Optimal. Leaf size=20 \[ \frac {x^{1-3 n}}{(a+b)^3 (1-3 n)} \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6, 12, 30}
\begin {gather*} \frac {x^{1-3 n}}{(1-3 n) (a+b)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 30
Rubi steps
\begin {align*} \int \frac {1}{\left (a x^n+b x^n\right )^3} \, dx &=\int \frac {x^{-3 n}}{(a+b)^3} \, dx\\ &=\frac {\int x^{-3 n} \, dx}{(a+b)^3}\\ &=\frac {x^{1-3 n}}{(a+b)^3 (1-3 n)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{1-3 n}}{(a+b)^3 (1-3 n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 1.05
method | result | size |
gosper | \(-\frac {x \,x^{-3 n}}{\left (-1+3 n \right ) \left (a +b \right )^{3}}\) | \(21\) |
norman | \(-\frac {x \,{\mathrm e}^{-3 n \ln \left (x \right )}}{\left (3 a n +3 b n -a -b \right ) \left (a +b \right )^{2}}\) | \(33\) |
risch | \(-\frac {x \,x^{-3 n}}{\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (-1+3 n \right )}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (21) = 42\).
time = 0.30, size = 53, normalized size = 2.65 \begin {gather*} -\frac {x}{{\left (a^{3} {\left (3 \, n - 1\right )} + 3 \, a^{2} b {\left (3 \, n - 1\right )} + 3 \, a b^{2} {\left (3 \, n - 1\right )} + b^{3} {\left (3 \, n - 1\right )}\right )} x^{3 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs.
\(2 (21) = 42\).
time = 3.24, size = 52, normalized size = 2.60 \begin {gather*} \frac {x}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} n\right )} x^{3 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (15) = 30\).
time = 0.46, size = 119, normalized size = 5.95 \begin {gather*} \begin {cases} - \frac {x}{3 a^{3} n x^{3 n} - a^{3} x^{3 n} + 9 a^{2} b n x^{3 n} - 3 a^{2} b x^{3 n} + 9 a b^{2} n x^{3 n} - 3 a b^{2} x^{3 n} + 3 b^{3} n x^{3 n} - b^{3} x^{3 n}} & \text {for}\: n \neq \frac {1}{3} \\\frac {\log {\left (x \right )}}{a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}} & \text {otherwise} \end {cases} \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 [B]
time = 5.12, size = 21, normalized size = 1.05 \begin {gather*} -\frac {x^{1-3\,n}}{{\left (a+b\right )}^3\,\left (3\,n-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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