3.309 \(\int (g x)^m \left (d^2-e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=75 \[ \frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1)} \]

[Out]

((g*x)^(1 + m)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, (e^
2*x^2)/d^2])/(g*(1 + m)*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.0534599, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(g*x)^m*(d^2 - e^2*x^2)^p,x]

[Out]

((g*x)^(1 + m)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, (e^
2*x^2)/d^2])/(g*(1 + m)*(1 - (e^2*x^2)/d^2)^p)

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Rubi in Sympy [A]  time = 11.4165, size = 60, normalized size = 0.8 \[ \frac{\left (g x\right )^{m + 1} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x)**m*(-e**2*x**2+d**2)**p,x)

[Out]

(g*x)**(m + 1)*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p, m/2 +
1/2), (m/2 + 3/2,), e**2*x**2/d**2)/(g*(m + 1))

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Mathematica [A]  time = 0.0405495, size = 73, normalized size = 0.97 \[ \frac{x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+1}{2}+1;\frac{e^2 x^2}{d^2}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(g*x)^m*(d^2 - e^2*x^2)^p,x]

[Out]

(x*(g*x)^m*(d^2 - e^2*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, 1 + (1 + m)/2, (e^
2*x^2)/d^2])/((1 + m)*(1 - (e^2*x^2)/d^2)^p)

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x)^m*(-e^2*x^2+d^2)^p,x)

[Out]

int((g*x)^m*(-e^2*x^2+d^2)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*(g*x)^m,x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^p*(g*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*(g*x)^m,x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p*(g*x)^m, x)

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Sympy [A]  time = 15.2955, size = 61, normalized size = 0.81 \[ \frac{d^{2 p} g^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x)**m*(-e**2*x**2+d**2)**p,x)

[Out]

d**(2*p)*g**m*x*x**m*gamma(m/2 + 1/2)*hyper((-p, m/2 + 1/2), (m/2 + 3/2,), e**2*
x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*(g*x)^m,x, algorithm="giac")

[Out]

integrate((-e^2*x^2 + d^2)^p*(g*x)^m, x)