Optimal. Leaf size=46 \[ -\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n}+\frac {b^2 \log \left (b+c x^n\right )}{c^3 n} \]
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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1598, 272, 45}
\begin {gather*} \frac {b^2 \log \left (b+c x^n\right )}{c^3 n}-\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{-1+4 n}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1+3 n}}{b+c x^n} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{b+c x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {b}{c^2}+\frac {x}{c}+\frac {b^2}{c^2 (b+c x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n}+\frac {b^2 \log \left (b+c x^n\right )}{c^3 n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.83 \begin {gather*} \frac {c x^n \left (-2 b+c x^n\right )+2 b^2 \log \left (b+c x^n\right )}{2 c^3 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 47, normalized size = 1.02
method | result | size |
risch | \(\frac {x^{2 n}}{2 c n}-\frac {b \,x^{n}}{c^{2} n}+\frac {b^{2} \ln \left (x^{n}+\frac {b}{c}\right )}{c^{3} n}\) | \(47\) |
norman | \(\left (\frac {{\mathrm e}^{3 n \ln \left (x \right )}}{2 c n}-\frac {b \,{\mathrm e}^{2 n \ln \left (x \right )}}{c^{2} n}\right ) {\mathrm e}^{-n \ln \left (x \right )}+\frac {b^{2} \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}+b \right )}{c^{3} n}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 45, normalized size = 0.98 \begin {gather*} \frac {b^{2} \log \left (\frac {c x^{n} + b}{c}\right )}{c^{3} n} + \frac {c x^{2 \, n} - 2 \, b x^{n}}{2 \, c^{2} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 38, normalized size = 0.83 \begin {gather*} \frac {c^{2} x^{2 \, n} - 2 \, b c x^{n} + 2 \, b^{2} \log \left (c x^{n} + b\right )}{2 \, c^{3} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.25, size = 42, normalized size = 0.91 \begin {gather*} \frac {b^{2} \left (\begin {cases} \frac {x^{n}}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + c x^{n} \right )}}{c} & \text {otherwise} \end {cases}\right )}{c^{2} n} - \frac {b x^{n}}{c^{2} n} + \frac {x^{2 n}}{2 c n} \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{4\,n-1}}{b\,x^n+c\,x^{2\,n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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