log(x**2 + b/a)/(2*b)

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Giac [A]
time = 1.06, size = 24, normalized size = 1.60 \begin {gather*} \frac {\log \left (x^{2}\right )}{2 \, b} - \frac {\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 18, normalized size = 1.20 \begin {gather*} -\frac {\ln \left (a\,x^2+b\right )-2\,\ln \left (x\right )}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x^2)),x)

[Out]

-(log(b + a*x^2) - 2*log(x))/(2*b)

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