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

Optimal. Leaf size=128 \[ 2 d e x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{\left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 (p+1)} \]

[Out]

-(d^2 - e^2*x^2)^(1 + p)/(2*(1 + p)) + (2*d*e*x*(d^2 - e^2*x^2)^p*Hypergeometric
2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p - ((d^2 - e^2*x^2)^(1 +
p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(2*(1 + p))

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Rubi [A]  time = 0.185525, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ 2 d e x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{\left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 (p+1)}-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-(d^2 - e^2*x^2)^(1 + p)/(2*(1 + p)) + (2*d*e*x*(d^2 - e^2*x^2)^p*Hypergeometric
2F1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p - ((d^2 - e^2*x^2)^(1 +
p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2])/(2*(1 + p))

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Rubi in Sympy [A]  time = 37.8625, size = 102, normalized size = 0.8 \[ 2 d e x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 \left (p + 1\right )} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.419431, size = 147, normalized size = 1.15 \[ \frac{1}{2} \left (d^2-e^2 x^2\right )^p \left (\frac{d^2 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}+4 d e x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+\frac{d^2 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^{-p}-1\right )}{p+1}+\frac{e^2 x^2}{p+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.2874, size = 136, normalized size = 1.06 \[ - \frac{d^{2} e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (- p + 1\right )} + 2 d d^{2 p} e x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e^{2} \left (\begin{cases} \frac{x^{2} \left (d^{2}\right )^{p}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\begin{cases} \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (d^{2} - e^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 e^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**2*e**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p)*hyper((-p, -p), (-p + 1,), d**2/(e
**2*x**2))/(2*gamma(-p + 1)) + 2*d*d**(2*p)*e*x*hyper((1/2, -p), (3/2,), e**2*x*
*2*exp_polar(2*I*pi)/d**2) + e**2*Piecewise((x**2*(d**2)**p/2, Eq(e**2, 0)), (-P
iecewise(((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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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