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

Optimal. Leaf size=214 \[ -\frac{2 (m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},2-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g (m+1) (-m-2 p+1)}+\frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)}-\frac{2 e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},2-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 g^2 (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.482796, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{2 (m+p) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},2-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g (m+1) (-m-2 p+1)}+\frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p-1}}{g (-m-2 p+1)}-\frac{2 e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},2-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^3 g^2 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 83.4721, size = 199, normalized size = 0.93 \[ \frac{\left (g x\right )^{m + 1} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{2} g \left (m + 1\right )} - \frac{2 e \left (g x\right )^{m + 2} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{3} g^{2} \left (m + 2\right )} + \frac{e^{2} \left (g x\right )^{m + 3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{4} g^{3} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.480381, size = 166, normalized size = 0.78 \[ \frac{d (m+2) x (g x)^m (d-e x)^p (d+e x)^{p-2} F_1\left (m+1;-p,2-p;m+2;\frac{e x}{d},-\frac{e x}{d}\right )}{(m+1) \left (d (m+2) F_1\left (m+1;-p,2-p;m+2;\frac{e x}{d},-\frac{e x}{d}\right )+e x \left ((p-2) F_1\left (m+2;-p,3-p;m+3;\frac{e x}{d},-\frac{e x}{d}\right )-p F_1\left (m+2;1-p,2-p;m+3;\frac{e x}{d},-\frac{e x}{d}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(d*(2 + m)*x*(g*x)^m*(d - e*x)^p*(d + e*x)^(-2 + p)*AppellF1[1 + m, -p, 2 - p, 2
+ m, (e*x)/d, -((e*x)/d)])/((1 + m)*(d*(2 + m)*AppellF1[1 + m, -p, 2 - p, 2 + m
, (e*x)/d, -((e*x)/d)] + e*x*(-(p*AppellF1[2 + m, 1 - p, 2 - p, 3 + m, (e*x)/d,
-((e*x)/d)]) + (-2 + p)*AppellF1[2 + m, -p, 3 - p, 3 + m, (e*x)/d, -((e*x)/d)]))
)

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{ \left ( gx \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((g*x)**m*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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