Optimal. Leaf size=48 \[ -\frac{a^2}{b^3 n \left (a+b x^n\right )}-\frac{2 a \log \left (a+b x^n\right )}{b^3 n}+\frac{x^n}{b^2 n} \]
[Out]
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Rubi [A] time = 0.0783434, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^2}{b^3 n \left (a+b x^n\right )}-\frac{2 a \log \left (a+b x^n\right )}{b^3 n}+\frac{x^n}{b^2 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 3*n)/(a + b*x^n)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{b^{3} n \left (a + b x^{n}\right )} - \frac{2 a \log{\left (a + b x^{n} \right )}}{b^{3} n} + \frac{\int ^{x^{n}} \frac{1}{b^{2}}\, dx}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+3*n)/(a+b*x**n)**2,x)
[Out]
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Mathematica [A] time = 0.0511669, size = 38, normalized size = 0.79 \[ \frac{-\frac{a^2}{a+b x^n}-2 a \log \left (a+b x^n\right )+b x^n}{b^3 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 3*n)/(a + b*x^n)^2,x]
[Out]
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Maple [A] time = 0.031, size = 59, normalized size = 1.2 \[{\frac{1}{a+b{{\rm e}^{n\ln \left ( x \right ) }}} \left ({\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{bn}}-2\,{\frac{{a}^{2}}{{b}^{3}n}} \right ) }-2\,{\frac{a\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+3*n)/(a+b*x^n)^2,x)
[Out]
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Maxima [A] time = 1.4509, size = 82, normalized size = 1.71 \[ \frac{b^{2} x^{2 \, n} + a b x^{n} - a^{2}}{b^{4} n x^{n} + a b^{3} n} - \frac{2 \, a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/(b*x^n + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228412, size = 80, normalized size = 1.67 \[ \frac{b^{2} x^{2 \, n} + a b x^{n} - a^{2} - 2 \,{\left (a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{b^{4} n x^{n} + a b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/(b*x^n + a)^2,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)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/(b*x^n + a)^2,x, algorithm="giac")
[Out]