3.3.41 \(\int \frac {1}{\frac {b}{x}+a x} \, dx\)

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

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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 = {1593, 260} \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 260

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 1593

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 (a x^2+b\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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\frac {b}{x}+a x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b/x + a*x)^(-1),x]

[Out]

IntegrateAlgebraic[(b/x + a*x)^(-1), x]

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fricas [A]  time = 0.37, 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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giac [A]  time = 0.15, 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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maple [A]  time = 0.04, size = 14, normalized size = 0.93 \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/(b/x+a*x),x)

[Out]

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

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maxima [A]  time = 1.29, 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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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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sympy [A]  time = 0.12, 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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