3.974 \(\int x^m (1-a x)^n (1+a x)^n \, dx\)

Optimal. Leaf size=36 \[ \frac{x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};a^2 x^2\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.0444997, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};a^2 x^2\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(1 - a*x)^n*(1 + a*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.40879, size = 27, normalized size = 0.75 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{a^{2} x^{2}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(-a*x+1)**n*(a*x+1)**n,x)

[Out]

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

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Mathematica [A]  time = 0.0380133, size = 38, normalized size = 1.06 \[ \frac{x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+1}{2}+1;a^2 x^2\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(1 - a*x)^n*(1 + a*x)^n,x]

[Out]

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

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Maple [F]  time = 0.199, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( -ax+1 \right ) ^{n} \left ( ax+1 \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(-a*x+1)^n*(a*x+1)^n,x)

[Out]

int(x^m*(-a*x+1)^n*(a*x+1)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + 1\right )}^{n}{\left (-a x + 1\right )}^{n} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)^n*(-a*x + 1)^n*x^m,x, algorithm="maxima")

[Out]

integrate((a*x + 1)^n*(-a*x + 1)^n*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (a x + 1\right )}^{n}{\left (-a x + 1\right )}^{n} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)^n*(-a*x + 1)^n*x^m,x, algorithm="fricas")

[Out]

integral((a*x + 1)^n*(-a*x + 1)^n*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(x**m*(-a*x+1)**n*(a*x+1)**n,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + 1\right )}^{n}{\left (-a x + 1\right )}^{n} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)^n*(-a*x + 1)^n*x^m,x, algorithm="giac")

[Out]

integrate((a*x + 1)^n*(-a*x + 1)^n*x^m, x)