Optimal. Leaf size=46 \[ \frac {a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac {a x^n}{b^2 n}+\frac {x^{2 n}}{2 b n} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {266, 43} \[ \frac {a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac {a x^n}{b^2 n}+\frac {x^{2 n}}{2 b n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{2+3 (-1+n)}}{a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a x^n}{b^2 n}+\frac {x^{2 n}}{2 b n}+\frac {a^2 \log \left (a+b x^n\right )}{b^3 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.83 \[ \frac {2 a^2 \log \left (a+b x^n\right )+b x^n \left (b x^n-2 a\right )}{2 b^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 38, normalized size = 0.83 \[ \frac {b^{2} x^{2 \, n} - 2 \, a b x^{n} + 2 \, a^{2} \log \left (b x^{n} + a\right )}{2 \, b^{3} n} \]
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 \[ \int \frac {x^{3 \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 51, normalized size = 1.11 \[ \frac {a^{2} \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{b^{3} n}-\frac {a \,{\mathrm e}^{n \ln \relax (x )}}{b^{2} n}+\frac {{\mathrm e}^{2 n \ln \relax (x )}}{2 b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 45, normalized size = 0.98 \[ \frac {a^{2} \log \left (\frac {b x^{n} + a}{b}\right )}{b^{3} n} + \frac {b x^{2 \, n} - 2 \, a x^{n}}{2 \, b^{2} n} \]
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 \[ \int \frac {x^{3\,n-1}}{a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.72, size = 56, normalized size = 1.22 \[ \begin {cases} \frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b} & \text {for}\: n = 0 \\\frac {x^{3 n}}{3 a n} & \text {for}\: b = 0 \\\frac {a^{2} \log {\left (\frac {a}{b} + x^{n} \right )}}{b^{3} n} - \frac {a x^{n}}{b^{2} n} + \frac {x^{2 n}}{2 b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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