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

Optimal. Leaf size=90 \[ -\frac{e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},1-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^2}-\frac{d \left (d^2-e^2 x^2\right )^p}{2 e^2 p} \]

[Out]

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

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Rubi [A]  time = 0.166434, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},1-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^2}-\frac{d \left (d^2-e^2 x^2\right )^p}{2 e^2 p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.389, size = 73, normalized size = 0.81 \[ - \frac{d \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{2} p} - \frac{e x^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 1, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.144862, size = 147, normalized size = 1.63 \[ \frac{2^{p-1} \left (\frac{e x}{d}+1\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (2 e (p+1) x \left (\frac{e x}{2 d}+\frac{1}{2}\right )^p \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+(d-e x) \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )\right )}{e^2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2^(-1 + p)*(d^2 - e^2*x^2)^p*(2*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*Hypergeometr
ic2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)]))/(e^2*(1 + p)*(1 + (e*x)/d)^p*(1 -
(e^2*x^2)/d^2)^p)

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ex+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 15.0963, size = 427, normalized size = 4.74 \[ \begin{cases} \frac{0^{p} d d^{2 p} \log{\left (\frac{d^{2}}{e^{2} x^{2}} \right )}}{2 e^{2}} - \frac{0^{p} d d^{2 p} \log{\left (\frac{d^{2}}{e^{2} x^{2}} - 1 \right )}}{2 e^{2}} - \frac{0^{p} d d^{2 p} \operatorname{acoth}{\left (\frac{d}{e x} \right )}}{e^{2}} + \frac{0^{p} d^{2 p} x}{e} - \frac{e^{2 p} p x x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p - \frac{1}{2} \\ - p + \frac{1}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e \Gamma \left (- p + \frac{1}{2}\right ) \Gamma \left (p + 1\right )} - \frac{d^{2 p} x^{2} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{3}F_{2}\left (\begin{matrix} 2, 1, - p + 1 \\ 2, 2 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{0^{p} d d^{2 p} \log{\left (\frac{d^{2}}{e^{2} x^{2}} \right )}}{2 e^{2}} - \frac{0^{p} d d^{2 p} \log{\left (- \frac{d^{2}}{e^{2} x^{2}} + 1 \right )}}{2 e^{2}} - \frac{0^{p} d d^{2 p} \operatorname{atanh}{\left (\frac{d}{e x} \right )}}{e^{2}} + \frac{0^{p} d^{2 p} x}{e} - \frac{e^{2 p} p x x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p - \frac{1}{2} \\ - p + \frac{1}{2} \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 e \Gamma \left (- p + \frac{1}{2}\right ) \Gamma \left (p + 1\right )} - \frac{d^{2 p} x^{2} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{3}F_{2}\left (\begin{matrix} 2, 1, - p + 1 \\ 2, 2 \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((0**p*d*d**(2*p)*log(d**2/(e**2*x**2))/(2*e**2) - 0**p*d*d**(2*p)*log(
d**2/(e**2*x**2) - 1)/(2*e**2) - 0**p*d*d**(2*p)*acoth(d/(e*x))/e**2 + 0**p*d**(
2*p)*x/e - e**(2*p)*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((-p
+ 1, -p - 1/2), (-p + 1/2,), d**2/(e**2*x**2))/(2*e*gamma(-p + 1/2)*gamma(p + 1)
) - d**(2*p)*x**2*gamma(p)*gamma(-p + 1)*hyper((2, 1, -p + 1), (2, 2), e**2*x**2
*exp_polar(2*I*pi)/d**2)/(2*d*gamma(-p)*gamma(p + 1)), Abs(d**2/(e**2*x**2)) > 1
), (0**p*d*d**(2*p)*log(d**2/(e**2*x**2))/(2*e**2) - 0**p*d*d**(2*p)*log(-d**2/(
e**2*x**2) + 1)/(2*e**2) - 0**p*d*d**(2*p)*atanh(d/(e*x))/e**2 + 0**p*d**(2*p)*x
/e - e**(2*p)*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((-p + 1, -
p - 1/2), (-p + 1/2,), d**2/(e**2*x**2))/(2*e*gamma(-p + 1/2)*gamma(p + 1)) - d*
*(2*p)*x**2*gamma(p)*gamma(-p + 1)*hyper((2, 1, -p + 1), (2, 2), e**2*x**2*exp_p
olar(2*I*pi)/d**2)/(2*d*gamma(-p)*gamma(p + 1)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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