Optimal. Leaf size=71 \[ \frac{a^2 b x^{3 (m+1)}}{m+1}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{5 (m+1)}}{5 (m+1)}+\frac{b^3 x^{7 (m+1)}}{7 (m+1)} \]
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Rubi [A] time = 0.0361272, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {270} \[ \frac{a^2 b x^{3 (m+1)}}{m+1}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{5 (m+1)}}{5 (m+1)}+\frac{b^3 x^{7 (m+1)}}{7 (m+1)} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin{align*} \int x^m \left (a+b x^{2+2 m}\right )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{2+3 m}+3 a b^2 x^{4+5 m}+b^3 x^{6+7 m}\right ) \, dx\\ &=\frac{a^3 x^{1+m}}{1+m}+\frac{a^2 b x^{3 (1+m)}}{1+m}+\frac{3 a b^2 x^{5 (1+m)}}{5 (1+m)}+\frac{b^3 x^{7 (1+m)}}{7 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0450491, size = 57, normalized size = 0.8 \[ \frac{35 a^2 b x^{3 m+3}+35 a^3 x^{m+1}+21 a b^2 x^{5 m+5}+5 b^3 x^{7 m+7}}{35 m+35} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 70, normalized size = 1. \begin{align*}{\frac{{b}^{3}{x}^{7} \left ({x}^{m} \right ) ^{7}}{7+7\,m}}+{\frac{3\,{x}^{5}a{b}^{2} \left ({x}^{m} \right ) ^{5}}{5+5\,m}}+{\frac{{a}^{2}b{x}^{3} \left ({x}^{m} \right ) ^{3}}{1+m}}+{\frac{{a}^{3}x{x}^{m}}{1+m}} \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 = 1.3323, size = 128, normalized size = 1.8 \begin{align*} \frac{5 \, b^{3} x^{7} x^{7 \, m} + 21 \, a b^{2} x^{5} x^{5 \, m} + 35 \, a^{2} b x^{3} x^{3 \, m} + 35 \, a^{3} x x^{m}}{35 \,{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13911, size = 78, normalized size = 1.1 \begin{align*} \frac{5 \, b^{3} x^{7} x^{7 \, m} + 21 \, a b^{2} x^{5} x^{5 \, m} + 35 \, a^{2} b x^{3} x^{3 \, m} + 35 \, a^{3} x x^{m}}{35 \,{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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