Optimal. Leaf size=93 \[ -\frac{a^2 b c^2 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^3 c^2 (e x)^{m+1}}{e (m+1)}-\frac{a b^2 c^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{b^3 c^2 (e x)^{m+4}}{e^4 (m+4)} \]
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Rubi [A] time = 0.0468879, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {75} \[ -\frac{a^2 b c^2 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^3 c^2 (e x)^{m+1}}{e (m+1)}-\frac{a b^2 c^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{b^3 c^2 (e x)^{m+4}}{e^4 (m+4)} \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin{align*} \int (e x)^m (a+b x) (a c-b c x)^2 \, dx &=\int \left (a^3 c^2 (e x)^m-\frac{a^2 b c^2 (e x)^{1+m}}{e}-\frac{a b^2 c^2 (e x)^{2+m}}{e^2}+\frac{b^3 c^2 (e x)^{3+m}}{e^3}\right ) \, dx\\ &=\frac{a^3 c^2 (e x)^{1+m}}{e (1+m)}-\frac{a^2 b c^2 (e x)^{2+m}}{e^2 (2+m)}-\frac{a b^2 c^2 (e x)^{3+m}}{e^3 (3+m)}+\frac{b^3 c^2 (e x)^{4+m}}{e^4 (4+m)}\\ \end{align*}
Mathematica [A] time = 0.06659, size = 88, normalized size = 0.95 \[ \frac{c^2 x (e x)^m \left (\frac{a (2 m+5) \left (a^2 \left (m^2+5 m+6\right )-2 a b \left (m^2+4 m+3\right ) x+b^2 \left (m^2+3 m+2\right ) x^2\right )}{(m+1) (m+2) (m+3)}+(b x-a)^3\right )}{m+4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 174, normalized size = 1.9 \begin{align*}{\frac{{c}^{2} \left ( ex \right ) ^{m} \left ({b}^{3}{m}^{3}{x}^{3}-a{b}^{2}{m}^{3}{x}^{2}+6\,{b}^{3}{m}^{2}{x}^{3}-{a}^{2}b{m}^{3}x-7\,a{b}^{2}{m}^{2}{x}^{2}+11\,{b}^{3}m{x}^{3}+{a}^{3}{m}^{3}-8\,{a}^{2}b{m}^{2}x-14\,a{b}^{2}m{x}^{2}+6\,{b}^{3}{x}^{3}+9\,{a}^{3}{m}^{2}-19\,{a}^{2}bmx-8\,a{b}^{2}{x}^{2}+26\,{a}^{3}m-12\,{a}^{2}bx+24\,{a}^{3} \right ) x}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \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 [B] time = 1.81561, size = 423, normalized size = 4.55 \begin{align*} \frac{{\left ({\left (b^{3} c^{2} m^{3} + 6 \, b^{3} c^{2} m^{2} + 11 \, b^{3} c^{2} m + 6 \, b^{3} c^{2}\right )} x^{4} -{\left (a b^{2} c^{2} m^{3} + 7 \, a b^{2} c^{2} m^{2} + 14 \, a b^{2} c^{2} m + 8 \, a b^{2} c^{2}\right )} x^{3} -{\left (a^{2} b c^{2} m^{3} + 8 \, a^{2} b c^{2} m^{2} + 19 \, a^{2} b c^{2} m + 12 \, a^{2} b c^{2}\right )} x^{2} +{\left (a^{3} c^{2} m^{3} + 9 \, a^{3} c^{2} m^{2} + 26 \, a^{3} c^{2} m + 24 \, a^{3} c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.14576, size = 821, normalized size = 8.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19902, size = 410, normalized size = 4.41 \begin{align*} \frac{b^{3} c^{2} m^{3} x^{4} x^{m} e^{m} - a b^{2} c^{2} m^{3} x^{3} x^{m} e^{m} + 6 \, b^{3} c^{2} m^{2} x^{4} x^{m} e^{m} - a^{2} b c^{2} m^{3} x^{2} x^{m} e^{m} - 7 \, a b^{2} c^{2} m^{2} x^{3} x^{m} e^{m} + 11 \, b^{3} c^{2} m x^{4} x^{m} e^{m} + a^{3} c^{2} m^{3} x x^{m} e^{m} - 8 \, a^{2} b c^{2} m^{2} x^{2} x^{m} e^{m} - 14 \, a b^{2} c^{2} m x^{3} x^{m} e^{m} + 6 \, b^{3} c^{2} x^{4} x^{m} e^{m} + 9 \, a^{3} c^{2} m^{2} x x^{m} e^{m} - 19 \, a^{2} b c^{2} m x^{2} x^{m} e^{m} - 8 \, a b^{2} c^{2} x^{3} x^{m} e^{m} + 26 \, a^{3} c^{2} m x x^{m} e^{m} - 12 \, a^{2} b c^{2} x^{2} x^{m} e^{m} + 24 \, a^{3} c^{2} x x^{m} e^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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