Optimal. Leaf size=212 \[ -\frac{3 a^2 b^2 x^{n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left (a b+b^2 x^n\right )}-\frac{3 a b^3 x^{2 n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-2 n) \left (a b+b^2 x^n\right )}-\frac{b^4 x^{3 n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-3 n) \left (a b+b^2 x^n\right )}-\frac{a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )} \]
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Rubi [A] time = 0.0698802, antiderivative size = 212, normalized size of antiderivative = 1., 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 1355
Rule 270
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*}
Mathematica [A] time = 0.0941215, size = 124, normalized size = 0.58 \[ \frac{\sqrt{\left (a+b x^n\right )^2} \left (3 a^2 b \left (6 n^2-5 n+1\right ) x^n+a^3 \left (-6 n^3+11 n^2-6 n+1\right )+3 a b^2 \left (3 n^2-4 n+1\right ) x^{2 n}+b^3 \left (2 n^2-3 n+1\right ) x^{3 n}\right )}{(n-1) (2 n-1) (3 n-1) x \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 147, normalized size = 0.7 \begin{align*} -{\frac{{a}^{3}}{ \left ( a+b{x}^{n} \right ) x}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{b}^{3} \left ({x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n} \right ) \left ( -1+3\,n \right ) x}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}a{b}^{2} \left ({x}^{n} \right ) ^{2}}{ \left ( a+b{x}^{n} \right ) \left ( -1+2\,n \right ) x}}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{a}^{2}b{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( -1+n \right ) x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03693, size = 136, normalized size = 0.64 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58915, size = 270, normalized size = 1.27 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{n}\right )^{2}\right )^{\frac{3}{2}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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