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

Optimal. Leaf size=184 \[ \frac{2 (p+4) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2 (2 p+3)}-\frac{x^5 \left (d^2-e^2 x^2\right )^{p-1}}{2 p+3}+\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}-\frac{d^5 \left (d^2-e^2 x^2\right )^{p-1}}{e^5 (1-p)}-\frac{2 d^3 \left (d^2-e^2 x^2\right )^p}{e^5 p} \]

[Out]

-((d^5*(d^2 - e^2*x^2)^(-1 + p))/(e^5*(1 - p))) - (x^5*(d^2 - e^2*x^2)^(-1 + p))
/(3 + 2*p) - (2*d^3*(d^2 - e^2*x^2)^p)/(e^5*p) + (d*(d^2 - e^2*x^2)^(1 + p))/(e^
5*(1 + p)) + (2*(4 + p)*x^5*(d^2 - e^2*x^2)^p*Hypergeometric2F1[5/2, 2 - p, 7/2,
(e^2*x^2)/d^2])/(5*d^2*(3 + 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.430833, antiderivative size = 184, 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 \[ \frac{2 (p+4) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^2 (2 p+3)}-\frac{x^5 \left (d^2-e^2 x^2\right )^{p-1}}{2 p+3}+\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}-\frac{d^5 \left (d^2-e^2 x^2\right )^{p-1}}{e^5 (1-p)}-\frac{2 d^3 \left (d^2-e^2 x^2\right )^p}{e^5 p} \]

Antiderivative was successfully verified.

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

[Out]

-((d^5*(d^2 - e^2*x^2)^(-1 + p))/(e^5*(1 - p))) - (x^5*(d^2 - e^2*x^2)^(-1 + p))
/(3 + 2*p) - (2*d^3*(d^2 - e^2*x^2)^p)/(e^5*p) + (d*(d^2 - e^2*x^2)^(1 + p))/(e^
5*(1 + p)) + (2*(4 + p)*x^5*(d^2 - e^2*x^2)^p*Hypergeometric2F1[5/2, 2 - p, 7/2,
(e^2*x^2)/d^2])/(5*d^2*(3 + 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi in Sympy [A]  time = 87.2523, size = 172, normalized size = 0.93 \[ - \frac{d^{5} \left (d^{2} - e^{2} x^{2}\right )^{p - 1}}{e^{5} \left (- p + 1\right )} - \frac{2 d^{3} \left (d^{2} - e^{2} x^{2}\right )^{p}}{e^{5} p} + \frac{d \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{e^{5} \left (p + 1\right )} + \frac{x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5 d^{2}} + \frac{e^{2} x^{7} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{7 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**5*(d**2 - e**2*x**2)**(p - 1)/(e**5*(-p + 1)) - 2*d**3*(d**2 - e**2*x**2)**p
/(e**5*p) + d*(d**2 - e**2*x**2)**(p + 1)/(e**5*(p + 1)) + 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**2) + e**2*x**7*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p +
2, 7/2), (9/2,), e**2*x**2/d**2)/(7*d**4)

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Mathematica [C]  time = 0.42535, size = 140, normalized size = 0.76 \[ -\frac{6 d x^5 (d-e x)^p (d+e x)^{p-2} F_1\left (5;-p,2-p;6;\frac{e x}{d},-\frac{e x}{d}\right )}{5 \left (e x \left (p F_1\left (6;1-p,2-p;7;\frac{e x}{d},-\frac{e x}{d}\right )-(p-2) F_1\left (6;-p,3-p;7;\frac{e x}{d},-\frac{e x}{d}\right )\right )-6 d F_1\left (5;-p,2-p;6;\frac{e x}{d},-\frac{e x}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-6*d*x^5*(d - e*x)^p*(d + e*x)^(-2 + p)*AppellF1[5, -p, 2 - p, 6, (e*x)/d, -((e
*x)/d)])/(5*(-6*d*AppellF1[5, -p, 2 - p, 6, (e*x)/d, -((e*x)/d)] + e*x*(p*Appell
F1[6, 1 - p, 2 - p, 7, (e*x)/d, -((e*x)/d)] - (-2 + p)*AppellF1[6, -p, 3 - p, 7,
(e*x)/d, -((e*x)/d)])))

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Maple [F]  time = 0.116, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^4*(-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. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*(-(-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. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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