Optimal. Leaf size=85 \[ \frac {\log (x) \left (a+b x^n\right )}{a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1355, 266, 36, 29, 31} \[ \frac {\log (x) \left (a+b x^n\right )}{a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx &=\frac {\left (a b+b^2 x^n\right ) \int \frac {1}{x \left (a b+b^2 x^n\right )} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (a b+b^2 x^n\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )} \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (a b+b^2 x^n\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{a b n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (b \left (a b+b^2 x^n\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x} \, dx,x,x^n\right )}{a n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (a+b x^n\right ) \log (x)}{a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.49 \[ \frac {\left (a+b x^n\right ) \left (n \log (x)-\log \left (a+b x^n\right )\right )}{a n \sqrt {\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 22, normalized size = 0.26 \[ \frac {n \log \relax (x) - \log \left (b x^{n} + a\right )}{a n} \]
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 \[ \int \frac {1}{\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.78 \[ \frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, \ln \relax (x )}{\left (b \,x^{n}+a \right ) a}-\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, \ln \left (x^{n}+\frac {a}{b}\right )}{\left (b \,x^{n}+a \right ) a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 27, normalized size = 0.32 \[ \frac {\log \relax (x)}{a} - \frac {\log \left (\frac {b x^{n} + a}{b}\right )}{a n} \]
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 \[ \int \frac {1}{x\,\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\left (a + b x^{n}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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