Optimal. Leaf size=58 \[ \frac{b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac{c^2 (d x)^{m+5}}{d^5 (m+5)} \]
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Rubi [A] time = 0.0408282, antiderivative size = 58, 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 = {647, 43} \[ \frac{b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac{c^2 (d x)^{m+5}}{d^5 (m+5)} \]
Antiderivative was successfully verified.
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Rule 647
Rule 43
Rubi steps
\begin{align*} \int (d x)^m \left (b x+c x^2\right )^2 \, dx &=\frac{\int (d x)^{2+m} (b+c x)^2 \, dx}{d^2}\\ &=\frac{\int \left (b^2 (d x)^{2+m}+\frac{2 b c (d x)^{3+m}}{d}+\frac{c^2 (d x)^{4+m}}{d^2}\right ) \, dx}{d^2}\\ &=\frac{b^2 (d x)^{3+m}}{d^3 (3+m)}+\frac{2 b c (d x)^{4+m}}{d^4 (4+m)}+\frac{c^2 (d x)^{5+m}}{d^5 (5+m)}\\ \end{align*}
Mathematica [A] time = 0.0306284, size = 41, normalized size = 0.71 \[ x^3 (d x)^m \left (\frac{b^2}{m+3}+\frac{2 b c x}{m+4}+\frac{c^2 x^2}{m+5}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 90, normalized size = 1.6 \begin{align*}{\frac{ \left ( dx \right ) ^{m} \left ({c}^{2}{m}^{2}{x}^{2}+2\,bc{m}^{2}x+7\,{c}^{2}m{x}^{2}+{b}^{2}{m}^{2}+16\,bcmx+12\,{c}^{2}{x}^{2}+9\,{b}^{2}m+30\,bcx+20\,{b}^{2} \right ){x}^{3}}{ \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16304, size = 74, normalized size = 1.28 \begin{align*} \frac{c^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac{2 \, b c d^{m} x^{4} x^{m}}{m + 4} + \frac{b^{2} d^{m} x^{3} x^{m}}{m + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1322, size = 193, normalized size = 3.33 \begin{align*} \frac{{\left ({\left (c^{2} m^{2} + 7 \, c^{2} m + 12 \, c^{2}\right )} x^{5} + 2 \,{\left (b c m^{2} + 8 \, b c m + 15 \, b c\right )} x^{4} +{\left (b^{2} m^{2} + 9 \, b^{2} m + 20 \, b^{2}\right )} x^{3}\right )} \left (d x\right )^{m}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.60889, size = 345, normalized size = 5.95 \begin{align*} \begin{cases} \frac{- \frac{b^{2}}{2 x^{2}} - \frac{2 b c}{x} + c^{2} \log{\left (x \right )}}{d^{5}} & \text{for}\: m = -5 \\\frac{- \frac{b^{2}}{x} + 2 b c \log{\left (x \right )} + c^{2} x}{d^{4}} & \text{for}\: m = -4 \\\frac{b^{2} \log{\left (x \right )} + 2 b c x + \frac{c^{2} x^{2}}{2}}{d^{3}} & \text{for}\: m = -3 \\\frac{b^{2} d^{m} m^{2} x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{9 b^{2} d^{m} m x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{20 b^{2} d^{m} x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{2 b c d^{m} m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{16 b c d^{m} m x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{30 b c d^{m} x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{c^{2} d^{m} m^{2} x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{7 c^{2} d^{m} m x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{12 c^{2} d^{m} x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29118, size = 190, normalized size = 3.28 \begin{align*} \frac{\left (d x\right )^{m} c^{2} m^{2} x^{5} + 2 \, \left (d x\right )^{m} b c m^{2} x^{4} + 7 \, \left (d x\right )^{m} c^{2} m x^{5} + \left (d x\right )^{m} b^{2} m^{2} x^{3} + 16 \, \left (d x\right )^{m} b c m x^{4} + 12 \, \left (d x\right )^{m} c^{2} x^{5} + 9 \, \left (d x\right )^{m} b^{2} m x^{3} + 30 \, \left (d x\right )^{m} b c x^{4} + 20 \, \left (d x\right )^{m} b^{2} x^{3}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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