Optimal. Leaf size=46 \[ \frac{b^2 \log \left (b+c x^n\right )}{c^3 n}-\frac{b x^n}{c^2 n}+\frac{x^{2 n}}{2 c n} \]
[Out]
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Rubi [A] time = 0.0716772, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{b^2 \log \left (b+c x^n\right )}{c^3 n}-\frac{b x^n}{c^2 n}+\frac{x^{2 n}}{2 c n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 4*n)/(b*x^n + c*x^(2*n)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} \log{\left (b + c x^{n} \right )}}{c^{3} n} + \frac{\int ^{x^{n}} x\, dx}{c n} - \frac{\int ^{x^{n}} b\, dx}{c^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+4*n)/(b*x**n+c*x**(2*n)),x)
[Out]
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Mathematica [A] time = 0.0326373, size = 38, normalized size = 0.83 \[ \frac{2 b^2 \log \left (b+c x^n\right )+c x^n \left (c x^n-2 b\right )}{2 c^3 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 4*n)/(b*x^n + c*x^(2*n)),x]
[Out]
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Maple [A] time = 0.04, size = 62, normalized size = 1.4 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ({\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,cn}}-{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{c}^{2}n}} \right ) }+{\frac{{b}^{2}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{{c}^{3}n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+4*n)/(b*x^n+c*x^(2*n)),x)
[Out]
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Maxima [A] time = 0.754667, size = 61, normalized size = 1.33 \[ \frac{b^{2} \log \left (\frac{c x^{n} + b}{c}\right )}{c^{3} n} + \frac{c x^{2 \, n} - 2 \, b x^{n}}{2 \, c^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.303133, size = 51, normalized size = 1.11 \[ \frac{c^{2} x^{2 \, n} - 2 \, b c x^{n} + 2 \, b^{2} \log \left (c x^{n} + b\right )}{2 \, c^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n),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)/(b*x**n+c*x**(2*n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4 \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(4*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="giac")
[Out]