Optimal. Leaf size=18 \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]
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Rubi [A] time = 0.0116419, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^n)^(-1 - n^(-1)),x]
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Rubi in Sympy [A] time = 1.39143, size = 12, normalized size = 0.67 \[ \frac{x \left (c + d x^{n}\right )^{- \frac{1}{n}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*x**n)**(-1-1/n),x)
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Mathematica [A] time = 0.0399012, size = 18, normalized size = 1. \[ \frac{x \left (c+d x^n\right )^{-1/n}}{c} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^n)^(-1 - n^(-1)),x]
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Maple [B] time = 0.036, size = 53, normalized size = 2.9 \[ x{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{dx{{\rm e}^{n\ln \left ( x \right ) }}}{c}{{\rm e}^{ \left ( -1-{n}^{-1} \right ) \ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*x^n)^(-1-1/n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n - 1),x, algorithm="maxima")
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Fricas [A] time = 0.248533, size = 42, normalized size = 2.33 \[ \frac{d x x^{n} + c x}{{\left (d x^{n} + c\right )}^{\frac{n + 1}{n}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n - 1),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d*x**n)**(-1-1/n),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n - 1),x, algorithm="giac")
[Out]