\(\int (d+e x)^3 (1-\frac {e^2 x^2}{d^2})^p \, dx\) [960]

Optimal result

Integrand size = 23, antiderivative size = 57 \[ \int (d+e x)^3 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=-\frac {2^{3+p} d^4 \left (\frac {d-e x}{d}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-3-p,1+p,2+p,\frac {d-e x}{2 d}\right )}{e (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {690, 71} \[ \int (d+e x)^3 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=-\frac {d^4 2^{p+3} \left (\frac {d-e x}{d}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p-3,p+1,p+2,\frac {d-e x}{2 d}\right )}{e (p+1)} \]

[In]

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

[Out]

-((2^(3 + p)*d^4*((d - e*x)/d)^(1 + p)*Hypergeometric2F1[-3 - 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 690

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/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]) && GtQ[a, 0] &&  !(IGtQ[m, 0] &&
(IntegerQ[3*p] || IntegerQ[4*p]))

Rubi steps \begin{align*} \text {integral}& = \left (d^2 \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 )^{3+p} \, dx \\ & = -\frac {2^{3+p} d^4 \left (\frac {d-e x}{d}\right )^{1+p} \, _2F_1\left (-3-p,1+p;2+p;\frac {d-e x}{2 d}\right )}{e (1+p)} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(116\) vs. \(2(57)=114\).

Time = 0.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.04 \[ \int (d+e x)^3 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=-\frac {2 d^4 \left (1-\frac {e^2 x^2}{d^2}\right )^{1+p}}{e (1+p)}+\frac {d^4 \left (1-\frac {e^2 x^2}{d^2}\right )^{2+p}}{2 e (2+p)}+d^3 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )+d e^2 x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right ) \]

[In]

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

[Out]

(-2*d^4*(1 - (e^2*x^2)/d^2)^(1 + p))/(e*(1 + p)) + (d^4*(1 - (e^2*x^2)/d^2)^(2 + p))/(2*e*(2 + p)) + d^3*x*Hyp
ergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + d*e^2*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2]

Maple [A] (verified)

Time = 4.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.82

method	result	size
meijerg	\(\frac {e^{3} x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (2,-p ;3;\frac {e^{2} x^{2}}{d^{2}}\right )}{4}+d \,e^{2} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{2},-p ;\frac {5}{2};\frac {e^{2} x^{2}}{d^{2}}\right )+\frac {3 e \,d^{2} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (1,-p ;2;\frac {e^{2} x^{2}}{d^{2}}\right )}{2}+d^{3} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},-p ;\frac {3}{2};\frac {e^{2} x^{2}}{d^{2}}\right )\)	\(104\)

[In]

int((e*x+d)^3*(1-e^2*x^2/d^2)^p,x,method=_RETURNVERBOSE)

[Out]

1/4*e^3*x^4*hypergeom([2,-p],[3],e^2*x^2/d^2)+d*e^2*x^3*hypergeom([3/2,-p],[5/2],e^2*x^2/d^2)+3/2*e*d^2*x^2*hy
pergeom([1,-p],[2],e^2*x^2/d^2)+d^3*x*hypergeom([1/2,-p],[3/2],e^2*x^2/d^2)

Fricas [F]

\[ \int (d+e x)^3 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p} \,d x } \]

[In]

integrate((e*x+d)^3*(1-e^2*x^2/d^2)^p,x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(-(e^2*x^2 - d^2)/d^2)^p, x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (42) = 84\).

Time = 2.19 (sec) , antiderivative size = 479, normalized size of antiderivative = 8.40 \[ \int (d+e x)^3 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=d^{3} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + 3 d^{2} e \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: e^{2} = 0 \\- \frac {d^{2} \left (\begin {cases} \frac {\left (1 - \frac {e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (1 - \frac {e^{2} x^{2}}{d^{2}} \right )} & \text {otherwise} \end {cases}\right )}{2 e^{2}} & \text {otherwise} \end {cases}\right ) + d e^{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e^{3} \left (\begin {cases} \frac {x^{4}}{4} & \text {for}\: e = 0 \\- \frac {d^{6} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{6} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac {d^{6}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {d^{4} e^{2} x^{2} \log {\left (- \frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac {d^{4} e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text {for}\: p = -2 \\- \frac {d^{4} \log {\left (- \frac {d}{e} + x \right )}}{2 e^{4}} - \frac {d^{4} \log {\left (\frac {d}{e} + x \right )}}{2 e^{4}} - \frac {d^{2} x^{2}}{2 e^{2}} & \text {for}\: p = -1 \\- \frac {d^{4} \left (1 - \frac {e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac {d^{2} e^{2} p x^{2} \left (1 - \frac {e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} p x^{4} \left (1 - \frac {e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac {e^{4} x^{4} \left (1 - \frac {e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**3*x*hyper((1/2, -p), (3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + 3*d**2*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)
, True)) + d*e**2*x**3*hyper((3/2, -p), (5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + e**3*Piecewise((x**4/4, Eq
(e, 0)), (-d**6*log(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**6*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) -
d**6/(-2*d**2*e**4 + 2*e**6*x**2) + d**4*e**2*x**2*log(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) + d**4*e**2*x**2
*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2), Eq(p, -2)), (-d**4*log(-d/e + x)/(2*e**4) - d**4*log(d/e + x)/(2*e
**4) - d**2*x**2/(2*e**2), Eq(p, -1)), (-d**4*(1 - e**2*x**2/d**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) - d**2
*e**2*p*x**2*(1 - e**2*x**2/d**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*p*x**4*(1 - e**2*x**2/d**2)**p/(
2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*x**4*(1 - e**2*x**2/d**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4), True))

Maxima [F]

\[ \int (d+e x)^3 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p} \,d x } \]

[In]

integrate((e*x+d)^3*(1-e^2*x^2/d^2)^p,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int (d+e x)^3 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-\frac {e^{2} x^{2}}{d^{2}} + 1\right )}^{p} \,d x } \]

[In]

integrate((e*x+d)^3*(1-e^2*x^2/d^2)^p,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx=\int {\left (1-\frac {e^2\,x^2}{d^2}\right )}^p\,{\left (d+e\,x\right )}^3 \,d x \]

[In]

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

[Out]

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