Optimal. Leaf size=212 \[ -\frac {3 a^2 b^2 x^{n-1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left (a b+b^2 x^n\right )}-\frac {b^4 x^{3 n-1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-3 n) \left (a b+b^2 x^n\right )}-\frac {3 a b^3 x^{2 n-1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-2 n) \left (a b+b^2 x^n\right )}-\frac {a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 212, 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, 270} \[ -\frac {3 a^2 b^2 x^{n-1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left (a b+b^2 x^n\right )}-\frac {3 a b^3 x^{2 n-1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-2 n) \left (a b+b^2 x^n\right )}-\frac {b^4 x^{3 n-1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-3 n) \left (a b+b^2 x^n\right )}-\frac {a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \frac {\left (a b+b^2 x^n\right )^3}{x^2} \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \left (\frac {a^3 b^3}{x^2}+3 a^2 b^4 x^{-2+n}+3 a b^5 x^{2 (-1+n)}+b^6 x^{-2+3 n}\right ) \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )}-\frac {3 a^2 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 )}-\frac {3 a b^3 x^{-1+2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-2 n) \left (a b+b^2 x^n\right )}-\frac {b^4 x^{-1+3 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1-3 n) \left (a b+b^2 x^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 124, normalized size = 0.58 \[ \frac {\sqrt {\left (a+b x^n\right )^2} \left (a^3 \left (-6 n^3+11 n^2-6 n+1\right )+3 a^2 b \left (6 n^2-5 n+1\right ) x^n+3 a b^2 \left (3 n^2-4 n+1\right ) x^{2 n}+b^3 \left (2 n^2-3 n+1\right ) x^{3 n}\right )}{(n-1) (2 n-1) (3 n-1) x \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 131, normalized size = 0.62 \[ -\frac {6 \, a^{3} n^{3} - 11 \, a^{3} n^{2} + 6 \, a^{3} n - a^{3} - {\left (2 \, b^{3} n^{2} - 3 \, b^{3} n + b^{3}\right )} x^{3 \, n} - 3 \, {\left (3 \, a b^{2} n^{2} - 4 \, a b^{2} n + a b^{2}\right )} x^{2 \, n} - 3 \, {\left (6 \, a^{2} b n^{2} - 5 \, a^{2} b n + a^{2} b\right )} x^{n}}{{\left (6 \, n^{3} - 11 \, n^{2} + 6 \, n - 1\right )} x} \]
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 \[ \int \frac {{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 147, normalized size = 0.69 \[ \frac {3 \sqrt {\left (b \,x^{n}+a \right )^{2}}\, a^{2} b \,x^{n}}{\left (b \,x^{n}+a \right ) \left (n -1\right ) x}+\frac {3 \sqrt {\left (b \,x^{n}+a \right )^{2}}\, a \,b^{2} x^{2 n}}{\left (b \,x^{n}+a \right ) \left (2 n -1\right ) x}+\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, b^{3} x^{3 n}}{\left (b \,x^{n}+a \right ) \left (3 n -1\right ) x}-\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, a^{3}}{\left (b \,x^{n}+a \right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 101, normalized size = 0.48 \[ \frac {{\left (2 \, n^{2} - 3 \, n + 1\right )} b^{3} x^{3 \, n} + 3 \, {\left (3 \, n^{2} - 4 \, n + 1\right )} a b^{2} x^{2 \, n} + 3 \, {\left (6 \, n^{2} - 5 \, n + 1\right )} a^{2} b x^{n} - {\left (6 \, n^{3} - 11 \, n^{2} + 6 \, n - 1\right )} a^{3}}{{\left (6 \, n^{3} - 11 \, n^{2} + 6 \, n - 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}}{x^2} \,d x \]
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 \[ \int \frac {\left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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