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 = 0.02, 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 = {1905, 251, 266}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 266
Rule 1905
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 = 0.08, size = 48, normalized size = 1.14 \begin {gather*} \frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a}+\frac {d \left (n \log (x)+\log \left (a-a x^n\right )\right )}{b n} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {c +d \,x^{-1+n}}{a +b \,x^{n}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.38, size = 21, normalized size = 0.50 \begin {gather*} {\rm integral}\left (\frac {d x^{n - 1} + c}{b x^{n} + a}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.20, size = 65, normalized size = 1.55 \begin {gather*} d \left (\begin {cases} \frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{n}}{a n} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b} & \text {for}\: n = 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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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