Optimal. Leaf size=159 \[ \frac {1}{a^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {1}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {\log (x) \left (a+b x^n\right )}{a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a^3 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.08, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1355, 266, 44} \[ \frac {1}{a^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {1}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {\log (x) \left (a+b x^n\right )}{a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a^3 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac {1}{x \left (a b+b^2 x^n\right )^3} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )^3} \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^3 b^3 x}-\frac {1}{a b^2 (a+b x)^3}-\frac {1}{a^2 b^2 (a+b x)^2}-\frac {1}{a^3 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 {1}{a^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {1}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {\left (a+b x^n\right ) \log (x)}{a^3 \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^3 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 76, normalized size = 0.48 \[ \frac {\left (a+b x^n\right )^3 \left (-\frac {\log \left (a+b x^n\right )}{a^3}+\frac {n \log (x)}{a^3}+\frac {1}{a^2 \left (a+b x^n\right )}+\frac {1}{2 a \left (a+b x^n\right )^2}\right )}{n \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 106, normalized size = 0.67 \[ \frac {2 \, b^{2} n x^{2 \, n} \log \relax (x) + 2 \, a^{2} n \log \relax (x) + 3 \, a^{2} + 2 \, {\left (2 \, a b n \log \relax (x) + a b\right )} x^{n} - 2 \, {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}} \]
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 \[ \int \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 104, normalized size = 0.65 \[ \frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, \ln \relax (x )}{\left (b \,x^{n}+a \right ) a^{3}}+\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, \left (2 b \,x^{n}+3 a \right )}{2 \left (b \,x^{n}+a \right )^{3} a^{2} n}-\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, \ln \left (x^{n}+\frac {a}{b}\right )}{\left (b \,x^{n}+a \right ) a^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 70, normalized size = 0.44 \[ \frac {2 \, b x^{n} + 3 \, a}{2 \, {\left (a^{2} b^{2} n x^{2 \, n} + 2 \, a^{3} b n x^{n} + a^{4} n\right )}} + \frac {\log \relax (x)}{a^{3}} - \frac {\log \left (\frac {b x^{n} + a}{b}\right )}{a^{3} n} \]
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 \[ \int \frac {1}{x\,{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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