Optimal. Leaf size=211 \[ -\frac{2 a b (e x)^{m+2} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{e^2 (m+2)}+\frac{2 a^2 (m+n+2) (e x)^{m+1} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{e (m+1) (m+2 n+3)}-\frac{(e x)^{m+1} (a-b x)^{n+1} (a+b x)^{n+1}}{e (m+2 n+3)} \]
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Rubi [A] time = 0.173635, antiderivative size = 238, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {127, 126, 365, 364} \[ -\frac{2 a b (e x)^{m+2} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{e^2 (m+2)}+\frac{b^2 (e x)^{m+3} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+3}{2},-n;\frac{m+5}{2};\frac{b^2 x^2}{a^2}\right )}{e^3 (m+3)}+\frac{a^2 (e x)^{m+1} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 127
Rule 126
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (e x)^m (a-b x)^{2+n} (a+b x)^n \, dx &=\int \left (a^2 (e x)^m (a-b x)^n (a+b x)^n-\frac{2 a b (e x)^{1+m} (a-b x)^n (a+b x)^n}{e}+\frac{b^2 (e x)^{2+m} (a-b x)^n (a+b x)^n}{e^2}\right ) \, dx\\ &=a^2 \int (e x)^m (a-b x)^n (a+b x)^n \, dx+\frac{b^2 \int (e x)^{2+m} (a-b x)^n (a+b x)^n \, dx}{e^2}-\frac{(2 a b) \int (e x)^{1+m} (a-b x)^n (a+b x)^n \, dx}{e}\\ &=\left (a^2 (a-b x)^n (a+b x)^n \left (a^2-b^2 x^2\right )^{-n}\right ) \int (e x)^m \left (a^2-b^2 x^2\right )^n \, dx+\frac{\left (b^2 (a-b x)^n (a+b x)^n \left (a^2-b^2 x^2\right )^{-n}\right ) \int (e x)^{2+m} \left (a^2-b^2 x^2\right )^n \, dx}{e^2}-\frac{\left (2 a b (a-b x)^n (a+b x)^n \left (a^2-b^2 x^2\right )^{-n}\right ) \int (e x)^{1+m} \left (a^2-b^2 x^2\right )^n \, dx}{e}\\ &=\left (a^2 (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n}\right ) \int (e x)^m \left (1-\frac{b^2 x^2}{a^2}\right )^n \, dx+\frac{\left (b^2 (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n}\right ) \int (e x)^{2+m} \left (1-\frac{b^2 x^2}{a^2}\right )^n \, dx}{e^2}-\frac{\left (2 a b (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n}\right ) \int (e x)^{1+m} \left (1-\frac{b^2 x^2}{a^2}\right )^n \, dx}{e}\\ &=\frac{a^2 (e x)^{1+m} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{1+m}{2},-n;\frac{3+m}{2};\frac{b^2 x^2}{a^2}\right )}{e (1+m)}-\frac{2 a b (e x)^{2+m} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{2+m}{2},-n;\frac{4+m}{2};\frac{b^2 x^2}{a^2}\right )}{e^2 (2+m)}+\frac{b^2 (e x)^{3+m} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{3+m}{2},-n;\frac{5+m}{2};\frac{b^2 x^2}{a^2}\right )}{e^3 (3+m)}\\ \end{align*}
Mathematica [A] time = 0.11497, size = 172, normalized size = 0.82 \[ \frac{x (e x)^m (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \left (a^2 \left (m^2+5 m+6\right ) \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )-b (m+1) x \left (2 a (m+3) \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )-b (m+2) x \, _2F_1\left (\frac{m+3}{2},-n;\frac{m+5}{2};\frac{b^2 x^2}{a^2}\right )\right )\right )}{(m+1) (m+2) (m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( -bx+a \right ) ^{2+n} \left ( bx+a \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n}{\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}^{n}{\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n}{\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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