\(\int (1-e x)^m (1+e x)^m (a+c x^2)^p \, dx\) [802]

Optimal result

Integrand size = 25, antiderivative size = 54 \[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},e^2 x^2\right ) \]

[Out]

x*(c*x^2+a)^p*AppellF1(1/2,-m,-p,3/2,e^2*x^2,-c*x^2/a)/((1+c*x^2/a)^p)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {531, 441, 440} \[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},e^2 x^2\right ) \]

[In]

Int[(1 - e*x)^m*(1 + e*x)^m*(a + c*x^2)^p,x]

[Out]

(x*(a + c*x^2)^p*AppellF1[1/2, -p, -m, 3/2, -((c*x^2)/a), e^2*x^2])/(1 + (c*x^2)/a)^p

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 531

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^
(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x]
&& EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rubi steps \begin{align*} \text {integral}& = \int \left (a+c x^2\right )^p \left (1-e^2 x^2\right )^m \, dx \\ & = \left (\left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \left (1-e^2 x^2\right )^m \, dx \\ & = x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,-m;\frac {3}{2};-\frac {c x^2}{a},e^2 x^2\right ) \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(167\) vs. \(2(54)=108\).

Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.09 \[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=\frac {3 a x \left (a+c x^2\right )^p \left (1-e^2 x^2\right )^m \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},e^2 x^2\right )}{3 a \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},e^2 x^2\right )+2 x^2 \left (c p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-m,\frac {5}{2},-\frac {c x^2}{a},e^2 x^2\right )-a e^2 m \operatorname {AppellF1}\left (\frac {3}{2},-p,1-m,\frac {5}{2},-\frac {c x^2}{a},e^2 x^2\right )\right )} \]

[In]

Integrate[(1 - e*x)^m*(1 + e*x)^m*(a + c*x^2)^p,x]

[Out]

(3*a*x*(a + c*x^2)^p*(1 - e^2*x^2)^m*AppellF1[1/2, -p, -m, 3/2, -((c*x^2)/a), e^2*x^2])/(3*a*AppellF1[1/2, -p,
-m, 3/2, -((c*x^2)/a), e^2*x^2] + 2*x^2*(c*p*AppellF1[3/2, 1 - p, -m, 5/2, -((c*x^2)/a), e^2*x^2] - a*e^2*m*A
ppellF1[3/2, -p, 1 - m, 5/2, -((c*x^2)/a), e^2*x^2]))

Maple [F]

\[\int \left (-e x +1\right )^{m} \left (e x +1\right )^{m} \left (c \,x^{2}+a \right )^{p}d x\]

[In]

int((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x)

[Out]

int((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x)

Fricas [F]

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

[In]

integrate((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(e*x + 1)^m*(-e*x + 1)^m, x)

Sympy [F(-1)]

Timed out. \[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=\text {Timed out} \]

[In]

integrate((-e*x+1)**m*(e*x+1)**m*(c*x**2+a)**p,x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(e*x + 1)^m*(-e*x + 1)^m, x)

Giac [F]

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

[In]

integrate((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(e*x + 1)^m*(-e*x + 1)^m, x)

Mupad [F(-1)]

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

[In]

int((a + c*x^2)^p*(1 - e*x)^m*(e*x + 1)^m,x)

[Out]

int((a + c*x^2)^p*(1 - e*x)^m*(e*x + 1)^m, x)