3.945 \(\int (e x)^m (a-b x)^{2+n} (a+b x)^n \, dx\)

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]

-(((e*x)^(1 + m)*(a - b*x)^(1 + n)*(a + b*x)^(1 + n))/(e*(3 + m + 2*n))) + (2*a^
2*(2 + m + n)*(e*x)^(1 + m)*(a - b*x)^n*(a + b*x)^n*Hypergeometric2F1[(1 + m)/2,
-n, (3 + m)/2, (b^2*x^2)/a^2])/(e*(1 + m)*(3 + m + 2*n)*(1 - (b^2*x^2)/a^2)^n)
- (2*a*b*(e*x)^(2 + m)*(a - b*x)^n*(a + b*x)^n*Hypergeometric2F1[(2 + m)/2, -n,
(4 + m)/2, (b^2*x^2)/a^2])/(e^2*(2 + m)*(1 - (b^2*x^2)/a^2)^n)

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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]

(a^2*(e*x)^(1 + m)*(a - b*x)^n*(a + b*x)^n*Hypergeometric2F1[(1 + m)/2, -n, (3 +
m)/2, (b^2*x^2)/a^2])/(e*(1 + m)*(1 - (b^2*x^2)/a^2)^n) - (2*a*b*(e*x)^(2 + m)*
(a - b*x)^n*(a + b*x)^n*Hypergeometric2F1[(2 + m)/2, -n, (4 + m)/2, (b^2*x^2)/a^
2])/(e^2*(2 + m)*(1 - (b^2*x^2)/a^2)^n) + (b^2*(e*x)^(3 + m)*(a - b*x)^n*(a + b*
x)^n*Hypergeometric2F1[(3 + m)/2, -n, (5 + m)/2, (b^2*x^2)/a^2])/(e^3*(3 + m)*(1
- (b^2*x^2)/a^2)^n)

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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]

a**2*(e*x)**(m + 1)*(1 - b**2*x**2/a**2)**(-n)*(a - b*x)**n*(a + b*x)**n*hyper((
-n, m/2 + 1/2), (m/2 + 3/2,), b**2*x**2/a**2)/(e*(m + 1)) - 2*a*b*(e*x)**(m + 2)
*(1 - b**2*x**2/a**2)**(-n)*(a - b*x)**n*(a + b*x)**n*hyper((-n, m/2 + 1), (m/2
+ 2,), b**2*x**2/a**2)/(e**2*(m + 2)) + b**2*(e*x)**(m + 3)*(1 - b**2*x**2/a**2)
**(-n)*(a - b*x)**n*(a + b*x)**n*hyper((-n, m/2 + 3/2), (m/2 + 5/2,), b**2*x**2/
a**2)/(e**3*(m + 3))

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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]

(x*(e*x)^m*(a - b*x)^n*(a + b*x)^n*(-2*a*b*(3 + 4*m + m^2)*x*Hypergeometric2F1[1
+ m/2, -n, 2 + m/2, (b^2*x^2)/a^2] + (2 + m)*(a^2*(3 + m)*Hypergeometric2F1[(1
+ m)/2, -n, (3 + m)/2, (b^2*x^2)/a^2] + b^2*(1 + m)*x^2*Hypergeometric2F1[(3 + m
)/2, -n, (5 + m)/2, (b^2*x^2)/a^2])))/((1 + m)*(2 + m)*(3 + m)*(1 - (b^2*x^2)/a^
2)^n)

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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]

int((e*x)^m*(-b*x+a)^(2+n)*(b*x+a)^n,x)

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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]

integrate((b*x + a)^n*(-b*x + a)^(n + 2)*(e*x)^m, x)

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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]

integral((b*x + a)^n*(-b*x + a)^(n + 2)*(e*x)^m, x)

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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]

Timed out

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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]

integrate((b*x + a)^n*(-b*x + a)^(n + 2)*(e*x)^m, x)