3.140 \(\int \frac{a+b x^2+c x^4}{\sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (8 a e^4+4 b d^2 e^2+3 c d^4\right )}{4 e^5}-\frac{x \sqrt{d-e x} \sqrt{d+e x} \left (4 b e^2+3 c d^2\right )}{8 e^4}+\frac{c x^3 (e x-d) \sqrt{d+e x}}{4 e^2 \sqrt{d-e x}} \]

[Out]

-((3*c*d^2 + 4*b*e^2)*x*Sqrt[d - e*x]*Sqrt[d + e*x])/(8*e^4) + (c*x^3*(-d + e*x)
*Sqrt[d + e*x])/(4*e^2*Sqrt[d - e*x]) - ((3*c*d^4 + 4*b*d^2*e^2 + 8*a*e^4)*ArcTa
n[Sqrt[d - e*x]/Sqrt[d + e*x]])/(4*e^5)

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Rubi [A]  time = 0.259615, antiderivative size = 179, normalized size of antiderivative = 1.4, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{\sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (8 a e^4+4 b d^2 e^2+3 c d^4\right )}{8 e^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (4 b e^2+3 c d^2\right )}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

-((3*c*d^2 + 4*b*e^2)*x*(d^2 - e^2*x^2))/(8*e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - (
c*x^3*(d^2 - e^2*x^2))/(4*e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + ((3*c*d^4 + 4*b*d^2
*e^2 + 8*a*e^4)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^5*Sq
rt[d - e*x]*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 19.0009, size = 141, normalized size = 1.1 \[ - \frac{c x^{3} \sqrt{d - e x} \sqrt{d + e x}}{4 e^{2}} - \frac{x \sqrt{d - e x} \sqrt{d + e x} \left (4 b e^{2} + 3 c d^{2}\right )}{8 e^{4}} + \frac{\sqrt{d - e x} \sqrt{d + e x} \left (8 a e^{4} + 4 b d^{2} e^{2} + 3 c d^{4}\right ) \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e^{5} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-c*x**3*sqrt(d - e*x)*sqrt(d + e*x)/(4*e**2) - x*sqrt(d - e*x)*sqrt(d + e*x)*(4*
b*e**2 + 3*c*d**2)/(8*e**4) + sqrt(d - e*x)*sqrt(d + e*x)*(8*a*e**4 + 4*b*d**2*e
**2 + 3*c*d**4)*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e**5*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.14877, size = 99, normalized size = 0.77 \[ \frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d-e x} \sqrt{d+e x}}\right ) \left (8 a e^4+4 b d^2 e^2+3 c d^4\right )-e x \sqrt{d-e x} \sqrt{d+e x} \left (4 b e^2+3 c d^2+2 c e^2 x^2\right )}{8 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

(-(e*x*Sqrt[d - e*x]*Sqrt[d + e*x]*(3*c*d^2 + 4*b*e^2 + 2*c*e^2*x^2)) + (3*c*d^4
+ 4*b*d^2*e^2 + 8*a*e^4)*ArcTan[(e*x)/(Sqrt[d - e*x]*Sqrt[d + e*x])])/(8*e^5)

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Maple [C]  time = 0.029, size = 191, normalized size = 1.5 \[ -{\frac{{\it csgn} \left ( e \right ) }{8\,{e}^{5}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 2\,{\it csgn} \left ( e \right ){x}^{3}c{e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+4\,bx\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{e}^{3}{\it csgn} \left ( e \right ) +3\,c{d}^{2}x\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( e \right ) e-8\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) a{e}^{4}-4\,b{d}^{2}\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ){e}^{2}-3\,c{d}^{4}\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/8*(-e*x+d)^(1/2)*(e*x+d)^(1/2)*(2*csgn(e)*x^3*c*e^3*(-e^2*x^2+d^2)^(1/2)+4*b*
x*(-e^2*x^2+d^2)^(1/2)*e^3*csgn(e)+3*c*d^2*x*(-e^2*x^2+d^2)^(1/2)*csgn(e)*e-8*ar
ctan(csgn(e)*e*x/(-e^2*x^2+d^2)^(1/2))*a*e^4-4*b*d^2*arctan(csgn(e)*e*x/(-e^2*x^
2+d^2)^(1/2))*e^2-3*c*d^4*arctan(csgn(e)*e*x/(-e^2*x^2+d^2)^(1/2)))*csgn(e)/(-e^
2*x^2+d^2)^(1/2)/e^5

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Maxima [A]  time = 0.795528, size = 201, normalized size = 1.57 \[ -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{3}}{4 \, e^{2}} + \frac{a \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} + \frac{3 \, c d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{4}} + \frac{b d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x}{8 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x}{2 \, e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="maxima")

[Out]

-1/4*sqrt(-e^2*x^2 + d^2)*c*x^3/e^2 + a*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) +
3/8*c*d^4*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^4) + 1/2*b*d^2*arcsin(e^2*x/s
qrt(d^2*e^2))/(sqrt(e^2)*e^2) - 3/8*sqrt(-e^2*x^2 + d^2)*c*d^2*x/e^4 - 1/2*sqrt(
-e^2*x^2 + d^2)*b*x/e^2

