Optimal. Leaf size=206 \[ \frac {b^6 x^{3 n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(3 n+1) \left (a b+b^2 x^n\right )^3}+\frac {3 a b^5 x^{2 n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(2 n+1) \left (a b+b^2 x^n\right )^3}+\frac {3 a^2 b^4 x^{n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(n+1) \left (a b+b^2 x^n\right )^3}+\frac {a^3 x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{\left (a+b x^n\right )^3} \]
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Rubi [A] time = 0.05, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1343, 244} \[ \frac {3 a^2 b^4 x^{n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(n+1) \left (a b+b^2 x^n\right )^3}+\frac {3 a b^5 x^{2 n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(2 n+1) \left (a b+b^2 x^n\right )^3}+\frac {b^6 x^{3 n+1} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(3 n+1) \left (a b+b^2 x^n\right )^3}+\frac {a^3 x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{\left (a+b x^n\right )^3} \]
Antiderivative was successfully verified.
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Rule 244
Rule 1343
Rubi steps
\begin {align*} \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx &=\frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \int \left (2 a b+2 b^2 x^n\right )^3 \, dx}{\left (2 a b+2 b^2 x^n\right )^3}\\ &=\frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \int \left (8 a^3 b^3+24 a^2 b^4 x^n+24 a b^5 x^{2 n}+8 b^6 x^{3 n}\right ) \, dx}{\left (2 a b+2 b^2 x^n\right )^3}\\ &=\frac {a^3 x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{\left (a+b x^n\right )^3}+\frac {3 a^2 b^4 x^{1+n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+n) \left (a b+b^2 x^n\right )^3}+\frac {3 a b^5 x^{1+2 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+2 n) \left (a b+b^2 x^n\right )^3}+\frac {b^6 x^{1+3 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+3 n) \left (a b+b^2 x^n\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 122, normalized size = 0.59 \[ \frac {x \sqrt {\left (a+b x^n\right )^2} \left (a^3 \left (6 n^3+11 n^2+6 n+1\right )+3 a^2 b \left (6 n^2+5 n+1\right ) x^n+3 a b^2 \left (3 n^2+4 n+1\right ) x^{2 n}+b^3 \left (2 n^2+3 n+1\right ) x^{3 n}\right )}{(n+1) (2 n+1) (3 n+1) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 130, normalized size = 0.63 \[ \frac {{\left (2 \, b^{3} n^{2} + 3 \, b^{3} n + b^{3}\right )} x x^{3 \, n} + 3 \, {\left (3 \, a b^{2} n^{2} + 4 \, a b^{2} n + a b^{2}\right )} x x^{2 \, n} + 3 \, {\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} + {\left (6 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 6 \, a^{3} n + a^{3}\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 263, normalized size = 1.28 \[ \frac {6 \, a^{3} n^{3} x \mathrm {sgn}\left (b x^{n} + a\right ) + 2 \, b^{3} n^{2} x x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 9 \, a b^{2} n^{2} x x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 18 \, a^{2} b n^{2} x x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + 11 \, a^{3} n^{2} x \mathrm {sgn}\left (b x^{n} + a\right ) + 3 \, b^{3} n x x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 12 \, a b^{2} n x x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 15 \, a^{2} b n x x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + 6 \, a^{3} n x \mathrm {sgn}\left (b x^{n} + a\right ) + b^{3} x x^{3 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 3 \, a b^{2} x x^{2 \, n} \mathrm {sgn}\left (b x^{n} + a\right ) + 3 \, a^{2} b x x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + a^{3} x \mathrm {sgn}\left (b x^{n} + a\right )}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 138, normalized size = 0.67 \[ \frac {3 \sqrt {\left (b \,x^{n}+a \right )^{2}}\, a^{2} b x \,x^{n}}{\left (b \,x^{n}+a \right ) \left (n +1\right )}+\frac {3 \sqrt {\left (b \,x^{n}+a \right )^{2}}\, a \,b^{2} x \,x^{2 n}}{\left (b \,x^{n}+a \right ) \left (2 n +1\right )}+\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, b^{3} x \,x^{3 n}}{\left (b \,x^{n}+a \right ) \left (3 n +1\right )}+\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, a^{3} x}{b \,x^{n}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 101, normalized size = 0.49 \[ \frac {{\left (2 \, n^{2} + 3 \, n + 1\right )} b^{3} x x^{3 \, n} + 3 \, {\left (3 \, n^{2} + 4 \, n + 1\right )} a b^{2} x x^{2 \, n} + 3 \, {\left (6 \, n^{2} + 5 \, n + 1\right )} a^{2} b x x^{n} + {\left (6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1\right )} a^{3} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]
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 \[ \int {\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2} \,d x \]
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 \[ \int \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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