Optimal. Leaf size=52 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{3 (m+1)}}{3 (m+1)}+\frac{b^2 x^{5 (m+1)}}{5 (m+1)} \]
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Rubi [A] time = 0.021432, antiderivative size = 52, 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} \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{3 (m+1)}}{3 (m+1)}+\frac{b^2 x^{5 (m+1)}}{5 (m+1)} \]
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 )^2 \, dx &=\int \left (a^2 x^m+2 a b x^{2+3 m}+b^2 x^{4+5 m}\right ) \, dx\\ &=\frac{a^2 x^{1+m}}{1+m}+\frac{2 a b x^{3 (1+m)}}{3 (1+m)}+\frac{b^2 x^{5 (1+m)}}{5 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0238763, size = 42, normalized size = 0.81 \[ \frac{15 a^2 x^{m+1}+10 a b x^{3 m+3}+3 b^2 x^{5 m+5}}{15 m+15} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 50, normalized size = 1. \begin{align*}{\frac{{b}^{2}{x}^{5} \left ({x}^{m} \right ) ^{5}}{5+5\,m}}+{\frac{2\,{x}^{3}ab \left ({x}^{m} \right ) ^{3}}{3+3\,m}}+{\frac{{a}^{2}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 = 1.32061, size = 95, normalized size = 1.83 \begin{align*} \frac{3 \, b^{2} x^{5} x^{5 \, m} + 10 \, a b x^{3} x^{3 \, m} + 15 \, a^{2} x x^{m}}{15 \,{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 140.369, size = 61, normalized size = 1.17 \begin{align*} \begin{cases} \frac{15 a^{2} x x^{m}}{15 m + 15} + \frac{10 a b x^{3} x^{3 m}}{15 m + 15} + \frac{3 b^{2} x^{5} x^{5 m}}{15 m + 15} & \text{for}\: m \neq -1 \\\left (a + b\right )^{2} \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 = 1.12214, size = 57, normalized size = 1.1 \begin{align*} \frac{3 \, b^{2} x^{5} x^{5 \, m} + 10 \, a b x^{3} x^{3 \, m} + 15 \, a^{2} x x^{m}}{15 \,{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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