Optimal. Leaf size=211 \[ \frac{3 a b^3 x^{2 (n+1)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (n+1) \left (a b+b^2 x^n\right )}+\frac{3 a^2 b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(3 n+2) \left (a b+b^2 x^n\right )}+\frac{a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )} \]
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Rubi [A] time = 0.0581511, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1355, 270} \[ \frac{3 a b^3 x^{2 (n+1)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (n+1) \left (a b+b^2 x^n\right )}+\frac{3 a^2 b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(3 n+2) \left (a b+b^2 x^n\right )}+\frac{a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 270
Rubi steps
\begin{align*} \int x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int x \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+3 a^2 b^4 x^{1+n}+3 a b^5 x^{1+2 n}+b^6 x^{1+3 n}\right ) \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=\frac{a^3 x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )}+\frac{3 a b^3 x^{2 (1+n)} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 (1+n) \left (a b+b^2 x^n\right )}+\frac{3 a^2 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 )}+\frac{b^4 x^{2+3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2+3 n) \left (a b+b^2 x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.0726074, size = 124, normalized size = 0.59 \[ \frac{x^2 \sqrt{\left (a+b x^n\right )^2} \left (6 a^2 b \left (3 n^2+5 n+2\right ) x^n+a^3 \left (3 n^3+11 n^2+12 n+4\right )+3 a b^2 \left (3 n^2+8 n+4\right ) x^{2 n}+2 b^3 \left (n^2+3 n+2\right ) x^{3 n}\right )}{2 (n+1) (n+2) (3 n+2) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 145, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}{a}^{3}}{2\,a+2\,b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{b}^{3}{x}^{2} \left ({x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n} \right ) \left ( 2+3\,n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{3\,a{b}^{2}{x}^{2} \left ({x}^{n} \right ) ^{2}}{ \left ( 2\,a+2\,b{x}^{n} \right ) \left ( 1+n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{a}^{2}b{x}^{2}{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( 2+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00437, size = 147, normalized size = 0.7 \begin{align*} \frac{2 \,{\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{2} x^{3 \, n} + 3 \,{\left (3 \, n^{2} + 8 \, n + 4\right )} a b^{2} x^{2} x^{2 \, n} + 6 \,{\left (3 \, n^{2} + 5 \, n + 2\right )} a^{2} b x^{2} x^{n} +{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )} a^{3} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57025, size = 306, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{2} x^{3 \, n} + 3 \,{\left (3 \, a b^{2} n^{2} + 8 \, a b^{2} n + 4 \, a b^{2}\right )} x^{2} x^{2 \, n} + 6 \,{\left (3 \, a^{2} b n^{2} + 5 \, a^{2} b n + 2 \, a^{2} b\right )} x^{2} x^{n} +{\left (3 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 12 \, a^{3} n + 4 \, a^{3}\right )} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14155, size = 394, normalized size = 1.87 \begin{align*} \frac{2 \, b^{3} n^{2} x^{2} x^{3 \, n} \mathrm{sgn}\left (b x^{n} + a\right ) + 9 \, a b^{2} n^{2} x^{2} x^{2 \, n} \mathrm{sgn}\left (b x^{n} + a\right ) + 18 \, a^{2} b n^{2} x^{2} x^{n} \mathrm{sgn}\left (b x^{n} + a\right ) + 3 \, a^{3} n^{3} x^{2} \mathrm{sgn}\left (b x^{n} + a\right ) + 6 \, b^{3} n x^{2} x^{3 \, n} \mathrm{sgn}\left (b x^{n} + a\right ) + 24 \, a b^{2} n x^{2} x^{2 \, n} \mathrm{sgn}\left (b x^{n} + a\right ) + 30 \, a^{2} b n x^{2} x^{n} \mathrm{sgn}\left (b x^{n} + a\right ) + 11 \, a^{3} n^{2} x^{2} \mathrm{sgn}\left (b x^{n} + a\right ) + 4 \, b^{3} x^{2} x^{3 \, n} \mathrm{sgn}\left (b x^{n} + a\right ) + 12 \, a b^{2} x^{2} x^{2 \, n} \mathrm{sgn}\left (b x^{n} + a\right ) + 12 \, a^{2} b x^{2} x^{n} \mathrm{sgn}\left (b x^{n} + a\right ) + 12 \, a^{3} n x^{2} \mathrm{sgn}\left (b x^{n} + a\right ) + 4 \, a^{3} x^{2} \mathrm{sgn}\left (b x^{n} + a\right )}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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