Optimal. Leaf size=206 \[ \frac {a^3 x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{\left (a+b x^n\right )^3}+\frac {3 a^2 b^4 x^{1+n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+n) \left (a b+b^2 x^n\right )^3}+\frac {3 a b^5 x^{1+2 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+2 n) \left (a b+b^2 x^n\right )^3}+\frac {b^6 x^{1+3 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{(1+3 n) \left (a b+b^2 x^n\right )^3} \]
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Rubi [A]
time = 0.04, 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 = {1357, 250}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 250
Rule 1357
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 = 0.07, size = 122, normalized size = 0.59 \begin {gather*} \frac {x \sqrt {\left (a+b x^n\right )^2} \left (a^3 \left (1+6 n+11 n^2+6 n^3\right )+3 a^2 b \left (1+5 n+6 n^2\right ) x^n+3 a b^2 \left (1+4 n+3 n^2\right ) x^{2 n}+b^3 \left (1+3 n+2 n^2\right ) x^{3 n}\right )}{(1+n) (1+2 n) (1+3 n) \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 138, normalized size = 0.67
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{3} x}{a +b \,x^{n}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{3} x \,x^{3 n}}{\left (a +b \,x^{n}\right ) \left (1+3 n \right )}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, a \,b^{2} x \,x^{2 n}}{\left (a +b \,x^{n}\right ) \left (1+2 n \right )}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{2} b x \,x^{n}}{\left (a +b \,x^{n}\right ) \left (1+n \right )}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 101, normalized size = 0.49 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 130, normalized size = 0.63 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \int \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.47, size = 263, normalized size = 1.28 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \int {\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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