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

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 = 0.113934, 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 \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-m;\frac{3}{2};-\frac{c x^2}{a},e^2 x^2\right ) \]

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

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Rubi in Sympy [A]  time = 23.4026, size = 41, normalized size = 0.76 \[ x \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2},- m,- p,\frac{3}{2},e^{2} x^{2},- \frac{c x^{2}}{a} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + c*x**2/a)**(-p)*(a + c*x**2)**p*appellf1(1/2, -m, -p, 3/2, e**2*x**2, -c*
x**2/a)

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Mathematica [B]  time = 0.35557, 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*Appe
llF1[3/2, 1 - p, -m, 5/2, -((c*x^2)/a), e^2*x^2] - a*e^2*m*AppellF1[3/2, -p, 1 -
m, 5/2, -((c*x^2)/a), e^2*x^2]))

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Maple [F]  time = 0.226, size = 0, normalized size = 0. \[ \int \left ( -ex+1 \right ) ^{m} \left ( ex+1 \right ) ^{m} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + 1)^m*(-e*x + 1)^m,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. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + 1\right )}^{m}{\left (-e x + 1\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + 1)^m*(-e*x + 1)^m,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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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