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

Optimal. Leaf size=93 \[ -\frac{a \tanh ^{-1}\left (\frac{\sqrt{d-e x} \sqrt{d+e x}}{d}\right )}{d}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (b e^2+c d^2\right )}{e^4}+\frac{c (d-e x)^{3/2} (d+e x)^{3/2}}{3 e^4} \]

[Out]

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

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Rubi [A]  time = 0.482115, antiderivative size = 151, normalized size of antiderivative = 1.62, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{a \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (b e^2+c d^2\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.8527, size = 114, normalized size = 1.23 \[ - \frac{a \sqrt{d - e x} \sqrt{d + e x} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d \sqrt{d^{2} - e^{2} x^{2}}} + \frac{c \sqrt{d - e x} \sqrt{d + e x} \left (d^{2} - e^{2} x^{2}\right )}{3 e^{4}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (b e^{2} + c d^{2}\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.209577, size = 84, normalized size = 0.9 \[ -\frac{a \log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right )}{d}+\frac{a \log (x)}{d}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (3 b e^2+2 c d^2+c e^2 x^2\right )}{3 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.048, size = 143, normalized size = 1.5 \[ -{\frac{{\it csgn} \left ( d \right ) }{3\,d{e}^{4}}\sqrt{-ex+d}\sqrt{ex+d} \left ({\it csgn} \left ( d \right ){x}^{2}cd{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+3\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ) a{e}^{4}+3\,{\it csgn} \left ( d \right ) \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}bd{e}^{2}+2\,{\it csgn} \left ( d \right ) \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}c{d}^{3} \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x/(-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*(csgn(d)*x^2*c*d*e^2*(-e^2*x^2+d^2)^(1/2)+3*
ln(2*d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+d)/x)*a*e^4+3*csgn(d)*(-e^2*x^2+d^2)^(1/2)*
b*d*e^2+2*csgn(d)*(-e^2*x^2+d^2)^(1/2)*c*d^3)*csgn(d)/(-e^2*x^2+d^2)^(1/2)/e^4

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279206, size = 270, normalized size = 2.9 \[ -\frac{c d e^{2} x^{6} - 6 \, b d^{3} x^{2} - 3 \,{\left (c d^{3} - b d e^{2}\right )} x^{4} + 3 \,{\left (c d^{2} x^{4} + 2 \, b d^{2} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d} - 3 \,{\left (3 \, a d e^{2} x^{2} - 4 \, a d^{3} -{\left (a e^{2} x^{2} - 4 \, a d^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )} \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right )}{3 \,{\left (3 \, d^{2} e^{2} x^{2} - 4 \, d^{4} -{\left (d e^{2} x^{2} - 4 \, d^{3}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

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),x, algorithm="fricas")

[Out]

-1/3*(c*d*e^2*x^6 - 6*b*d^3*x^2 - 3*(c*d^3 - b*d*e^2)*x^4 + 3*(c*d^2*x^4 + 2*b*d
^2*x^2)*sqrt(e*x + d)*sqrt(-e*x + d) - 3*(3*a*d*e^2*x^2 - 4*a*d^3 - (a*e^2*x^2 -
4*a*d^2)*sqrt(e*x + d)*sqrt(-e*x + d))*log((sqrt(e*x + d)*sqrt(-e*x + d) - d)/x
))/(3*d^2*e^2*x^2 - 4*d^4 - (d*e^2*x^2 - 4*d^3)*sqrt(e*x + d)*sqrt(-e*x + d))

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Sympy [A]  time = 101.102, size = 304, normalized size = 3.27 \[ \frac{i a{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i b d{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{2}} - \frac{b d{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{2}} - \frac{i c d^{3}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{4}} - \frac{c d^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.670355, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

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),x, algorithm="giac")

[Out]

sage0*x