Optimal. Leaf size=85 \[ \frac {b^2 x^n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac {a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \log (x)}{a+b x^n} \]
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Rubi [A]
time = 0.02, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1369, 14}
\begin {gather*} \frac {b^2 x^n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac {a \log (x) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1369
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{x} \, dx &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \frac {a b+b^2 x^n}{x} \, dx}{a b+b^2 x^n}\\ &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \left (\frac {a b}{x}+b^2 x^{-1+n}\right ) \, dx}{a b+b^2 x^n}\\ &=\frac {b^2 x^n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac {a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \log (x)}{a+b x^n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 38, normalized size = 0.45 \begin {gather*} \frac {\sqrt {\left (a+b x^n\right )^2} \left (b x^n+a \log \left (x^n\right )\right )}{n \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 54, normalized size = 0.64
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a \ln \left (x \right )}{a +b \,x^{n}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b \,x^{n}}{\left (a +b \,x^{n}\right ) n}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 13, normalized size = 0.15 \begin {gather*} a \log \left (x\right ) + \frac {b x^{n}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 15, normalized size = 0.18 \begin {gather*} \frac {a n \log \left (x\right ) + b x^{n}}{n} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {\left (a + b x^{n}\right )^{2}}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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