3.28.38 \(\int \frac {x^m}{a+b x^{1+m}} \, dx\) [2738]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {266} \begin {gather*} \frac {\log \left (a+b x^{m+1}\right )}{b (m+1)} \end {gather*}

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

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Mathematica [A]
time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log \left (a+b x^{1+m}\right )}{b (1+m)} \end {gather*}

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 = 0.24, size = 21, normalized size = 1.11

method	result	size
norman	\(\frac {\ln \left (a +b x \,{\mathrm e}^{m \ln \left (x \right )}\right )}{b \left (1+m \right )}\)	\(21\)
risch	\(\frac {\ln \left (x \right )}{b}-\frac {m \ln \left (x \right )}{b \left (1+m \right )}+\frac {\ln \left (x^{m}+\frac {a}{b x}\right )}{b \left (1+m \right )}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x^{m + 1} + a\right )}{b {\left (m + 1\right )}} \end {gather*}

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 = 0.39, size = 18, normalized size = 0.95 \begin {gather*} \frac {\log \left (b x^{m + 1} + a\right )}{b m + b} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (14) = 28\).
time = 0.39, size = 37, normalized size = 1.95 \begin {gather*} \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 {gather*}

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 = 0.61, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log \left ({\left | b x^{m + 1} + a \right |}\right )}{b m + b} \end {gather*}

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)

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Mupad [B]
time = 1.25, size = 19, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,x^{m+1}\right )}{b\,\left (m+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a + b*x^(m + 1)),x)

[Out]

log(a + b*x^(m + 1))/(b*(m + 1))

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