3.977 \(\int x^m \left (1-\frac{a x}{2}\right )^n (2+a x)^n \, dx\)

Optimal. Leaf size=42 \[ \frac{2^n x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{a^2 x^2}{4}\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.0436796, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2^n x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{a^2 x^2}{4}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.48796, size = 32, normalized size = 0.76 \[ \frac{2^{n} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{a^{2} x^{2}}{4}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(1-1/2*a*x)**n*(a*x+2)**n,x)

[Out]

2**n*x**(m + 1)*hyper((-n, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2/4)/(m + 1)

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Mathematica [A]  time = 0.0753358, size = 74, normalized size = 1.76 \[ \frac{x^{m+1} \left (1-\frac{a x}{2}\right )^n (a x+2)^n \left (1-\frac{a^2 x^2}{4}\right )^{-n} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+1}{2}+1;\frac{a^2 x^2}{4}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.224, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( 1-{\frac{ax}{2}} \right ) ^{n} \left ( ax+2 \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^m*(1-1/2*a*x)^n*(a*x+2)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + 2\right )}^{n}{\left (-\frac{1}{2} \, a x + 1\right )}^{n} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 2)^n*(-1/2*a*x + 1)^n*x^m,x, algorithm="maxima")

[Out]

integrate((a*x + 2)^n*(-1/2*a*x + 1)^n*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (a x + 2\right )}^{n}{\left (-\frac{1}{2} \, a x + 1\right )}^{n} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 2)^n*(-1/2*a*x + 1)^n*x^m,x, algorithm="fricas")

[Out]

integral((a*x + 2)^n*(-1/2*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*(1-1/2*a*x)**n*(a*x+2)**n,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + 2\right )}^{n}{\left (-\frac{1}{2} \, a x + 1\right )}^{n} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 2)^n*(-1/2*a*x + 1)^n*x^m,x, algorithm="giac")

[Out]

integrate((a*x + 2)^n*(-1/2*a*x + 1)^n*x^m, x)