3.229 \(\int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=80 \[ \frac{d^4 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

[Out]

(d^4*(g*x)^(1 + m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m
)/2, (e^2*x^2)/d^2])/(g*(1 + m)*Sqrt[1 - (e^2*x^2)/d^2])

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Rubi [A]  time = 0.0723568, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{d^4 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(g*x)^m*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(d^4*(g*x)^(1 + m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m
)/2, (e^2*x^2)/d^2])/(g*(1 + m)*Sqrt[1 - (e^2*x^2)/d^2])

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Rubi in Sympy [A]  time = 11.674, size = 68, normalized size = 0.85 \[ \frac{d^{4} \left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x)**m*(-e**2*x**2+d**2)**(5/2),x)

[Out]

d**4*(g*x)**(m + 1)*sqrt(d**2 - e**2*x**2)*hyper((-5/2, m/2 + 1/2), (m/2 + 3/2,)
, e**2*x**2/d**2)/(g*sqrt(1 - e**2*x**2/d**2)*(m + 1))

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Mathematica [B]  time = 0.165613, size = 183, normalized size = 2.29 \[ \frac{x \sqrt{d^2-e^2 x^2} (g x)^m \left (d^4 \left (m^2+8 m+15\right ) \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )-e^2 (m+1) x^2 \left (2 d^2 (m+5) \, _2F_1\left (-\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )-e^2 (m+3) x^2 \, _2F_1\left (-\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\frac{e^2 x^2}{d^2}\right )\right )\right )}{(m+1) (m+3) (m+5) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(g*x)^m*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*(d^4*(15 + 8*m + m^2)*Hypergeometric2F1[-1/2, (1
+ m)/2, (3 + m)/2, (e^2*x^2)/d^2] - e^2*(1 + m)*x^2*(2*d^2*(5 + m)*Hypergeometri
c2F1[-1/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2] - e^2*(3 + m)*x^2*Hypergeometric
2F1[-1/2, (5 + m)/2, (7 + m)/2, (e^2*x^2)/d^2])))/((1 + m)*(3 + m)*(5 + m)*Sqrt[
1 - (e^2*x^2)/d^2])

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x)^m*(-e^2*x^2+d^2)^(5/2),x)

[Out]

int((g*x)^m*(-e^2*x^2+d^2)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} \left (g x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m,x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{4} x^{4} - 2 \, d^{2} e^{2} x^{2} + d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m,x, algorithm="fricas")

[Out]

integral((e^4*x^4 - 2*d^2*e^2*x^2 + d^4)*sqrt(-e^2*x^2 + d^2)*(g*x)^m, x)

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Sympy [A]  time = 134.14, size = 61, normalized size = 0.76 \[ \frac{d^{5} g^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x)**m*(-e**2*x**2+d**2)**(5/2),x)

[Out]

d**5*g**m*x*x**m*gamma(m/2 + 1/2)*hyper((-5/2, 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. \[ \int{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} \left (g x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m,x, algorithm="giac")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m, x)