3.2738 \(\int \frac{x^m}{a+b x^{1+m}} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log \left (a+b x^{m+1}\right )}{b (m+1)} \]

[Out]

Log[a + b*x^(1 + m)]/(b*(1 + m))

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Rubi [A]  time = 0.0051135, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {260} \[ \frac{\log \left (a+b x^{m+1}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m/(a + b*x^(1 + m)),x]

[Out]

Log[a + b*x^(1 + m)]/(b*(1 + m))

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]

Rubi steps

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

Mathematica [A]  time = 0.0054525, size = 19, normalized size = 1. \[ \frac{\log \left (a+b x^{m+1}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/(a + b*x^(1 + m)),x]

[Out]

Log[a + b*x^(1 + m)]/(b*(1 + m))

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Maple [A]  time = 0.012, size = 21, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+bx{{\rm e}^{m\ln \left ( x \right ) }} \right ) }{b \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b/(1+m)*ln(a+b*x*exp(m*ln(x)))

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Maxima [A]  time = 0.966832, size = 26, normalized size = 1.37 \begin{align*} \frac{\log \left (b x^{m + 1} + a\right )}{b{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x^(m + 1) + a)/(b*(m + 1))

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Fricas [A]  time = 1.35528, size = 43, normalized size = 2.26 \begin{align*} \frac{\log \left (b x^{m + 1} + a\right )}{b m + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x^(m + 1) + a)/(b*m + b)

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Sympy [A]  time = 0.773872, size = 37, normalized size = 1.95 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge m = -1 \\\frac{x x^{m}}{a \left (m + 1\right )} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: m = -1 \\\frac{\log{\left (\frac{a}{b} + x x^{m} \right )}}{b m + b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(x)/a, Eq(b, 0) & Eq(m, -1)), (x*x**m/(a*(m + 1)), Eq(b, 0)), (log(x)/(a + b), Eq(m, -1)), (log(
a/b + x*x**m)/(b*m + b), True))

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Giac [A]  time = 1.0839, size = 26, normalized size = 1.37 \begin{align*} \frac{\log \left ({\left | b x^{m + 1} + a \right |}\right )}{b m + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*x^(m + 1) + a))/(b*m + b)