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 = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 245
Rule 388
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*}
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Mathematica [A] time = 0.01, 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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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{n} + a}{d x^{n} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b x^{n} + a}{d x^{n} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {b \,x^{n}+a}{d \,x^{n}+c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -{\left (b c - a d\right )} \int \frac {1}{d^{2} x^{n} + c d}\,{d x} + \frac {b x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,x^n}{c+d\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.33, size = 73, normalized size = 1.70 \[ \frac {a x \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{c n^{2} \Gamma \left (1 + \frac {1}{n}\right )} - \frac {b x \Phi \left (\frac {c x^{- n} e^{i \pi }}{d}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{d n^{2} \Gamma \left (1 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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