Optimal. Leaf size=96 \[ -\frac {b^2 x^{n-2} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac {a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 96, 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} \[ -\frac {b^2 x^{n-2} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac {a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \]
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^3} \, dx &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \frac {a b+b^2 x^n}{x^3} \, 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^3}+b^2 x^{-3+n}\right ) \, dx}{a b+b^2 x^n}\\ &=-\frac {a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )}-\frac {b^2 x^{-2+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.49 \[ \frac {\sqrt {\left (a+b x^n\right )^2} \left (2 b x^n-a (n-2)\right )}{2 (n-2) x^2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 23, normalized size = 0.24 \[ -\frac {a n - 2 \, b x^{n} - 2 \, a}{2 \, {\left (n - 2\right )} x^{2}} \]
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 \[ \int \frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 61, normalized size = 0.64 \[ \frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, b \,x^{n}}{\left (b \,x^{n}+a \right ) \left (n -2\right ) x^{2}}-\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, a}{2 \left (b \,x^{n}+a \right ) x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 22, normalized size = 0.23 \[ -\frac {a {\left (n - 2\right )} - 2 \, b x^{n}}{2 \, {\left (n - 2\right )} x^{2}} \]
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 \[ \int \frac {\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {\left (a + b x^{n}\right )^{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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