3.306 \(\int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=264 \[ \frac{2 d^2 e (2 m+3 p+7) (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+2 p+4)}-\frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^{p+1}}{g^2 (m+2 p+4)}-\frac{3 d (g x)^{m+1} \left (d^2-e^2 x^2\right )^{p+1}}{g (m+2 p+3)}+\frac{2 d^3 (2 m+p+3) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+2 p+3)} \]

[Out]

(-3*d*(g*x)^(1 + m)*(d^2 - e^2*x^2)^(1 + p))/(g*(3 + m + 2*p)) - (e*(g*x)^(2 + m
)*(d^2 - e^2*x^2)^(1 + p))/(g^2*(4 + m + 2*p)) + (2*d^3*(3 + 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^2*e*(7 + 2*m + 3*p)*(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)*(4 + m + 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.654608, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{2 d^2 e (2 m+3 p+7) (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+2}{2},-p;\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+2 p+4)}-\frac{e (g x)^{m+2} \left (d^2-e^2 x^2\right )^{p+1}}{g^2 (m+2 p+4)}-\frac{3 d (g x)^{m+1} \left (d^2-e^2 x^2\right )^{p+1}}{g (m+2 p+3)}+\frac{2 d^3 (2 m+p+3) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+2 p+3)} \]

Antiderivative was successfully verified.

[In]  Int[(g*x)^m*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]

[Out]

(-3*d*(g*x)^(1 + m)*(d^2 - e^2*x^2)^(1 + p))/(g*(3 + m + 2*p)) - (e*(g*x)^(2 + m
)*(d^2 - e^2*x^2)^(1 + p))/(g^2*(4 + m + 2*p)) + (2*d^3*(3 + 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^2*e*(7 + 2*m + 3*p)*(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)*(4 + m + 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi in Sympy [A]  time = 68.3166, size = 262, normalized size = 0.99 \[ \frac{d^{3} \left (g x\right )^{m + 1} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g \left (m + 1\right )} + \frac{3 d^{2} e \left (g x\right )^{m + 2} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{2} \left (m + 2\right )} + \frac{3 d e^{2} \left (g x\right )^{m + 3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{3} \left (m + 3\right )} + \frac{e^{3} \left (g x\right )^{m + 4} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{g^{4} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x)**m*(e*x+d)**3*(-e**2*x**2+d**2)**p,x)

[Out]

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

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Mathematica [A]  time = 0.240428, size = 198, normalized size = 0.75 \[ x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (\frac{3 d^2 e x \, _2F_1\left (\frac{m}{2}+1,-p;\frac{m}{2}+2;\frac{e^2 x^2}{d^2}\right )}{m+2}+\frac{3 d e^2 x^2 \, _2F_1\left (\frac{m+3}{2},-p;\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )}{m+3}+\frac{e^3 x^3 \, _2F_1\left (\frac{m}{2}+2,-p;\frac{m}{2}+3;\frac{e^2 x^2}{d^2}\right )}{m+4}+\frac{d^3 \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(g*x)^m*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]

[Out]

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

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Maple [F]  time = 0.19, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x)^m*(e*x+d)^3*(-e^2*x^2+d^2)^p,x)

[Out]

int((g*x)^m*(e*x+d)^3*(-e^2*x^2+d^2)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{3}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*(g*x)^m,x, algorithm="maxima")

[Out]

integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*(g*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*(g*x)^m,x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(-e^2*x^2 + d^2)^p*(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*x+d)**3*(-e**2*x**2+d**2)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{3}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*(g*x)^m,x, algorithm="giac")

[Out]

integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*(g*x)^m, x)