Optimal. Leaf size=63 \[ \frac{3 a^2 b x^{5 n}}{5 n}+\frac{a^3 x^{4 n}}{4 n}+\frac{a b^2 x^{6 n}}{2 n}+\frac{b^3 x^{7 n}}{7 n} \]
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Rubi [A] time = 0.0271848, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac{3 a^2 b x^{5 n}}{5 n}+\frac{a^3 x^{4 n}}{4 n}+\frac{a b^2 x^{6 n}}{2 n}+\frac{b^3 x^{7 n}}{7 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 (a+b x)^3 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3 x^3+3 a^2 b x^4+3 a b^2 x^5+b^3 x^6\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a^3 x^{4 n}}{4 n}+\frac{3 a^2 b x^{5 n}}{5 n}+\frac{a b^2 x^{6 n}}{2 n}+\frac{b^3 x^{7 n}}{7 n}\\ \end{align*}
Mathematica [A] time = 0.0213119, size = 48, normalized size = 0.76 \[ \frac{x^{4 n} \left (84 a^2 b x^n+35 a^3+70 a b^2 x^{2 n}+20 b^3 x^{3 n}\right )}{140 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 56, normalized size = 0.9 \begin{align*}{\frac{{b}^{3} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{{b}^{2}a \left ({x}^{n} \right ) ^{6}}{2\,n}}+{\frac{3\,b{a}^{2} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{{a}^{3} \left ({x}^{n} \right ) ^{4}}{4\,n}} \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 = 1.27533, size = 109, normalized size = 1.73 \begin{align*} \frac{20 \, b^{3} x^{7 \, n} + 70 \, a b^{2} x^{6 \, n} + 84 \, a^{2} b x^{5 \, n} + 35 \, a^{3} x^{4 \, n}}{140 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 29.7732, size = 60, normalized size = 0.95 \begin{align*} \begin{cases} \frac{a^{3} x^{4 n}}{4 n} + \frac{3 a^{2} b x^{5 n}}{5 n} + \frac{a b^{2} x^{6 n}}{2 n} + \frac{b^{3} x^{7 n}}{7 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{3} x^{4 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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