3.2731 \(\int \frac{x^m}{a+b x^{1+m}} \, dx\)

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 = 0.0242368, 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 \[ \frac{\log \left (a+b x^{m+1}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^(1 + m)),x]

[Out]

Log[a + b*x^(1 + m)]/(b*(1 + m))

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Rubi in Sympy [A]  time = 2.79787, size = 14, normalized size = 0.74 \[ \frac{\log{\left (a + b x^{m + 1} \right )}}{b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(a+b*x**(1+m)),x)

[Out]

log(a + b*x**(m + 1))/(b*(m + 1))

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Mathematica [A]  time = 0.0101044, 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 = 0.022, size = 21, normalized size = 1.1 \[{\frac{\ln \left ( a+bx{{\rm e}^{m\ln \left ( x \right ) }} \right ) }{b \left ( 1+m \right ) }} \]

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 = 1.40971, size = 26, normalized size = 1.37 \[ \frac{\log \left (b x^{m + 1} + a\right )}{b{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x^(m + 1) + a)/(b*(m + 1))

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Fricas [A]  time = 0.229625, size = 24, normalized size = 1.26 \[ \frac{\log \left (b x^{m + 1} + a\right )}{b m + b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x^(m + 1) + a)/(b*m + b)

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Sympy [A]  time = 2.60728, size = 37, normalized size = 1.95 \[ \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge m = -1 \\\frac{x x^{m}}{a \left (m + 1\right )} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: m = -1 \\\frac{\log{\left (\frac{a}{b} + x x^{m} \right )}}{b m + b} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.21729, size = 26, normalized size = 1.37 \[ \frac{{\rm ln}\left ({\left | b x^{m + 1} + a \right |}\right )}{b m + b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*x^(m + 1) + a))/(b*m + b)