Optimal. Leaf size=43 \[ \frac{b x}{d}-\frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d} \]
[Out]
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Rubi [A] time = 0.0551257, antiderivative size = 43, 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 \[ \frac{b x}{d}-\frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)/(c + d*x^n),x]
[Out]
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Rubi in Sympy [A] time = 7.20277, size = 31, normalized size = 0.72 \[ \frac{b x}{d} + \frac{x \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)/(c+d*x**n),x)
[Out]
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Mathematica [A] time = 0.0369414, size = 40, normalized size = 0.93 \[ \frac{x \left ((a d-b c) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+b c\right )}{c d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)/(c + d*x^n),x]
[Out]
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Maple [F] time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{a+b{x}^{n}}{c+d{x}^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)/(c+d*x^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -{\left (b c - a d\right )} \int \frac{1}{d^{2} x^{n} + c d}\,{d x} + \frac{b x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)/(d*x^n + c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b x^{n} + a}{d x^{n} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)/(d*x^n + c),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((a+b*x**n)/(c+d*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{n} + a}{d x^{n} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)/(d*x^n + c),x, algorithm="giac")
[Out]