3.722 \(\int (d+e x)^2 \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=133 \[ -\frac{x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (a e^2-c d^2 (2 p+3)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )}{c (2 p+3)}+\frac{e (d+e x) \left (a+c x^2\right )^{p+1}}{c (2 p+3)}+\frac{d e (p+2) \left (a+c x^2\right )^{p+1}}{c (p+1) (2 p+3)} \]

[Out]

(d*e*(2 + p)*(a + c*x^2)^(1 + p))/(c*(1 + p)*(3 + 2*p)) + (e*(d + e*x)*(a + c*x^
2)^(1 + p))/(c*(3 + 2*p)) - ((a*e^2 - c*d^2*(3 + 2*p))*x*(a + c*x^2)^p*Hypergeom
etric2F1[1/2, -p, 3/2, -((c*x^2)/a)])/(c*(3 + 2*p)*(1 + (c*x^2)/a)^p)

_______________________________________________________________________________________

Rubi [A]  time = 0.193603, antiderivative size = 125, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d^2-\frac{a e^2}{2 c p+3 c}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+\frac{e (d+e x) \left (a+c x^2\right )^{p+1}}{c (2 p+3)}+\frac{d e (p+2) \left (a+c x^2\right )^{p+1}}{c (p+1) (2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

(d*e*(2 + p)*(a + c*x^2)^(1 + p))/(c*(1 + p)*(3 + 2*p)) + (e*(d + e*x)*(a + c*x^
2)^(1 + p))/(c*(3 + 2*p)) + ((d^2 - (a*e^2)/(3*c + 2*c*p))*x*(a + c*x^2)^p*Hyper
geometric2F1[1/2, -p, 3/2, -((c*x^2)/a)])/(1 + (c*x^2)/a)^p

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.8794, size = 107, normalized size = 0.8 \[ \frac{d e \left (a + c x^{2}\right )^{p + 1} \left (p + 2\right )}{c \left (p + 1\right ) \left (2 p + 3\right )} + \frac{e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )}{c \left (2 p + 3\right )} - \frac{x \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (a e^{2} - c d^{2} \left (2 p + 3\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{c \left (2 p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.16578, size = 133, normalized size = 1. \[ \frac{\left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (3 c d^2 (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+e \left (3 d \left (c x^2 \left (\frac{c x^2}{a}+1\right )^p+a \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )\right )+c e (p+1) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right )\right )\right )}{3 c (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2*(c*x^2+a)^p,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 42.4672, size = 97, normalized size = 0.73 \[ a^{p} d^{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )} + \frac{a^{p} e^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{3} + 2 d e \left (\begin{cases} \frac{a^{p} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\begin{cases} \frac{\left (a + c x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + c x^{2} \right )} & \text{otherwise} \end{cases}}{2 c} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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