Optimal. Leaf size=46 \[ \frac{a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]
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Rubi [A] time = 0.0658019, antiderivative size = 46, 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 \[ \frac{a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]
Antiderivative was successfully verified.
[In] Int[x^(2 + 3*(-1 + n))/(a + b*x^n),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \log{\left (a + b x^{n} \right )}}{b^{3} n} + \frac{\int ^{x^{n}} x\, dx}{b n} - \frac{\int ^{x^{n}} a\, dx}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+3*n)/(a+b*x**n),x)
[Out]
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Mathematica [A] time = 0.0252838, size = 38, normalized size = 0.83 \[ \frac{2 a^2 \log \left (a+b x^n\right )+b x^n \left (b x^n-2 a\right )}{2 b^3 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(2 + 3*(-1 + n))/(a + b*x^n),x]
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Maple [A] time = 0.001, size = 51, normalized size = 1.1 \[{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,bn}}-{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}}+{\frac{{a}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+3*n)/(a+b*x^n),x)
[Out]
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Maxima [A] time = 1.45392, size = 61, normalized size = 1.33 \[ \frac{a^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} + \frac{b x^{2 \, n} - 2 \, a x^{n}}{2 \, b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/(b*x^n + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226673, size = 51, normalized size = 1.11 \[ \frac{b^{2} x^{2 \, n} - 2 \, a b x^{n} + 2 \, a^{2} \log \left (b x^{n} + a\right )}{2 \, b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/(b*x^n + a),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+3*n)/(a+b*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3 \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/(b*x^n + a),x, algorithm="giac")
[Out]