Integrand size = 28, antiderivative size = 212 \[ \int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {a^3 x^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left (a+b x^n\right )}+\frac {b^4 x^{3 (1+n)} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 (1+n) \left (a b+b^2 x^n\right )}+\frac {3 a^2 b^2 x^{3+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(3+n) \left (a b+b^2 x^n\right )}+\frac {3 a b^3 x^{3+2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(3+2 n) \left (a b+b^2 x^n\right )} \]
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Time = 0.05 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1369, 276} \[ \int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {3 a^2 b^2 x^{n+3} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(n+3) \left (a b+b^2 x^n\right )}+\frac {b^4 x^{3 (n+1)} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 (n+1) \left (a b+b^2 x^n\right )}+\frac {3 a b^3 x^{2 n+3} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(2 n+3) \left (a b+b^2 x^n\right )}+\frac {a^3 x^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left (a+b x^n\right )} \]
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Rule 276
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int x^2 \left (a b+b^2 x^n\right )^3 \, 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 (a^3 b^3 x^2+3 a b^5 x^{2 (1+n)}+3 a^2 b^4 x^{2+n}+b^6 x^{2+3 n}\right ) \, dx}{b^2 \left (a b+b^2 x^n\right )} \\ & = \frac {a^3 x^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 \left (a+b x^n\right )}+\frac {b^4 x^{3 (1+n)} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 (1+n) \left (a b+b^2 x^n\right )}+\frac {3 a^2 b^2 x^{3+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(3+n) \left (a b+b^2 x^n\right )}+\frac {3 a b^3 x^{3+2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(3+2 n) \left (a b+b^2 x^n\right )} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.58 \[ \int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {x^3 \sqrt {\left (a+b x^n\right )^2} \left (a^3 \left (9+18 n+11 n^2+2 n^3\right )+9 a^2 b \left (3+5 n+2 n^2\right ) x^n+9 a b^2 \left (3+4 n+n^2\right ) x^{2 n}+b^3 \left (9+9 n+2 n^2\right ) x^{3 n}\right )}{3 (1+n) (3+n) (3+2 n) \left (a+b x^n\right )} \]
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Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{3} x^{3}}{3 a +3 b \,x^{n}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{3} x^{3} x^{3 n}}{3 \left (a +b \,x^{n}\right ) \left (1+n \right )}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{2} a \,x^{3} x^{2 n}}{\left (a +b \,x^{n}\right ) \left (3+2 n \right )}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{2} b \,x^{3} x^{n}}{\left (a +b \,x^{n}\right ) \left (3+n \right )}\) | \(146\) |
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Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.68 \[ \int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {{\left (2 \, b^{3} n^{2} + 9 \, b^{3} n + 9 \, b^{3}\right )} x^{3} x^{3 \, n} + 9 \, {\left (a b^{2} n^{2} + 4 \, a b^{2} n + 3 \, a b^{2}\right )} x^{3} x^{2 \, n} + 9 \, {\left (2 \, a^{2} b n^{2} + 5 \, a^{2} b n + 3 \, a^{2} b\right )} x^{3} x^{n} + {\left (2 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 18 \, a^{3} n + 9 \, a^{3}\right )} x^{3}}{3 \, {\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \]
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\[ \int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\int x^{2} \left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.51 \[ \int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {{\left (2 \, n^{2} + 9 \, n + 9\right )} b^{3} x^{3} x^{3 \, n} + 9 \, {\left (n^{2} + 4 \, n + 3\right )} a b^{2} x^{3} x^{2 \, n} + 9 \, {\left (2 \, n^{2} + 5 \, n + 3\right )} a^{2} b x^{3} x^{n} + {\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )} a^{3} x^{3}}{3 \, {\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.38 \[ \int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\frac {2 \, b^{3} n^{2} x^{3} x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 9 \, a b^{2} n^{2} x^{3} x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 18 \, a^{2} b n^{2} x^{3} x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + 2 \, a^{3} n^{3} x^{3} \mathrm {sgn}\left (b x^{n} + a\right ) + 9 \, b^{3} n x^{3} x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 36 \, a b^{2} n x^{3} x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 45 \, a^{2} b n x^{3} x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + 11 \, a^{3} n^{2} x^{3} \mathrm {sgn}\left (b x^{n} + a\right ) + 9 \, b^{3} x^{3} x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 27 \, a b^{2} x^{3} x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 27 \, a^{2} b x^{3} x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + 18 \, a^{3} n x^{3} \mathrm {sgn}\left (b x^{n} + a\right ) + 9 \, a^{3} x^{3} \mathrm {sgn}\left (b x^{n} + a\right )}{3 \, {\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \]
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Timed out. \[ \int x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx=\int x^2\,{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2} \,d x \]
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