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

Optimal. Leaf size=206 \[ \frac{2 d e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2)}+\frac{2 d^2 (m+p+2) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+2 p+3)}-\frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p+1}}{g (m+2 p+3)} \]

[Out]

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

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Rubi [A]  time = 0.350055, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{2 d e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2)}+\frac{2 d^2 (m+p+2) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+2 p+3)}-\frac{(g x)^{m+1} \left (d^2-e^2 x^2\right )^{p+1}}{g (m+2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.2772, size = 194, normalized size = 0.94 \[ \frac{d^{2} \left (g x\right )^{m + 1} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \left (m + 1\right )} + \frac{2 d e \left (g x\right )^{m + 2} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{2} \left (m + 2\right )} + \frac{e^{2} \left (g x\right )^{m + 3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{3} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.168299, size = 173, normalized size = 0.84 \[ \frac{x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (2 d e \left (m^2+4 m+3\right ) x \, _2F_1\left (\frac{m}{2}+1,-p;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )+(m+2) \left (d^2 (m+3) \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )+e^2 (m+1) x^2 \, _2F_1\left (\frac{m+3}{2},-p;\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )\right )\right )}{(m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 100.798, size = 192, normalized size = 0.93 \[ \frac{d^{2} d^{2 p} g^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{d d^{2 p} e g^{m} x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac{m}{2} + 2\right )} + \frac{d^{2 p} e^{2} g^{m} x^{3} x^{m} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*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)) + d*d**(2*p)*e*g**m*x**2*
x**m*gamma(m/2 + 1)*hyper((-p, m/2 + 1), (m/2 + 2,), e**2*x**2*exp_polar(2*I*pi)
/d**2)/gamma(m/2 + 2) + d**(2*p)*e**2*g**m*x**3*x**m*gamma(m/2 + 3/2)*hyper((-p,
m/2 + 3/2), (m/2 + 5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 5/2)
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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