Optimal. Leaf size=24 \[ -\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n} \]
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Rubi [A] time = 0.0049612, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int x^{-1-4 n} \left (a+b x^n\right )^3 \, dx &=-\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n}\\ \end{align*}
Mathematica [A] time = 0.0066993, size = 24, normalized size = 1. \[ -\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 63, normalized size = 2.6 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ( -{\frac{{a}^{3}}{4\,n}}-{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-{\frac{b{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{n}}-{\frac{3\,{b}^{2}a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33464, size = 97, normalized size = 4.04 \begin{align*} -\frac{4 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} + a^{3}}{4 \, n x^{4 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 29.6429, size = 56, normalized size = 2.33 \begin{align*} \begin{cases} - \frac{a^{3} x^{- 4 n}}{4 n} - \frac{a^{2} b x^{- 3 n}}{n} - \frac{3 a b^{2} x^{- 2 n}}{2 n} - \frac{b^{3} x^{- n}}{n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14749, size = 62, normalized size = 2.58 \begin{align*} -\frac{4 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} + a^{3}}{4 \, n x^{4 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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