Optimal. Leaf size=42 \[ \frac {a^2 c (e x)^{m+1}}{e (m+1)}-\frac {b^2 c (e x)^{m+3}}{e^3 (m+3)} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {73, 14} \[ \frac {a^2 c (e x)^{m+1}}{e (m+1)}-\frac {b^2 c (e x)^{m+3}}{e^3 (m+3)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 73
Rubi steps
\begin {align*} \int (e x)^m (a+b x) (a c-b c x) \, dx &=\int (e x)^m \left (a^2 c-b^2 c x^2\right ) \, dx\\ &=\int \left (a^2 c (e x)^m-\frac {b^2 c (e x)^{2+m}}{e^2}\right ) \, dx\\ &=\frac {a^2 c (e x)^{1+m}}{e (1+m)}-\frac {b^2 c (e x)^{3+m}}{e^3 (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 0.74 \[ c x (e x)^m \left (\frac {a^2}{m+1}-\frac {b^2 x^2}{m+3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 50, normalized size = 1.19 \[ -\frac {{\left ({\left (b^{2} c m + b^{2} c\right )} x^{3} - {\left (a^{2} c m + 3 \, a^{2} c\right )} x\right )} \left (e x\right )^{m}}{m^{2} + 4 \, m + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 65, normalized size = 1.55 \[ -\frac {b^{2} c m x^{3} x^{m} e^{m} + b^{2} c x^{3} x^{m} e^{m} - a^{2} c m x x^{m} e^{m} - 3 \, a^{2} c x x^{m} e^{m}}{m^{2} + 4 \, m + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 47, normalized size = 1.12 \[ \frac {\left (-b^{2} m \,x^{2}-b^{2} x^{2}+a^{2} m +3 a^{2}\right ) c x \left (e x \right )^{m}}{\left (m +3\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 41, normalized size = 0.98 \[ -\frac {b^{2} c e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} a^{2} c}{e {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 46, normalized size = 1.10 \[ \frac {c\,x\,{\left (e\,x\right )}^m\,\left (a^2\,m+3\,a^2-b^2\,x^2-b^2\,m\,x^2\right )}{m^2+4\,m+3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 141, normalized size = 3.36 \[ \begin {cases} \frac {- \frac {a^{2} c}{2 x^{2}} - b^{2} c \log {\relax (x )}}{e^{3}} & \text {for}\: m = -3 \\\frac {a^{2} c \log {\relax (x )} - \frac {b^{2} c x^{2}}{2}}{e} & \text {for}\: m = -1 \\\frac {a^{2} c e^{m} m x x^{m}}{m^{2} + 4 m + 3} + \frac {3 a^{2} c e^{m} x x^{m}}{m^{2} + 4 m + 3} - \frac {b^{2} c e^{m} m x^{3} x^{m}}{m^{2} + 4 m + 3} - \frac {b^{2} c e^{m} x^{3} x^{m}}{m^{2} + 4 m + 3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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