Optimal. Leaf size=42 \[ \frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b}+\frac{d x}{b} \]
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Rubi [A] time = 0.0159626, 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, Rules used = {388, 245} \[ \frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b}+\frac{d x}{b} \]
Antiderivative was successfully verified.
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Rule 388
Rule 245
Rubi steps
\begin{align*} \int \frac{c+d x^n}{a+b x^n} \, dx &=\frac{d x}{b}-\frac{(-b c+a d) \int \frac{1}{a+b x^n} \, dx}{b}\\ &=\frac{d x}{b}+\frac{(b c-a d) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a b}\\ \end{align*}
Mathematica [A] time = 0.0102025, 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.
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Maple [F] time = 0.416, size = 0, normalized size = 0. \begin{align*} \int{\frac{c+d{x}^{n}}{a+b{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (b c - a d\right )} \int \frac{1}{b^{2} x^{n} + a b}\,{d x} + \frac{d x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d x^{n} + c}{b x^{n} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.69671, size = 73, normalized size = 1.74 \begin{align*} - \frac{d x \Phi \left (\frac{a x^{- n} e^{i \pi }}{b}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{b n^{2} \Gamma \left (1 + \frac{1}{n}\right )} + \frac{c x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a n^{2} \Gamma \left (1 + \frac{1}{n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{n} + c}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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