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

Optimal. Leaf size=204 \[ -\frac{2 d e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{d^2 (2 m+5) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+4) \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{\left (d^2-e^2 x^2\right )^{3/2} (g x)^{m+1}}{g (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.485507, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{2 d e \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{d^2 (2 m+5) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+4) \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{\left (d^2-e^2 x^2\right )^{3/2} (g x)^{m+1}}{g (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 77.7446, size = 209, normalized size = 1.02 \[ \frac{d^{2} \left (g x\right )^{m + 1} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 1\right )} - \frac{2 d e \left (g x\right )^{m + 2} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 2\right )} + \frac{e^{2} \left (g x\right )^{m + 3} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{3} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(g*x)**(m + 1)*sqrt(d**2 - e**2*x**2)*hyper((-1/2, 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)) - 2*d*e*(g*x)**(m + 2)*sq
rt(d**2 - e**2*x**2)*hyper((-1/2, m/2 + 1), (m/2 + 2,), e**2*x**2/d**2)/(g**2*sq
rt(1 - e**2*x**2/d**2)*(m + 2)) + e**2*(g*x)**(m + 3)*sqrt(d**2 - e**2*x**2)*hyp
er((-1/2, m/2 + 3/2), (m/2 + 5/2,), e**2*x**2/d**2)/(g**3*sqrt(1 - e**2*x**2/d**
2)*(m + 3))

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Mathematica [A]  time = 0.164661, size = 175, normalized size = 0.86 \[ \frac{x \sqrt{d^2-e^2 x^2} (g x)^m \left ((m+2) \left (d^2 (m+3) \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )+e^2 (m+1) x^2 \, _2F_1\left (-\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )\right )-2 d e \left (m^2+4 m+3\right ) x \, _2F_1\left (-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )\right )}{(m+1) (m+2) (m+3) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{ \left ( gx \right ) ^{m}}{ \left ( ex+d \right ) ^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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