\(\int (d+e x)^2 (d^2-e^2 x^2)^p \, dx\) [254]

Optimal result

Integrand size = 22, antiderivative size = 71 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {2^{2+p} d \left (1+\frac {e x}{d}\right )^{-1-p} \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-2-p,1+p,2+p,\frac {d-e x}{2 d}\right )}{e (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {692, 71} \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {d 2^{p+2} \left (\frac {e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p-2,p+1,p+2,\frac {d-e x}{2 d}\right )}{e (p+1)} \]

[In]

Int[(d + e*x)^2*(d^2 - e^2*x^2)^p,x]

[Out]

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

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(m - 1)*((a + c*x^2)^(p + 1)/((1
+ e*(x/d))^(p + 1)*(a/d + (c*x)/e)^(p + 1))), Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a,
c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (
IntegerQ[3*p] || IntegerQ[4*p]))

Rubi steps \begin{align*} \text {integral}& = \left (d (d-e x)^{-1-p} \left (1+\frac {e x}{d}\right )^{-1-p} \left (d^2-e^2 x^2\right )^{1+p}\right ) \int (d-e x)^p \left (1+\frac {e x}{d}\right )^{2+p} \, dx \\ & = -\frac {2^{2+p} d \left (1+\frac {e x}{d}\right )^{-1-p} \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (-2-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{e (1+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.89 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\frac {\left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (-3 d \left (d^2-e^2 x^2\right ) \left (1-\frac {e^2 x^2}{d^2}\right )^p+3 d^2 e (1+p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )+e^3 (1+p) x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )\right )}{3 e (1+p)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \,d x } \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.75 \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=d^{2} d^{2 p} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + 2 d e \left (\begin {cases} \frac {x^{2} \left (d^{2}\right )^{p}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\begin {cases} \frac {\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (d^{2} - e^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 e^{2}} & \text {otherwise} \end {cases}\right ) + \frac {d^{2 p} e^{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx=\int {\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]

[In]

int((d^2 - e^2*x^2)^p*(d + e*x)^2,x)

[Out]

int((d^2 - e^2*x^2)^p*(d + e*x)^2, x)