Optimal. Leaf size=42 \[ \frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b}+\frac{d x}{b} \]
[Out]
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Rubi [A] time = 0.0525895, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b}+\frac{d x}{b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^n)/(a + b*x^n),x]
[Out]
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Rubi in Sympy [A] time = 6.35457, size = 31, normalized size = 0.74 \[ \frac{d x}{b} - \frac{x \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*x**n)/(a+b*x**n),x)
[Out]
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Mathematica [A] time = 0.0332002, size = 40, normalized size = 0.95 \[ \frac{x \left ((b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+a d\right )}{a b} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^n)/(a + b*x^n),x]
[Out]
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Maple [F] time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{c+d{x}^{n}}{a+b{x}^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*x^n)/(a+b*x^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[{\left (b c - a d\right )} \int \frac{1}{b^{2} x^{n} + a b}\,{d x} + \frac{d x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)/(b*x^n + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{n} + c}{b x^{n} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + 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**n)/(a+b*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{n} + c}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)/(b*x^n + a),x, algorithm="giac")
[Out]