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Fricas [A]  time = 0.282719, size = 571, normalized size = 4.46 \[ \frac{8 \, c d e^{7} x^{7} - 4 \,{\left (3 \, c d^{3} e^{5} - 4 \, b d e^{7}\right )} x^{5} - 4 \,{\left (5 \, c d^{5} e^{3} + 12 \, b d^{3} e^{5}\right )} x^{3} -{\left (2 \, c e^{7} x^{7} -{\left (13 \, c d^{2} e^{5} - 4 \, b e^{7}\right )} x^{5} - 8 \,{\left (c d^{4} e^{3} + 4 \, b d^{2} e^{5}\right )} x^{3} + 8 \,{\left (3 \, c d^{6} e + 4 \, b d^{4} e^{3}\right )} x\right )} \sqrt{e x + d} \sqrt{-e x + d} + 8 \,{\left (3 \, c d^{7} e + 4 \, b d^{5} e^{3}\right )} x - 2 \,{\left (24 \, c d^{8} + 32 \, b d^{6} e^{2} + 64 \, a d^{4} e^{4} +{\left (3 \, c d^{4} e^{4} + 4 \, b d^{2} e^{6} + 8 \, a e^{8}\right )} x^{4} - 8 \,{\left (3 \, c d^{6} e^{2} + 4 \, b d^{4} e^{4} + 8 \, a d^{2} e^{6}\right )} x^{2} - 4 \,{\left (6 \, c d^{7} + 8 \, b d^{5} e^{2} + 16 \, a d^{3} e^{4} -{\left (3 \, c d^{5} e^{2} + 4 \, b d^{3} e^{4} + 8 \, a d e^{6}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right )}{8 \,{\left (e^{9} x^{4} - 8 \, d^{2} e^{7} x^{2} + 8 \, d^{4} e^{5} + 4 \,{\left (d e^{7} x^{2} - 2 \, d^{3} e^{5}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="fricas")

[Out]

1/8*(8*c*d*e^7*x^7 - 4*(3*c*d^3*e^5 - 4*b*d*e^7)*x^5 - 4*(5*c*d^5*e^3 + 12*b*d^3
*e^5)*x^3 - (2*c*e^7*x^7 - (13*c*d^2*e^5 - 4*b*e^7)*x^5 - 8*(c*d^4*e^3 + 4*b*d^2
*e^5)*x^3 + 8*(3*c*d^6*e + 4*b*d^4*e^3)*x)*sqrt(e*x + d)*sqrt(-e*x + d) + 8*(3*c
*d^7*e + 4*b*d^5*e^3)*x - 2*(24*c*d^8 + 32*b*d^6*e^2 + 64*a*d^4*e^4 + (3*c*d^4*e
^4 + 4*b*d^2*e^6 + 8*a*e^8)*x^4 - 8*(3*c*d^6*e^2 + 4*b*d^4*e^4 + 8*a*d^2*e^6)*x^
2 - 4*(6*c*d^7 + 8*b*d^5*e^2 + 16*a*d^3*e^4 - (3*c*d^5*e^2 + 4*b*d^3*e^4 + 8*a*d
*e^6)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d))*arctan((sqrt(e*x + d)*sqrt(-e*x + d) -
d)/(e*x)))/(e^9*x^4 - 8*d^2*e^7*x^2 + 8*d^4*e^5 + 4*(d*e^7*x^2 - 2*d^3*e^5)*sqrt
(e*x + d)*sqrt(-e*x + d))

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Sympy [A]  time = 104.666, size = 325, normalized size = 2.54 \[ - \frac{i a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e} + \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e} - \frac{i b d^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{3}} + \frac{b d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{3}} - \frac{i c d^{4}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & - \frac{3}{2}, - \frac{3}{2}, -1, 1 \\-2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{5}} + \frac{c d^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, -2, -2, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

-I*a*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), d**
2/(e**2*x**2))/(4*pi**(3/2)*e) + a*meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), (
(-1/4, 1/4), (-1/2, 0, 0, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)
*e) - I*b*d**2*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/
4, 0, 0), ()), d**2/(e**2*x**2))/(4*pi**(3/2)*e**3) + b*d**2*meijerg(((-3/2, -5/
4, -1, -3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -1, 0)), d**2*exp_polar(-2
*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**3) - I*c*d**4*meijerg(((-7/4, -5/4), (-3/2,
-3/2, -1, 1)), ((-2, -7/4, -3/2, -5/4, -1, 0), ()), d**2/(e**2*x**2))/(4*pi**(3/
2)*e**5) + c*d**4*meijerg(((-5/2, -9/4, -2, -7/4, -3/2, 1), ()), ((-9/4, -7/4),
(-5/2, -2, -2, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**5)

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GIAC/XCAS [A]  time = 0.308596, size = 170, normalized size = 1.33 \[ \frac{1}{114688} \,{\left ({\left (5 \, c d^{3} e^{16} + 4 \, b d e^{18} -{\left (9 \, c d^{2} e^{16} + 2 \,{\left ({\left (x e + d\right )} c e^{16} - 3 \, c d e^{16}\right )}{\left (x e + d\right )} + 4 \, b e^{18}\right )}{\left (x e + d\right )}\right )} \sqrt{x e + d} \sqrt{-x e + d} + 2 \,{\left (3 \, c d^{4} e^{16} + 4 \, b d^{2} e^{18} + 8 \, a e^{20}\right )} \arcsin \left (\frac{\sqrt{2} \sqrt{x e + d}}{2 \, \sqrt{d}}\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)/(sqrt(e*x + d)*sqrt(-e*x + d)),x, algorithm="giac")

[Out]

1/114688*((5*c*d^3*e^16 + 4*b*d*e^18 - (9*c*d^2*e^16 + 2*((x*e + d)*c*e^16 - 3*c
*d*e^16)*(x*e + d) + 4*b*e^18)*(x*e + d))*sqrt(x*e + d)*sqrt(-x*e + d) + 2*(3*c*
d^4*e^16 + 4*b*d^2*e^18 + 8*a*e^20)*arcsin(1/2*sqrt(2)*sqrt(x*e + d)/sqrt(d)))*e
^(-1)