3.2602 \(\int \frac{x^{1+2 (-1+n)}}{a+b x^n} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^n}{b n}-\frac{a \log \left (a+b x^n\right )}{b^2 n} \]

[Out]

x^n/(b*n) - (a*Log[a + b*x^n])/(b^2*n)

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Rubi [A]  time = 0.0473098, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^n}{b n}-\frac{a \log \left (a+b x^n\right )}{b^2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(1 + 2*(-1 + n))/(a + b*x^n),x]

[Out]

x^n/(b*n) - (a*Log[a + b*x^n])/(b^2*n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \log{\left (a + b x^{n} \right )}}{b^{2} n} + \frac{\int ^{x^{n}} \frac{1}{b}\, dx}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)/(a+b*x**n),x)

[Out]

-a*log(a + b*x**n)/(b**2*n) + Integral(1/b, (x, x**n))/n

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Mathematica [A]  time = 0.0135542, size = 24, normalized size = 0.86 \[ \frac{b x^n-a \log \left (a+b x^n\right )}{b^2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(1 + 2*(-1 + n))/(a + b*x^n),x]

[Out]

(b*x^n - a*Log[a + b*x^n])/(b^2*n)

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Maple [A]  time = 0., size = 33, normalized size = 1.2 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{bn}}-{\frac{a\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{2}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)/(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.44976, size = 43, normalized size = 1.54 \[ \frac{x^{n}}{b n} - \frac{a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

x^n/(b*n) - a*log((b*x^n + a)/b)/(b^2*n)

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Fricas [A]  time = 0.226984, size = 32, normalized size = 1.14 \[ \frac{b x^{n} - a \log \left (b x^{n} + a\right )}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/(b*x^n + a),x, algorithm="fricas")

[Out]

(b*x^n - a*log(b*x^n + a))/(b^2*n)

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Sympy [A]  time = 45.8575, size = 41, normalized size = 1.46 \[ \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{x^{2 n}}{2 a n} & \text{for}\: b = 0 \\- \frac{a \log{\left (\frac{a}{b} + x^{n} \right )}}{b^{2} n} + \frac{x^{n}}{b n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+2*n)/(a+b*x**n),x)

[Out]

Piecewise((log(x)/a, Eq(b, 0) & Eq(n, 0)), (log(x)/(a + b), Eq(n, 0)), (x**(2*n)
/(2*a*n), Eq(b, 0)), (-a*log(a/b + x**n)/(b**2*n) + x**n/(b*n), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)/(b*x^n + a), x)