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

Optimal. Leaf size=163 \[ \frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},1-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d g (m+1)}-\frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},1-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.374606, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},1-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{d g (m+1)}-\frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},1-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.4996, size = 126, normalized size = 0.77 \[ \frac{\left (g x\right )^{m + 1} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d g \left (m + 1\right )} - \frac{e \left (g x\right )^{m + 2} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{2} g^{2} \left (m + 2\right )} \]

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),x)

[Out]

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

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Mathematica [C]  time = 0.474102, size = 168, normalized size = 1.03 \[ \frac{d (m+2) x (g x)^m (d-e x)^p (d+e x)^{p-1} F_1\left (m+1;-p,1-p;m+2;\frac{e x}{d},-\frac{e x}{d}\right )}{(m+1) \left (e x \left ((p-1) F_1\left (m+2;-p,2-p;m+3;\frac{e x}{d},-\frac{e x}{d}\right )-p \, _2F_1\left (\frac{m}{2}+1,1-p;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )\right )+d (m+2) F_1\left (m+1;-p,1-p;m+2;\frac{e x}{d},-\frac{e x}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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

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),x)

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

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

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 44.2349, size = 337, normalized size = 2.07 \[ - \frac{0^{p} d d^{2 p} g^{m} m x^{m} \Phi \left (\frac{d^{2}}{e^{2} x^{2}}, 1, - \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right )}{4 e^{2} x \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right )} + \frac{0^{p} d d^{2 p} g^{m} x^{m} \Phi \left (\frac{d^{2}}{e^{2} x^{2}}, 1, - \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (- \frac{m}{2} + \frac{1}{2}\right )}{4 e^{2} x \Gamma \left (- \frac{m}{2} + \frac{3}{2}\right )} + \frac{0^{p} d^{2 p} g^{m} m x^{m} \Phi \left (\frac{d^{2}}{e^{2} x^{2}}, 1, \frac{m e^{i \pi }}{2}\right ) \Gamma \left (- \frac{m}{2}\right )}{4 e \Gamma \left (- \frac{m}{2} + 1\right )} + \frac{d e^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac{m}{2} - p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - \frac{m}{2} - p + \frac{1}{2} \\ - \frac{m}{2} - p + \frac{3}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e^{2} x \Gamma \left (p + 1\right ) \Gamma \left (- \frac{m}{2} - p + \frac{3}{2}\right )} - \frac{e^{2 p} g^{m} p x^{m} x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- \frac{m}{2} - p\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - \frac{m}{2} - p \\ - \frac{m}{2} - p + 1 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e \Gamma \left (p + 1\right ) \Gamma \left (- \frac{m}{2} - p + 1\right )} \]

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),x)

[Out]

-0**p*d*d**(2*p)*g**m*m*x**m*lerchphi(d**2/(e**2*x**2), 1, -m/2 + 1/2)*gamma(-m/
2 + 1/2)/(4*e**2*x*gamma(-m/2 + 3/2)) + 0**p*d*d**(2*p)*g**m*x**m*lerchphi(d**2/
(e**2*x**2), 1, -m/2 + 1/2)*gamma(-m/2 + 1/2)/(4*e**2*x*gamma(-m/2 + 3/2)) + 0**
p*d**(2*p)*g**m*m*x**m*lerchphi(d**2/(e**2*x**2), 1, m*exp_polar(I*pi)/2)*gamma(
-m/2)/(4*e*gamma(-m/2 + 1)) + d*e**(2*p)*g**m*p*x**m*x**(2*p)*exp(I*pi*p)*gamma(
p)*gamma(-m/2 - p + 1/2)*hyper((-p + 1, -m/2 - p + 1/2), (-m/2 - p + 3/2,), d**2
/(e**2*x**2))/(2*e**2*x*gamma(p + 1)*gamma(-m/2 - p + 3/2)) - e**(2*p)*g**m*p*x*
*m*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-m/2 - p)*hyper((-p + 1, -m/2 - p), (-m/2
- p + 1,), d**2/(e**2*x**2))/(2*e*gamma(p + 1)*gamma(-m/2 - p + 1))

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

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),x, algorithm="giac")

[Out]

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