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 = 0.17, 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 126
Rule 127
Rule 364
Rule 365
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*}
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Mathematica [A] time = 0.12, 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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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{n} {\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} {\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (-b x +a \right )^{n +2} \left (b x +a \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} {\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e\,x\right )}^m\,{\left (a+b\,x\right )}^n\,{\left (a-b\,x\right )}^{n+2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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