Optimal. Leaf size=94 \[ -\frac {b^2 x^{n-1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left (a b+b^2 x^n\right )}-\frac {a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 94, 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 = {1355, 14} \[ -\frac {b^2 x^{n-1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left (a b+b^2 x^n\right )}-\frac {a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1355
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \frac {a b+b^2 x^n}{x^2} \, 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^2}+b^2 x^{-2+n}\right ) \, dx}{a b+b^2 x^n}\\ &=-\frac {a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )}-\frac {b^2 x^{-1+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left (a b+b^2 x^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.45 \[ \frac {\sqrt {\left (a+b x^n\right )^2} \left (-a n+a+b x^n\right )}{(n-1) x \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 23, normalized size = 0.24 \[ -\frac {a n - b x^{n} - a}{{\left (n - 1\right )} x} \]
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 \[ \int \frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 61, normalized size = 0.65 \[ \frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, b \,x^{n}}{\left (b \,x^{n}+a \right ) \left (n -1\right ) x}-\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, a}{\left (b \,x^{n}+a \right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 22, normalized size = 0.23 \[ -\frac {a {\left (n - 1\right )} - b x^{n}}{{\left (n - 1\right )} x} \]
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 \[ \int \frac {\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {\left (a + b x^{n}\right )^{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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