Optimal. Leaf size=42 \[ \frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a}+\frac {d \log \left (a+b x^n\right )}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1891, 245, 260} \[ \frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a}+\frac {d \log \left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 245
Rule 260
Rule 1891
Rubi steps
\begin {align*} \int \frac {c+d x^{-1+n}}{a+b x^n} \, dx &=c \int \frac {1}{a+b x^n} \, dx+d \int \frac {x^{-1+n}}{a+b x^n} \, dx\\ &=\frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a}+\frac {d \log \left (a+b x^n\right )}{b n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 42, normalized size = 1.00 \[ \frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a}+\frac {d \log \left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d x^{n - 1} + c}{b x^{n} + a}, x\right ) \]
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 \[ \int \frac {d x^{n - 1} + c}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {d \,x^{n -1}+c}{b \,x^{n}+a}\, dx \]
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 \[ \frac {d \log \relax (x)}{b} + \int \frac {b c x - a d}{b^{2} x x^{n} + a b x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.33, size = 43, normalized size = 1.02 \[ \frac {c\,x\,{{}}_2{\mathrm {F}}_1\left (1,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{a}+\frac {d\,\ln \left (a+b\,x^n\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.15, size = 65, normalized size = 1.55 \[ d \left (\begin {cases} \frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b} & \text {for}\: n = 0 \\\frac {x^{n}}{a n} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{b n} & \text {otherwise} \end {cases}\right ) + \frac {c x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{2} \Gamma \left (1 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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