Optimal. Leaf size=42 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \log \left (a+b x^n\right )}{b n} \]
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Rubi [A] time = 0.0630661, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \log \left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^(-1 + n))/(a + b*x^n),x]
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Rubi in Sympy [A] time = 14.7596, size = 32, normalized size = 0.76 \[ \frac{d \log{\left (a + b x^{n} \right )}}{b n} + \frac{c x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*x**(-1+n))/(a+b*x**n),x)
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Mathematica [A] time = 0.0704356, size = 48, normalized size = 1.14 \[ \frac{c x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a}+\frac{d \left (\log \left (a-a x^n\right )+n \log (x)\right )}{b n} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + d*x^(-1 + n))/(a + b*x^n),x]
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Maple [F] time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{c+d{x}^{-1+n}}{a+b{x}^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*x^(-1+n))/(a+b*x^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{d \log \left (x\right )}{b} + \int \frac{b c x - a d}{b^{2} x x^{n} + a b x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)/(b*x^n + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{n - 1} + c}{b x^{n} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)/(b*x^n + a),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d*x**(-1+n))/(a+b*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{n - 1} + c}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^(n - 1) + c)/(b*x^n + a),x, algorithm="giac")
[Out]