Optimal. Leaf size=64 \[ -\frac{a^3 \log \left (a+b x^n\right )}{b^4 n}+\frac{a^2 x^n}{b^3 n}-\frac{a x^{2 n}}{2 b^2 n}+\frac{x^{3 n}}{3 b n} \]
[Out]
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Rubi [A] time = 0.0812594, antiderivative size = 64, 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 \[ -\frac{a^3 \log \left (a+b x^n\right )}{b^4 n}+\frac{a^2 x^n}{b^3 n}-\frac{a x^{2 n}}{2 b^2 n}+\frac{x^{3 n}}{3 b n} \]
Antiderivative was successfully verified.
[In] Int[x^(3 + 4*(-1 + n))/(a + b*x^n),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3} \log{\left (a + b x^{n} \right )}}{b^{4} n} - \frac{a \int ^{x^{n}} x\, dx}{b^{2} n} + \frac{x^{3 n}}{3 b n} + \frac{\int ^{x^{n}} a^{2}\, dx}{b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+4*n)/(a+b*x**n),x)
[Out]
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Mathematica [A] time = 0.0365072, size = 52, normalized size = 0.81 \[ \frac{b x^n \left (6 a^2-3 a b x^n+2 b^2 x^{2 n}\right )-6 a^3 \log \left (a+b x^n\right )}{6 b^4 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3 + 4*(-1 + n))/(a + b*x^n),x]
[Out]
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Maple [A] time = 0., size = 69, normalized size = 1.1 \[{\frac{{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{3}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,bn}}-{\frac{a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{2}n}}-{\frac{{a}^{3}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{4}n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+4*n)/(a+b*x^n),x)
[Out]
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Maxima [A] time = 1.42216, size = 81, normalized size = 1.27 \[ -\frac{a^{3} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{4} n} + \frac{2 \, b^{2} x^{3 \, n} - 3 \, a b x^{2 \, n} + 6 \, a^{2} x^{n}}{6 \, b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*n - 1)/(b*x^n + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227456, size = 70, normalized size = 1.09 \[ \frac{2 \, b^{3} x^{3 \, n} - 3 \, a b^{2} x^{2 \, n} + 6 \, a^{2} b x^{n} - 6 \, a^{3} \log \left (b x^{n} + a\right )}{6 \, b^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*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+4*n)/(a+b*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4 \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*n - 1)/(b*x^n + a),x, algorithm="giac")
[Out]