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

Optimal. Leaf size=54 \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},e^2 x^2\right ) \]

[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

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Rubi [A]  time = 0.0448021, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {517, 430, 429} \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},e^2 x^2\right ) \]

Antiderivative was successfully verified.

[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 517

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]))

Rule 430

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 429

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])

Rubi steps

\begin{align*} \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx &=\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]  time = 0.20552, size = 167, normalized size = 3.09 \[ \frac{3 a x \left (1-e^2 x^2\right )^m \left (a+c x^2\right )^p F_1\left (\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},e^2 x^2\right )}{2 x^2 \left (c p F_1\left (\frac{3}{2};1-p,-m;\frac{5}{2};-\frac{c x^2}{a},e^2 x^2\right )-a e^2 m F_1\left (\frac{3}{2};-p,1-m;\frac{5}{2};-\frac{c x^2}{a},e^2 x^2\right )\right )+3 a F_1\left (\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},e^2 x^2\right )} \]

Warning: Unable to verify antiderivative.

[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]))

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Maple [F]  time = 0.937, size = 0, normalized size = 0. \begin{align*} \int \left ( -ex+1 \right ) ^{m} \left ( ex+1 \right ) ^{m} \left ( c{x}^{2}+a \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + 1\right )}^{m}{\left (-e x + 1\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)