Optimal. Leaf size=43 \[ \frac{b x}{d}-\frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d} \]
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Rubi [A] time = 0.0170337, 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, Rules used = {388, 245} \[ \frac{b x}{d}-\frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d} \]
Antiderivative was successfully verified.
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Rule 388
Rule 245
Rubi steps
\begin{align*} \int \frac{a+b x^n}{c+d x^n} \, dx &=\frac{b x}{d}-\frac{(b c-a d) \int \frac{1}{c+d x^n} \, dx}{d}\\ &=\frac{b x}{d}-\frac{(b c-a d) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c d}\\ \end{align*}
Mathematica [A] time = 0.0117517, 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.
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Maple [F] time = 0.407, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{x}^{n}}{c+d{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}{d^{2} x^{n} + c d}\,{d x} + \frac{b x}{d} \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{b x^{n} + a}{d x^{n} + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.74438, size = 73, normalized size = 1.7 \begin{align*} \frac{a x \Phi \left (\frac{d x^{n} e^{i \pi }}{c}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{c n^{2} \Gamma \left (1 + \frac{1}{n}\right )} - \frac{b x \Phi \left (\frac{c x^{- n} e^{i \pi }}{d}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{d 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{b x^{n} + a}{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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