Optimal. Leaf size=20 \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]
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Rubi [A] time = 0.0089884, antiderivative size = 20, normalized size of antiderivative = 1., 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} \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]
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*}
Mathematica [A] time = 0.0032928, size = 20, normalized size = 1. \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 1.1 \begin{align*} -{\frac{x}{ \left ( -1+3\,n \right ) \left ({x}^{n} \right ) ^{3} \left ( a+b \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01413, size = 72, normalized size = 3.6 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.918571, size = 112, normalized size = 5.6 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.80554, size = 119, normalized size = 5.95 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x^{n} + b x^{n}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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