Optimal. Leaf size=211 \[ -\frac{2 a b (e x)^{m+2} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{e^2 (m+2)}+\frac{2 a^2 (m+n+2) (e x)^{m+1} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{e (m+1) (m+2 n+3)}-\frac{(e x)^{m+1} (a-b x)^{n+1} (a+b x)^{n+1}}{e (m+2 n+3)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.42897, 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 \[ \frac{b^2 (e x)^{m+3} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+3}{2},-n;\frac{m+5}{2};\frac{b^2 x^2}{a^2}\right )}{e^3 (m+3)}-\frac{2 a b (e x)^{m+2} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{b^2 x^2}{a^2}\right )}{e^2 (m+2)}+\frac{a^2 (e x)^{m+1} (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m*(a - b*x)^(2 + n)*(a + b*x)^n,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 60.3433, size = 199, normalized size = 0.94 \[ \frac{a^{2} \left (e x\right )^{m + 1} \left (1 - \frac{b^{2} x^{2}}{a^{2}}\right )^{- n} \left (a - b x\right )^{n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )}}{e \left (m + 1\right )} - \frac{2 a b \left (e x\right )^{m + 2} \left (1 - \frac{b^{2} x^{2}}{a^{2}}\right )^{- n} \left (a - b x\right )^{n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )}}{e^{2} \left (m + 2\right )} + \frac{b^{2} \left (e x\right )^{m + 3} \left (1 - \frac{b^{2} x^{2}}{a^{2}}\right )^{- n} \left (a - b x\right )^{n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{b^{2} x^{2}}{a^{2}}} \right )}}{e^{3} \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(-b*x+a)**(2+n)*(b*x+a)**n,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.238422, size = 174, normalized size = 0.82 \[ \frac{x (e x)^m (a-b x)^n (a+b x)^n \left (1-\frac{b^2 x^2}{a^2}\right )^{-n} \left ((m+2) \left (a^2 (m+3) \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{b^2 x^2}{a^2}\right )+b^2 (m+1) x^2 \, _2F_1\left (\frac{m+3}{2},-n;\frac{m+5}{2};\frac{b^2 x^2}{a^2}\right )\right )-2 a b \left (m^2+4 m+3\right ) x \, _2F_1\left (\frac{m}{2}+1,-n;\frac{m}{2}+2;\frac{b^2 x^2}{a^2}\right )\right )}{(m+1) (m+2) (m+3)} \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^m*(a - b*x)^(2 + n)*(a + b*x)^n,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.2, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( -bx+a \right ) ^{2+n} \left ( bx+a \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(-b*x+a)^(2+n)*(b*x+a)^n,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(-b*x + a)^(n + 2)*(e*x)^m,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{n}{\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(-b*x + a)^(n + 2)*(e*x)^m,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(-b*x+a)**(2+n)*(b*x+a)**n,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (-b x + a\right )}^{n + 2} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(-b*x + a)^(n + 2)*(e*x)^m,x, algorithm="giac")
[Out]