Optimal. Leaf size=27 \[ \frac{2 \log \left (c \sqrt{a+b x^n}+d\right )}{b c n} \]
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Rubi [A] time = 0.188998, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 \log \left (c \sqrt{a+b x^n}+d\right )}{b c n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + n)/(a*c + b*c*x^n + d*Sqrt[a + b*x^n]),x]
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Rubi in Sympy [A] time = 10.5515, size = 20, normalized size = 0.74 \[ \frac{2 \log{\left (c \sqrt{a + b x^{n}} + d \right )}}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+n)/(a*c+b*c*x**n+d*(a+b*x**n)**(1/2)),x)
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Mathematica [A] time = 0.0308307, size = 27, normalized size = 1. \[ \frac{2 \log \left (c \sqrt{a+b x^n}+d\right )}{b c n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + n)/(a*c + b*c*x^n + d*Sqrt[a + b*x^n]),x]
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Maple [F] time = 0.021, size = 0, normalized size = 0. \[ \int{{x}^{-1+n} \left ( ac+bc{x}^{n}+d\sqrt{a+b{x}^{n}} \right ) ^{-1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+n)/(a*c+b*c*x^n+d*(a+b*x^n)^(1/2)),x)
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Maxima [A] time = 0.770031, size = 82, normalized size = 3.04 \[ -\frac{\log \left (\frac{b x^{n} + a}{b}\right )}{b c n} + \frac{2 \, \log \left (\frac{b c x^{n} + a c + \sqrt{b x^{n} + a} d}{d}\right )}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(n - 1)/(b*c*x^n + a*c + sqrt(b*x^n + a)*d),x, algorithm="maxima")
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Fricas [A] time = 0.285102, size = 34, normalized size = 1.26 \[ \frac{2 \, \log \left (\sqrt{b x^{n} + a} c + d\right )}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(n - 1)/(b*c*x^n + a*c + sqrt(b*x^n + a)*d),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+n)/(a*c+b*c*x**n+d*(a+b*x**n)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.384767, size = 55, normalized size = 2.04 \[ \frac{2 \,{\rm ln}\left ({\left | \sqrt{b x^{n} + a} c + d \right |}\right )}{b c n} - \frac{2 \,{\rm ln}\left ({\left | d \right |}\right )}{b c n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(n - 1)/(b*c*x^n + a*c + sqrt(b*x^n + a)*d),x, algorithm="giac")
[Out]