Optimal. Leaf size=90 \[ \frac {x^n \left (a+b x^n\right )}{b n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {a \left (a+b x^n\right ) \log \left (a+b x^n\right )}{b^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.04, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1355, 266, 43} \[ \frac {x^n \left (a+b x^n\right )}{b n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {a \left (a+b x^n\right ) \log \left (a+b x^n\right )}{b^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\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 {x^{-1+2 n}}{a b+b^2 x^n} \, 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 {x}{a b+b^2 x} \, 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 \left (\frac {1}{b^2}-\frac {a}{b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {x^n \left (a+b x^n\right )}{b n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {a \left (a+b x^n\right ) \log \left (a+b x^n\right )}{b^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.51 \[ \frac {\left (a+b x^n\right ) \left (\frac {x^n}{b}-\frac {a \log \left (a+b x^n\right )}{b^2}\right )}{n \sqrt {\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 24, normalized size = 0.27 \[ \frac {b x^{n} - a \log \left (b x^{n} + a\right )}{b^{2} n} \]
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 \[ \int \frac {x^{2 \, n - 1}}{\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 71, normalized size = 0.79 \[ -\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, a \ln \left (x^{n}+\frac {a}{b}\right )}{\left (b \,x^{n}+a \right ) b^{2} n}+\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, x^{n}}{\left (b \,x^{n}+a \right ) b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 32, normalized size = 0.36 \[ \frac {x^{n}}{b n} - \frac {a \log \left (\frac {b x^{n} + a}{b}\right )}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{2\,n-1}}{\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}} \,d x \]
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 \[ \int \frac {x^{2 n - 1}}{\sqrt {\left (a + b x^{n}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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