Optimal. Leaf size=58 \[ -\frac {\log \left (a+b x^n\right )}{a^3 n}+\frac {\log (x)}{a^3}+\frac {1}{a^2 n \left (a+b x^n\right )}+\frac {1}{2 a n \left (a+b x^n\right )^2} \]
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Rubi [A] time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac {1}{a^2 n \left (a+b x^n\right )}-\frac {\log \left (a+b x^n\right )}{a^3 n}+\frac {\log (x)}{a^3}+\frac {1}{2 a n \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^n\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^3} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {1}{2 a n \left (a+b x^n\right )^2}+\frac {1}{a^2 n \left (a+b x^n\right )}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x^n\right )}{a^3 n}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 47, normalized size = 0.81 \[ \frac {\frac {a \left (3 a+2 b x^n\right )}{\left (a+b x^n\right )^2}-2 \log \left (a+b x^n\right )+2 n \log (x)}{2 a^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 106, normalized size = 1.83 \[ \frac {2 \, b^{2} n x^{2 \, n} \log \relax (x) + 2 \, a^{2} n \log \relax (x) + 3 \, a^{2} + 2 \, {\left (2 \, a b n \log \relax (x) + a b\right )} x^{n} - 2 \, {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}} \]
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 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 62, normalized size = 1.07 \[ \frac {1}{2 \left (b \,x^{n}+a \right )^{2} a n}+\frac {1}{\left (b \,x^{n}+a \right ) a^{2} n}+\frac {\ln \left (x^{n}\right )}{a^{3} n}-\frac {\ln \left (b \,x^{n}+a \right )}{a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 71, normalized size = 1.22 \[ \frac {2 \, b x^{n} + 3 \, a}{2 \, {\left (a^{2} b^{2} n x^{2 \, n} + 2 \, a^{3} b n x^{n} + a^{4} n\right )}} - \frac {\log \left (b x^{n} + a\right )}{a^{3} n} + \frac {\log \left (x^{n}\right )}{a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 69, normalized size = 1.19 \[ \frac {\ln \relax (x)}{a^3}+\frac {1}{a^2\,n\,\left (a+b\,x^n\right )}-\frac {\ln \left (a+b\,x^n\right )}{a^3\,n}+\frac {1}{2\,a\,n\,\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.80, size = 406, normalized size = 7.00 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {x^{- 3 n}}{3 b^{3} n} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{\left (a + b\right )^{3}} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{a^{3}} & \text {for}\: b = 0 \\\frac {2 a^{2} n \log {\relax (x )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac {2 a^{2} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac {3 a^{2}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac {4 a b n x^{n} \log {\relax (x )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac {4 a b x^{n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac {2 a b x^{n}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac {2 b^{2} n x^{2 n} \log {\relax (x )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac {2 b^{2} x^{2 n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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