Optimal. Leaf size=58 \[ \frac {a^2 (c x)^{m+1}}{c (m+1)}+\frac {2 a b (c x)^{m+2}}{c^2 (m+2)}+\frac {b^2 (c x)^{m+3}}{c^3 (m+3)} \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ \frac {a^2 (c x)^{m+1}}{c (m+1)}+\frac {2 a b (c x)^{m+2}}{c^2 (m+2)}+\frac {b^2 (c x)^{m+3}}{c^3 (m+3)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (c x)^m (a+b x)^2 \, dx &=\int \left (a^2 (c x)^m+\frac {2 a b (c x)^{1+m}}{c}+\frac {b^2 (c x)^{2+m}}{c^2}\right ) \, dx\\ &=\frac {a^2 (c x)^{1+m}}{c (1+m)}+\frac {2 a b (c x)^{2+m}}{c^2 (2+m)}+\frac {b^2 (c x)^{3+m}}{c^3 (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 39, normalized size = 0.67 \[ x (c x)^m \left (\frac {a^2}{m+1}+\frac {2 a b x}{m+2}+\frac {b^2 x^2}{m+3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 87, normalized size = 1.50 \[ \frac {{\left ({\left (b^{2} m^{2} + 3 \, b^{2} m + 2 \, b^{2}\right )} x^{3} + 2 \, {\left (a b m^{2} + 4 \, a b m + 3 \, a b\right )} x^{2} + {\left (a^{2} m^{2} + 5 \, a^{2} m + 6 \, a^{2}\right )} x\right )} \left (c x\right )^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 135, normalized size = 2.33 \[ \frac {\left (c x\right )^{m} b^{2} m^{2} x^{3} + 2 \, \left (c x\right )^{m} a b m^{2} x^{2} + 3 \, \left (c x\right )^{m} b^{2} m x^{3} + \left (c x\right )^{m} a^{2} m^{2} x + 8 \, \left (c x\right )^{m} a b m x^{2} + 2 \, \left (c x\right )^{m} b^{2} x^{3} + 5 \, \left (c x\right )^{m} a^{2} m x + 6 \, \left (c x\right )^{m} a b x^{2} + 6 \, \left (c x\right )^{m} a^{2} x}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 88, normalized size = 1.52 \[ \frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +3 b^{2} m \,x^{2}+a^{2} m^{2}+8 a b m x +2 b^{2} x^{2}+5 a^{2} m +6 a b x +6 a^{2}\right ) x \left (c x \right )^{m}}{\left (m +3\right ) \left (m +2\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 56, normalized size = 0.97 \[ \frac {b^{2} c^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, a b c^{m} x^{2} x^{m}}{m + 2} + \frac {\left (c x\right )^{m + 1} a^{2}}{c {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 95, normalized size = 1.64 \[ {\left (c\,x\right )}^m\,\left (\frac {a^2\,x\,\left (m^2+5\,m+6\right )}{m^3+6\,m^2+11\,m+6}+\frac {b^2\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {2\,a\,b\,x^2\,\left (m^2+4\,m+3\right )}{m^3+6\,m^2+11\,m+6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 338, normalized size = 5.83 \[ \begin {cases} \frac {- \frac {a^{2}}{2 x^{2}} - \frac {2 a b}{x} + b^{2} \log {\relax (x )}}{c^{3}} & \text {for}\: m = -3 \\\frac {- \frac {a^{2}}{x} + 2 a b \log {\relax (x )} + b^{2} x}{c^{2}} & \text {for}\: m = -2 \\\frac {a^{2} \log {\relax (x )} + 2 a b x + \frac {b^{2} x^{2}}{2}}{c} & \text {for}\: m = -1 \\\frac {a^{2} c^{m} m^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {5 a^{2} c^{m} m x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {6 a^{2} c^{m} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {2 a b c^{m} m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {8 a b c^{m} m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {6 a b c^{m} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {b^{2} c^{m} m^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {3 b^{2} c^{m} m x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {2 b^{2} c^{m} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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