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

Optimal. Leaf size=153 \[ \frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2)}+\frac{d (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1)} \]

[Out]

(d*(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)*(1 - (e^2*x^2)/d^2)^p) + (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 = 0.166817, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2)}+\frac{d (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(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)*(1 - (e^2*x^2)/d^2)^p) + (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 = 25.8605, size = 122, normalized size = 0.8 \[ \frac{d \left (g x\right )^{m + 1} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \left (m + 1\right )} + \frac{e \left (g x\right )^{m + 2} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{2} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(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)) + 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))

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Mathematica [A]  time = 0.108571, size = 116, normalized size = 0.76 \[ \frac{x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (e (m+1) x \, _2F_1\left (\frac{m}{2}+1,-p;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )+d (m+2) \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )\right )}{(m+1) (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 52.7187, size = 122, normalized size = 0.8 \[ \frac{d d^{2 p} g^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{d^{2 p} e g^{m} x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*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**(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
)/(2*gamma(m/2 + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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