3.961 \(\int (d+e x)^2 (1-\frac{e^2 x^2}{d^2})^p \, dx\)

Optimal. Leaf size=57 \[ -\frac{d^3 2^{p+2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0345681, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {676, 69} \[ -\frac{d^3 2^{p+2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 676

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

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), 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]))

Rubi steps

\begin{align*} \int (d+e x)^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \, dx &=\left (d \left (\frac{d-e x}{d}\right )^{1+p} \left (\frac{1}{d}-\frac{e x}{d^2}\right )^{-1-p}\right ) \int \left (\frac{1}{d}-\frac{e x}{d^2}\right )^p \left (1+\frac{e x}{d}\right )^{2+p} \, dx\\ &=-\frac{2^{2+p} d^3 \left (\frac{d-e x}{d}\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]  time = 0.0656816, size = 86, normalized size = 1.51 \[ d^2 x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+\frac{1}{3} e^2 x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^3 \left (1-\frac{e^2 x^2}{d^2}\right )^{p+1}}{e (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.495, size = 75, normalized size = 1.3 \begin{align*}{\frac{{e}^{2}{x}^{3}}{3}{\mbox{$_2$F$_1$}({\frac{3}{2}},-p;\,{\frac{5}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+ed{x}^{2}{\mbox{$_2$F$_1$}(1,-p;\,2;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}+{d}^{2}x{\mbox{$_2$F$_1$}({\frac{1}{2}},-p;\,{\frac{3}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(1-e^2*x^2/d^2)^p,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(1-e^2*x^2/d^2)^p,x, algorithm="maxima")

[Out]

integrate((e*x + d)^2*(-e^2*x^2/d^2 + 1)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(1-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)/d^2)^p, x)

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Sympy [C]  time = 3.89185, size = 116, normalized size = 2.04 \begin{align*} d^{2} 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}}{2} & \text{for}\: e^{2} = 0 \\- \frac{d^{2} \left (\begin{cases} \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (1 - \frac{e^{2} x^{2}}{d^{2}} \right )} & \text{otherwise} \end{cases}\right )}{2 e^{2}} & \text{otherwise} \end{cases}\right ) + \frac{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*x*hyper((1/2, -p), (3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + 2*d*e*Piecewise((x**2/2, Eq(e**2, 0)), (-d
**2*Piecewise(((1 - e**2*x**2/d**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(1 - e**2*x**2/d**2), True))/(2*e**2), T
rue)) + e**2*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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(1-e^2*x^2/d^2)^p,x, algorithm="giac")

[Out]

integrate((e*x + d)^2*(-e^2*x^2/d^2 + 1)^p, x)