3.277 \(\int \frac{x^3 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx\)

Optimal. Leaf size=150 \[ \frac{3 d^2 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)}+\frac{d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4 (1-p)}-\frac{2 e x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3} \]

[Out]

(d^4*(d^2 - e^2*x^2)^(-1 + p))/(e^4*(1 - p)) + (3*d^2*(d^2 - e^2*x^2)^p)/(2*e^4*
p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^4*(1 + p)) - (2*e*x^5*(d^2 - e^2*x^2)^p*Hyperg
eometric2F1[5/2, 2 - p, 7/2, (e^2*x^2)/d^2])/(5*d^3*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.36949, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{3 d^2 \left (d^2-e^2 x^2\right )^p}{2 e^4 p}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e^4 (p+1)}+\frac{d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4 (1-p)}-\frac{2 e x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},2-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(d^4*(d^2 - e^2*x^2)^(-1 + p))/(e^4*(1 - p)) + (3*d^2*(d^2 - e^2*x^2)^p)/(2*e^4*
p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^4*(1 + p)) - (2*e*x^5*(d^2 - e^2*x^2)^p*Hyperg
eometric2F1[5/2, 2 - p, 7/2, (e^2*x^2)/d^2])/(5*d^3*(1 - (e^2*x^2)/d^2)^p)

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Rubi in Sympy [A]  time = 88.3953, size = 122, normalized size = 0.81 \[ \frac{d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p - 1}}{e^{4} \left (- p + 1\right )} + \frac{3 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{4} \left (p + 1\right )} - \frac{2 e x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**4*(d**2 - e**2*x**2)**(p - 1)/(e**4*(-p + 1)) + 3*d**2*(d**2 - e**2*x**2)**p/
(2*e**4*p) - (d**2 - e**2*x**2)**(p + 1)/(2*e**4*(p + 1)) - 2*e*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**3)

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Mathematica [B]  time = 0.435732, size = 332, normalized size = 2.21 \[ \frac{2^{p-2} \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (-8 d e (p+1) x \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-6 d (d-e x) \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )+d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )-d e x \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )-2 d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p+2 e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p+2 d^2 \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p\right )}{e^4 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-2 + p)*(d^2 - e^2*x^2)^p*(2*d^2*(1/2 + (e*x)/(2*d))^p - 2*d^2*(1/2 + (e*x)/
(2*d))^p*(1 - (e^2*x^2)/d^2)^p + 2*e^2*x^2*(1/2 + (e*x)/(2*d))^p*(1 - (e^2*x^2)/
d^2)^p - 8*d*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hypergeometric2F1[1/2, -p, 3/2, (
e^2*x^2)/d^2] - 6*d*(d - e*x)*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[1 - p, 1 +
p, 2 + p, (d - e*x)/(2*d)] + d^2*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[2 - p,
1 + p, 2 + p, (d - e*x)/(2*d)] - d*e*x*(1 - (e^2*x^2)/d^2)^p*Hypergeometric2F1[
2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(e^4*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x
^2)/d^2)^p)

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Maple [F]  time = 0.107, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^3*(-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. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(-e**2*x**2+d**2)**p/(e*x+d)**2,x)

[Out]

Integral(x**3*(-(-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. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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