Optimal. Leaf size=218 \[ -\frac {a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )}-\frac {3 a b^3 x^{-2 (1-n)} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 (1-n) \left (a b+b^2 x^n\right )}-\frac {3 a^2 b^2 x^{-2+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac {b^4 x^{-2+3 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(2-3 n) \left (a b+b^2 x^n\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 218, normalized size of antiderivative = 1.00, 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 = {1369, 276}
\begin {gather*} -\frac {3 a^2 b^2 x^{n-2} \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^{3 n-2} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(2-3 n) \left (a b+b^2 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 {a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 1369
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}{x^3} \, 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^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 (\frac {a^3 b^3}{x^3}+3 a^2 b^4 x^{-3+n}+b^6 x^{3 (-1+n)}+3 a b^5 x^{-3+2 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}}}{2 x^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*}
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Mathematica [A]
time = 0.08, size = 124, normalized size = 0.57 \begin {gather*} \frac {\sqrt {\left (a+b x^n\right )^2} \left (a^3 \left (4-12 n+11 n^2-3 n^3\right )+6 a^2 b \left (2-5 n+3 n^2\right ) x^n+3 a b^2 \left (4-8 n+3 n^2\right ) x^{2 n}+2 b^3 \left (2-3 n+n^2\right ) x^{3 n}\right )}{2 (-2+n) (-1+n) (-2+3 n) x^2 \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 145, normalized size = 0.67
method | result | size |
risch | \(-\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{3}}{2 \left (a +b \,x^{n}\right ) x^{2}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{3} x^{3 n}}{\left (a +b \,x^{n}\right ) \left (-2+3 n \right ) x^{2}}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, a \,b^{2} x^{2 n}}{2 \left (a +b \,x^{n}\right ) \left (-1+n \right ) x^{2}}+\frac {3 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{2} b \,x^{n}}{\left (a +b \,x^{n}\right ) \left (-2+n \right ) x^{2}}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 101, normalized size = 0.46 \begin {gather*} \frac {2 \, {\left (n^{2} - 3 \, n + 2\right )} b^{3} x^{3 \, n} + 3 \, {\left (3 \, n^{2} - 8 \, n + 4\right )} a b^{2} x^{2 \, n} + 6 \, {\left (3 \, n^{2} - 5 \, n + 2\right )} a^{2} b x^{n} - {\left (3 \, n^{3} - 11 \, n^{2} + 12 \, n - 4\right )} a^{3}}{2 \, {\left (3 \, n^{3} - 11 \, n^{2} + 12 \, n - 4\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 134, normalized size = 0.61 \begin {gather*} -\frac {3 \, a^{3} n^{3} - 11 \, a^{3} n^{2} + 12 \, a^{3} n - 4 \, a^{3} - 2 \, {\left (b^{3} n^{2} - 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3 \, n} - 3 \, {\left (3 \, a b^{2} n^{2} - 8 \, a b^{2} n + 4 \, a b^{2}\right )} x^{2 \, n} - 6 \, {\left (3 \, a^{2} b n^{2} - 5 \, a^{2} b n + 2 \, a^{2} b\right )} x^{n}}{2 \, {\left (3 \, n^{3} - 11 \, n^{2} + 12 \, n - 4\right )} x^{2}} \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 \frac {\left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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