3.298 \(\int \frac{x^2 \left (d^2-e^2 x^2\right )^p}{(d+e x)^4} \, dx\)

Optimal. Leaf size=163 \[ \frac{\left (d^2-e^2 x^2\right )^{p+1}}{e^3 (1-2 p) (d+e x)^3}-\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^3 (3-p) (d+e x)^4}-\frac{2^{p-3} (p+7) \left (d^2-e^2 x^2\right )^{p+1} \left (\frac{e x}{d}+1\right )^{-p-1} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^3 e^3 (1-2 p) (3-p) (p+1)} \]

[Out]

-(d*(d^2 - e^2*x^2)^(1 + p))/(2*e^3*(3 - p)*(d + e*x)^4) + (d^2 - e^2*x^2)^(1 +
p)/(e^3*(1 - 2*p)*(d + e*x)^3) - (2^(-3 + p)*(7 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2
- e^2*x^2)^(1 + p)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^
3*e^3*(1 - 2*p)*(3 - p)*(1 + p))

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Rubi [A]  time = 0.396839, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\left (d^2-e^2 x^2\right )^{p+1}}{e^3 (1-2 p) (d+e x)^3}-\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^3 (3-p) (d+e x)^4}-\frac{2^{p-3} (p+7) \left (d^2-e^2 x^2\right )^{p+1} \left (\frac{e x}{d}+1\right )^{-p-1} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^3 e^3 (1-2 p) (3-p) (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x)^4,x]

[Out]

-(d*(d^2 - e^2*x^2)^(1 + p))/(2*e^3*(3 - p)*(d + e*x)^4) + (d^2 - e^2*x^2)^(1 +
p)/(e^3*(1 - 2*p)*(d + e*x)^3) - (2^(-3 + p)*(7 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2
- e^2*x^2)^(1 + p)*Hypergeometric2F1[3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(d^
3*e^3*(1 - 2*p)*(3 - p)*(1 + p))

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Rubi in Sympy [A]  time = 102.308, size = 202, normalized size = 1.24 \[ - \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (d - e x\right )^{- p} \left (d - e x\right )^{p + 1} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{16 d^{2} e^{3} \left (p + 1\right )} + \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (d - e x\right )^{- p} \left (d - e x\right )^{p + 2} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, p + 2 \\ p + 3 \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{8 d^{3} e^{3} \left (p + 2\right )} - \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (d - e x\right )^{- p} \left (d - e x\right )^{p + 3} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, p + 3 \\ p + 4 \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{16 d^{4} e^{3} \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(-e**2*x**2+d**2)**p/(e*x+d)**4,x)

[Out]

-((d/2 + e*x/2)/d)**(-p)*(d - e*x)**(-p)*(d - e*x)**(p + 1)*(d**2 - e**2*x**2)**
p*hyper((-p + 4, p + 1), (p + 2,), (d/2 - e*x/2)/d)/(16*d**2*e**3*(p + 1)) + ((d
/2 + e*x/2)/d)**(-p)*(d - e*x)**(-p)*(d - e*x)**(p + 2)*(d**2 - e**2*x**2)**p*hy
per((-p + 4, p + 2), (p + 3,), (d/2 - e*x/2)/d)/(8*d**3*e**3*(p + 2)) - ((d/2 +
e*x/2)/d)**(-p)*(d - e*x)**(-p)*(d - e*x)**(p + 3)*(d**2 - e**2*x**2)**p*hyper((
-p + 4, p + 3), (p + 4,), (d/2 - e*x/2)/d)/(16*d**4*e**3*(p + 3))

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Mathematica [A]  time = 0.129357, size = 130, normalized size = 0.8 \[ -\frac{2^{p-4} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (4 \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )-4 \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )+\, _2F_1\left (4-p,p+1;p+2;\frac{d-e x}{2 d}\right )\right )}{d^2 e^3 (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(d^2 - e^2*x^2)^p)/(d + e*x)^4,x]

[Out]

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

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Maple [F]  time = 0.151, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-e^2*x^2+d^2)^p/(e*x+d)^4,x)

[Out]

int(x^2*(-e^2*x^2+d^2)^p/(e*x+d)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{{\left (e x + d\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d)^4,x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d)^4,x, algorithm="fricas")

[Out]

integral((-e^2*x^2 + d^2)^p*x^2/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e
*x + d^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(-e**2*x**2+d**2)**p/(e*x+d)**4,x)

[Out]

Integral(x**2*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**4, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2}}{{\left (e x + d\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d)^4,x, algorithm="giac")

[Out]

integrate((-e^2*x^2 + d^2)^p*x^2/(e*x + d)^4, x)