3.244 \(\int x (d+e x) \left (d^2-e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=89 \[ \frac{1}{3} e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (p+1)} \]

[Out]

-(d*(d^2 - e^2*x^2)^(1 + p))/(2*e^2*(1 + p)) + (e*x^3*(d^2 - e^2*x^2)^p*Hypergeo
metric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(3*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.0939422, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{1}{3} e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d \left (d^2-e^2 x^2\right )^{p+1}}{2 e^2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(d^2 - e^2*x^2)^(1 + p))/(2*e^2*(1 + p)) + (e*x^3*(d^2 - e^2*x^2)^p*Hypergeo
metric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(3*(1 - (e^2*x^2)/d^2)^p)

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Rubi in Sympy [A]  time = 17.8137, size = 71, normalized size = 0.8 \[ - \frac{d \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{2} \left (p + 1\right )} + \frac{e x^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.108477, size = 121, normalized size = 1.36 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (3 d e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p+2 e^3 (p+1) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-3 d^3 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )\right )}{6 e^2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int x \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.98514, size = 85, normalized size = 0.96 \[ d \left (\begin{cases} \frac{x^{2} \left (d^{2}\right )^{p}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\begin{cases} \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (d^{2} - e^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 e^{2}} & \text{otherwise} \end{cases}\right ) + \frac{d^{2 p} e x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((x**2*(d**2)**p/2, Eq(e**2, 0)), (-Piecewise(((d**2 - e**2*x**2)**(p
+ 1)/(p + 1), Ne(p, -1)), (log(d**2 - e**2*x**2), True))/(2*e**2), True)) + d**
(2*p)*e*x**3*hyper((3/2, -p), (5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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