Optimal. Leaf size=42 \[ \frac {a^2 c (e x)^{1+m}}{e (1+m)}-\frac {b^2 c (e x)^{3+m}}{e^3 (3+m)} \]
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Rubi [A]
time = 0.01, 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 = {74, 14}
\begin {gather*} \frac {a^2 c (e x)^{m+1}}{e (m+1)}-\frac {b^2 c (e x)^{m+3}}{e^3 (m+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 74
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 = 0.04, size = 31, normalized size = 0.74 \begin {gather*} c x (e x)^m \left (\frac {a^2}{1+m}-\frac {b^2 x^2}{3+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 41, normalized size = 0.98
method | result | size |
norman | \(\frac {a^{2} c x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}-\frac {b^{2} c \,x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}\) | \(41\) |
gosper | \(\frac {c \left (e x \right )^{m} \left (-b^{2} m \,x^{2}-b^{2} x^{2}+a^{2} m +3 a^{2}\right ) x}{\left (3+m \right ) \left (1+m \right )}\) | \(47\) |
risch | \(\frac {c \left (e x \right )^{m} \left (-b^{2} m \,x^{2}-b^{2} x^{2}+a^{2} m +3 a^{2}\right ) x}{\left (3+m \right ) \left (1+m \right )}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 42, normalized size = 1.00 \begin {gather*} -\frac {b^{2} c x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} + \frac {\left (x e\right )^{m + 1} a^{2} c e^{\left (-1\right )}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 51, normalized size = 1.21 \begin {gather*} -\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 (x e\right )^{m}}{m^{2} + 4 \, m + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs.
\(2 (34) = 68\).
time = 0.14, size = 134, normalized size = 3.19 \begin {gather*} \begin {cases} \frac {- \frac {a^{2} c}{2 x^{2}} - b^{2} c \log {\left (x \right )}}{e^{3}} & \text {for}\: m = -3 \\\frac {a^{2} c \log {\left (x \right )} - \frac {b^{2} c x^{2}}{2}}{e} & \text {for}\: m = -1 \\\frac {a^{2} c m x \left (e x\right )^{m}}{m^{2} + 4 m + 3} + \frac {3 a^{2} c x \left (e x\right )^{m}}{m^{2} + 4 m + 3} - \frac {b^{2} c m x^{3} \left (e x\right )^{m}}{m^{2} + 4 m + 3} - \frac {b^{2} c x^{3} \left (e x\right )^{m}}{m^{2} + 4 m + 3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.41, size = 65, normalized size = 1.55 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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