3.4.38 \(\int \frac {1}{\frac {b}{x}+a x} \, dx\) [338]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 266} \begin {gather*} \frac {\log \left (a x^2+b\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b/x + a*x)^(-1),x]

[Out]

Log[b + a*x^2]/(2*a)

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]

Rule 1607

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (b+a x^2\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b/x + a*x)^(-1),x]

[Out]

Log[b + a*x^2]/(2*a)

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Maple [A]
time = 0.02, size = 14, normalized size = 0.93

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/x*b+a*x),x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a*x^2 + b)/a

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Fricas [A]
time = 1.82, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (a x^{2} + b\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x+a*x),x, algorithm="fricas")

[Out]

1/2*log(a*x^2 + b)/a

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log {\left (a x^{2} + b \right )}}{2 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x**2 + b)/(2*a)

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Giac [A]
time = 0.50, size = 14, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b/x+a*x),x, algorithm="giac")

[Out]

1/2*log(abs(a*x^2 + b))/a

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Mupad [B]
time = 0.05, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (a\,x^2+b\right )}{2\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + b/x),x)

[Out]

log(b + a*x^2)/(2*a)

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