Optimal. Leaf size=76 \[ \frac {b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac {b^3 \log (x)}{a^4}-\frac {b^2 x^{-n}}{a^3 n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {x^{-3 n}}{3 a n} \]
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Rubi [A] time = 0.03, antiderivative size = 76, 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, 44} \[ -\frac {b^2 x^{-n}}{a^3 n}+\frac {b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac {b^3 \log (x)}{a^4}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {x^{-3 n}}{3 a n} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-4-3 (-1+n)}}{a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 (a+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a x^4}-\frac {b}{a^2 x^3}+\frac {b^2}{a^3 x^2}-\frac {b^3}{a^4 x}+\frac {b^4}{a^4 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {b^2 x^{-n}}{a^3 n}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log \left (a+b x^n\right )}{a^4 n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.82 \[ -\frac {a x^{-3 n} \left (2 a^2-3 a b x^n+6 b^2 x^{2 n}\right )-6 b^3 \log \left (a+b x^n\right )+6 b^3 n \log (x)}{6 a^4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 72, normalized size = 0.95 \[ -\frac {6 \, b^{3} n x^{3 \, n} \log \relax (x) - 6 \, b^{3} x^{3 \, n} \log \left (b x^{n} + a\right ) + 6 \, a b^{2} x^{2 \, n} - 3 \, a^{2} b x^{n} + 2 \, a^{3}}{6 \, a^{4} n x^{3 \, n}} \]
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 = 88, normalized size = 1.16 \[ \left (-\frac {b^{3} {\mathrm e}^{3 n \ln \relax (x )} \ln \relax (x )}{a^{4}}+\frac {b \,{\mathrm e}^{n \ln \relax (x )}}{2 a^{2} n}-\frac {b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{a^{3} n}-\frac {1}{3 a n}\right ) {\mathrm e}^{-3 n \ln \relax (x )}+\frac {b^{3} \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{a^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 71, normalized size = 0.93 \[ -\frac {b^{3} \log \relax (x)}{a^{4}} + \frac {b^{3} \log \left (\frac {b x^{n} + a}{b}\right )}{a^{4} n} - \frac {6 \, b^{2} x^{2 \, n} - 3 \, a b x^{n} + 2 \, a^{2}}{6 \, a^{3} n x^{3 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{3\,n+1}\,\left (a+b\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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