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

Optimal. Leaf size=157 \[ -\frac{\left (d^2-e^2 x^2\right ) \left (2 a e^2+3 b d^2\right )}{3 d^4 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{3 d^2 x^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e \sqrt{d-e x} \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.417943, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{\left (d^2-e^2 x^2\right ) \left (2 a e^2+3 b d^2\right )}{3 d^4 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{3 d^2 x^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.8509, size = 133, normalized size = 0.85 \[ - \frac{a \sqrt{d - e x} \sqrt{d + e x}}{3 d^{2} x^{3}} - \frac{2 a e^{2} \sqrt{d - e x} \sqrt{d + e x}}{3 d^{4} x} - \frac{b \sqrt{d - e x} \sqrt{d + e x}}{d^{2} x} + \frac{c \sqrt{d - e x} \sqrt{d + e x} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.181367, size = 84, normalized size = 0.54 \[ \sqrt{d-e x} \sqrt{d+e x} \left (\frac{-2 a e^2-3 b d^2}{3 d^4 x}-\frac{a}{3 d^2 x^3}\right )+\frac{c \tan ^{-1}\left (\frac{e x}{\sqrt{d-e x} \sqrt{d+e x}}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.031, size = 146, normalized size = 0.9 \[ -{\frac{{\it csgn} \left ( e \right ) }{3\,{d}^{4}{x}^{3}e}\sqrt{-ex+d}\sqrt{ex+d} \left ( -3\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ){x}^{3}c{d}^{4}+2\,\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}a{e}^{3}{x}^{2}{\it csgn} \left ( e \right ) +3\,\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}b{d}^{2}{x}^{2}{\it csgn} \left ( e \right ) e+a\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{d}^{2}{\it csgn} \left ( e \right ) e \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288593, size = 396, normalized size = 2.52 \[ -\frac{4 \, a d^{6} e +{\left (3 \, b d^{2} e^{5} + 2 \, a e^{7}\right )} x^{6} - 3 \,{\left (5 \, b d^{4} e^{3} + 3 \, a d^{2} e^{5}\right )} x^{4} + 3 \,{\left (4 \, b d^{6} e + a d^{4} e^{3}\right )} x^{2} -{\left (4 \, a d^{5} e - 3 \,{\left (3 \, b d^{3} e^{3} + 2 \, a d e^{5}\right )} x^{4} +{\left (12 \, b d^{5} e + 5 \, a d^{3} e^{3}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d} + 6 \,{\left (3 \, c d^{5} e^{2} x^{5} - 4 \, c d^{7} x^{3} -{\left (c d^{4} e^{2} x^{5} - 4 \, c d^{6} x^{3}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right )}{3 \,{\left (3 \, d^{5} e^{3} x^{5} - 4 \, d^{7} e x^{3} -{\left (d^{4} e^{3} x^{5} - 4 \, d^{6} e x^{3}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

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^4),x, algorithm="fricas")

[Out]

-1/3*(4*a*d^6*e + (3*b*d^2*e^5 + 2*a*e^7)*x^6 - 3*(5*b*d^4*e^3 + 3*a*d^2*e^5)*x^
4 + 3*(4*b*d^6*e + a*d^4*e^3)*x^2 - (4*a*d^5*e - 3*(3*b*d^3*e^3 + 2*a*d*e^5)*x^4
+ (12*b*d^5*e + 5*a*d^3*e^3)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d) + 6*(3*c*d^5*e^2
*x^5 - 4*c*d^7*x^3 - (c*d^4*e^2*x^5 - 4*c*d^6*x^3)*sqrt(e*x + d)*sqrt(-e*x + d))
*arctan((sqrt(e*x + d)*sqrt(-e*x + d) - d)/(e*x)))/(3*d^5*e^3*x^5 - 4*d^7*e*x^3
- (d^4*e^3*x^5 - 4*d^6*e*x^3)*sqrt(e*x + d)*sqrt(-e*x + d))

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Sympy [A]  time = 167.912, size = 257, normalized size = 1.64 \[ \frac{i a e^{3}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{9}{4}, \frac{11}{4}, 1 & \frac{5}{2}, \frac{5}{2}, 3 \\2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3 & 0 \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} + \frac{a e^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, 1 & \\\frac{7}{4}, \frac{9}{4} & \frac{3}{2}, 2, 2, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} + \frac{i b e{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{b e{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e} + \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*a*e**3*meijerg(((9/4, 11/4, 1), (5/2, 5/2, 3)), ((2, 9/4, 5/2, 11/4, 3), (0,))
, d**2/(e**2*x**2))/(4*pi**(3/2)*d**4) + a*e**3*meijerg(((3/2, 7/4, 2, 9/4, 5/2,
1), ()), ((7/4, 9/4), (3/2, 2, 2, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*
pi**(3/2)*d**4) + I*b*e*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7
/4, 2), (0,)), d**2/(e**2*x**2))/(4*pi**(3/2)*d**2) + b*e*meijerg(((1/2, 3/4, 1,
5/4, 3/2, 1), ()), ((3/4, 5/4), (1/2, 1, 1, 0)), d**2*exp_polar(-2*I*pi)/(e**2*
x**2))/(4*pi**(3/2)*d**2) - I*c*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) + c*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)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError