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

Optimal. Leaf size=156 \[ \frac{2 (p+2) x^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{3}{2},2-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^2 (2 p+1)}-\frac{x^3 \left (d^2-e^2 x^2\right )^{p-1}}{2 p+1}-\frac{d \left (d^2-e^2 x^2\right )^p}{e^3 p}-\frac{d^3 \left (d^2-e^2 x^2\right )^{p-1}}{e^3 (1-p)} \]

[Out]

-((d^3*(d^2 - e^2*x^2)^(-1 + p))/(e^3*(1 - p))) - (x^3*(d^2 - e^2*x^2)^(-1 + p))
/(1 + 2*p) - (d*(d^2 - e^2*x^2)^p)/(e^3*p) + (2*(2 + p)*x^3*(d^2 - e^2*x^2)^p*Hy
pergeometric2F1[3/2, 2 - p, 5/2, (e^2*x^2)/d^2])/(3*d^2*(1 + 2*p)*(1 - (e^2*x^2)
/d^2)^p)

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Rubi [A]  time = 0.383072, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{2 (p+2) x^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{3}{2},2-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^2 (2 p+1)}-\frac{x^3 \left (d^2-e^2 x^2\right )^{p-1}}{2 p+1}-\frac{d \left (d^2-e^2 x^2\right )^p}{e^3 p}-\frac{d^3 \left (d^2-e^2 x^2\right )^{p-1}}{e^3 (1-p)} \]

Antiderivative was successfully verified.

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

[Out]

-((d^3*(d^2 - e^2*x^2)^(-1 + p))/(e^3*(1 - p))) - (x^3*(d^2 - e^2*x^2)^(-1 + p))
/(1 + 2*p) - (d*(d^2 - e^2*x^2)^p)/(e^3*p) + (2*(2 + p)*x^3*(d^2 - e^2*x^2)^p*Hy
pergeometric2F1[3/2, 2 - p, 5/2, (e^2*x^2)/d^2])/(3*d^2*(1 + 2*p)*(1 - (e^2*x^2)
/d^2)^p)

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Rubi in Sympy [A]  time = 73.4424, size = 146, normalized size = 0.94 \[ - \frac{d^{3} \left (d^{2} - e^{2} x^{2}\right )^{p - 1}}{e^{3} \left (- p + 1\right )} - \frac{d \left (d^{2} - e^{2} x^{2}\right )^{p}}{e^{3} p} + \frac{x^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3 d^{2}} + \frac{e^{2} x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5 d^{4}} \]

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)**2,x)

[Out]

-d**3*(d**2 - e**2*x**2)**(p - 1)/(e**3*(-p + 1)) - d*(d**2 - e**2*x**2)**p/(e**
3*p) + x**3*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p + 2, 3/2)
, (5/2,), e**2*x**2/d**2)/(3*d**2) + e**2*x**5*(1 - e**2*x**2/d**2)**(-p)*(d**2
- e**2*x**2)**p*hyper((-p + 2, 5/2), (7/2,), e**2*x**2/d**2)/(5*d**4)

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Mathematica [A]  time = 0.195556, size = 177, normalized size = 1.13 \[ \frac{2^{p-2} \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (4 e (p+1) x \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+(d-e x) \left (1-\frac{e^2 x^2}{d^2}\right )^p \left (4 \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )-\, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )\right )\right )}{e^3 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-2 + p)*(d^2 - e^2*x^2)^p*(4*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hypergeometri
c2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (d - e*x)*(1 - (e^2*x^2)/d^2)^p*(4*Hypergeom
etric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - Hypergeometric2F1[2 - p, 1 + p,
2 + p, (d - e*x)/(2*d)])))/(e^3*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^2)^p)

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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