3.136 \(\int \frac{a+b x^2+c x^4}{x^3 \sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{\left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d-e x} \sqrt{d+e x}}{d}\right )}{2 d^3}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{2 d^2 x^2}-\frac{c \sqrt{d-e x} \sqrt{d+e x}}{e^2} \]

[Out]

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

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Rubi [A]  time = 0.55525, antiderivative size = 155, normalized size of antiderivative = 1.57, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.2349, size = 114, normalized size = 1.15 \[ - \frac{a \sqrt{d - e x} \sqrt{d + e x}}{2 d^{2} x^{2}} - \frac{c \sqrt{d - e x} \sqrt{d + e x}}{e^{2}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (a e^{2} + 2 b d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{3} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**3/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.272164, size = 102, normalized size = 1.03 \[ \frac{\log (x) \left (a e^2+2 b d^2\right )}{2 d^3}-\frac{\left (a e^2+2 b d^2\right ) \log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right )}{2 d^3}+\sqrt{d-e x} \sqrt{d+e x} \left (-\frac{a}{2 d^2 x^2}-\frac{c}{e^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.032, size = 163, normalized size = 1.7 \[ -{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{3}{x}^{2}{e}^{2}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 2\,{\it csgn} \left ( d \right ){x}^{2}c{d}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{2}a{e}^{4}+2\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{2}b{d}^{2}{e}^{2}+{\it csgn} \left ( d \right ) ad{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265945, size = 378, normalized size = 3.82 \[ -\frac{2 \, c d^{3} e^{2} x^{6} - 5 \, a d^{3} e^{2} x^{2} + 4 \, a d^{5} -{\left (4 \, c d^{5} - a d e^{4}\right )} x^{4} +{\left (4 \, c d^{4} x^{4} + 3 \, a d^{2} e^{2} x^{2} - 4 \, a d^{4}\right )} \sqrt{e x + d} \sqrt{-e x + d} -{\left (3 \,{\left (2 \, b d^{3} e^{2} + a d e^{4}\right )} x^{4} - 4 \,{\left (2 \, b d^{5} + a d^{3} e^{2}\right )} x^{2} -{\left ({\left (2 \, b d^{2} e^{2} + a e^{4}\right )} x^{4} - 4 \,{\left (2 \, b d^{4} + a d^{2} e^{2}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )} \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right )}{2 \,{\left (3 \, d^{4} e^{2} x^{4} - 4 \, d^{6} x^{2} -{\left (d^{3} e^{2} x^{4} - 4 \, d^{5} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

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^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 144.224, size = 270, normalized size = 2.73 \[ \frac{i a e^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{a e^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{i b{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i c d{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{2}} - \frac{c d{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)/x**3/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.684055, 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^3),x, algorithm="giac")

[Out]

sage0*x