3.137 \(\int \frac{a+b x^2+c x^4}{x^5 \sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d-e x} \sqrt{d+e x}}{d}\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )}{8 d^5}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (3 a e^2+4 b d^2\right )}{8 d^4 x^2}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{4 d^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.62503, antiderivative size = 182, normalized size of antiderivative = 1.44, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{\sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )}{8 d^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (3 a e^2+4 b d^2\right )}{8 d^4 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{4 d^2 x^4 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)/(x^5*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 38.3906, size = 143, normalized size = 1.13 \[ - \frac{a \sqrt{d - e x} \sqrt{d + e x}}{4 d^{2} x^{4}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (3 a e^{2} + 4 b d^{2}\right )}{8 d^{4} x^{2}} - \frac{\sqrt{d - e x} \sqrt{d + e x} \left (3 a e^{4} + 4 b d^{2} e^{2} + 8 c d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d^{5} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)/x**5/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.247853, size = 135, normalized size = 1.07 \[ -\frac{x^4 \log (x) \left (-\left (3 a e^4+4 b d^2 e^2+8 c d^4\right )\right )+x^4 \log \left (\sqrt{d-e x} \sqrt{d+e x}+d\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )+d \sqrt{d-e x} \sqrt{d+e x} \left (2 a d^2+3 a e^2 x^2+4 b d^2 x^2\right )}{8 d^5 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)/(x^5*Sqrt[d - e*x]*Sqrt[d + e*x]),x]

[Out]

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

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Maple [C]  time = 0.033, size = 222, normalized size = 1.8 \[ -{\frac{{\it csgn} \left ( d \right ) }{8\,{d}^{5}{x}^{4}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 3\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{4}a{e}^{4}+4\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{4}b{d}^{2}{e}^{2}+8\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{4}c{d}^{4}+3\,{\it csgn} \left ( d \right ){x}^{2}ad{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+4\,{\it csgn} \left ( d \right ){x}^{2}b{d}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+2\,{\it csgn} \left ( d \right ) a{d}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)/x^5/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-1/8*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/d^5*(3*ln(2*d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+d)
/x)*x^4*a*e^4+4*ln(2*d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+d)/x)*x^4*b*d^2*e^2+8*ln(2*
d*((-e^2*x^2+d^2)^(1/2)*csgn(d)+d)/x)*x^4*c*d^4+3*csgn(d)*x^2*a*d*e^2*(-e^2*x^2+
d^2)^(1/2)+4*csgn(d)*x^2*b*d^3*(-e^2*x^2+d^2)^(1/2)+2*csgn(d)*a*d^3*(-e^2*x^2+d^
2)^(1/2))*csgn(d)/(-e^2*x^2+d^2)^(1/2)/x^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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303873, size = 567, normalized size = 4.5 \[ \frac{32 \, b d^{8} x^{2} + 16 \, a d^{8} + 4 \,{\left (4 \, b d^{4} e^{4} + 3 \, a d^{2} e^{6}\right )} x^{6} - 4 \,{\left (12 \, b d^{6} e^{2} + 7 \, a d^{4} e^{4}\right )} x^{4} -{\left (16 \, a d^{7} +{\left (4 \, b d^{3} e^{4} + 3 \, a d e^{6}\right )} x^{6} - 2 \,{\left (16 \, b d^{5} e^{2} + 11 \, a d^{3} e^{4}\right )} x^{4} + 8 \,{\left (4 \, b d^{7} + a d^{5} e^{2}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d} +{\left ({\left (8 \, c d^{4} e^{4} + 4 \, b d^{2} e^{6} + 3 \, a e^{8}\right )} x^{8} - 8 \,{\left (8 \, c d^{6} e^{2} + 4 \, b d^{4} e^{4} + 3 \, a d^{2} e^{6}\right )} x^{6} + 8 \,{\left (8 \, c d^{8} + 4 \, b d^{6} e^{2} + 3 \, a d^{4} e^{4}\right )} x^{4} + 4 \,{\left ({\left (8 \, c d^{5} e^{2} + 4 \, b d^{3} e^{4} + 3 \, a d e^{6}\right )} x^{6} - 2 \,{\left (8 \, c d^{7} + 4 \, b d^{5} e^{2} + 3 \, a d^{3} e^{4}\right )} x^{4}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )} \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right )}{8 \,{\left (d^{5} e^{4} x^{8} - 8 \, d^{7} e^{2} x^{6} + 8 \, d^{9} x^{4} + 4 \,{\left (d^{6} e^{2} x^{6} - 2 \, d^{8} x^{4}\right )} \sqrt{e x + d} \sqrt{-e x + d}\right )}} \]

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^5),x, algorithm="fricas")

[Out]

1/8*(32*b*d^8*x^2 + 16*a*d^8 + 4*(4*b*d^4*e^4 + 3*a*d^2*e^6)*x^6 - 4*(12*b*d^6*e
^2 + 7*a*d^4*e^4)*x^4 - (16*a*d^7 + (4*b*d^3*e^4 + 3*a*d*e^6)*x^6 - 2*(16*b*d^5*
e^2 + 11*a*d^3*e^4)*x^4 + 8*(4*b*d^7 + a*d^5*e^2)*x^2)*sqrt(e*x + d)*sqrt(-e*x +
d) + ((8*c*d^4*e^4 + 4*b*d^2*e^6 + 3*a*e^8)*x^8 - 8*(8*c*d^6*e^2 + 4*b*d^4*e^4
+ 3*a*d^2*e^6)*x^6 + 8*(8*c*d^8 + 4*b*d^6*e^2 + 3*a*d^4*e^4)*x^4 + 4*((8*c*d^5*e
^2 + 4*b*d^3*e^4 + 3*a*d*e^6)*x^6 - 2*(8*c*d^7 + 4*b*d^5*e^2 + 3*a*d^3*e^4)*x^4)
*sqrt(e*x + d)*sqrt(-e*x + d))*log((sqrt(e*x + d)*sqrt(-e*x + d) - d)/x))/(d^5*e
^4*x^8 - 8*d^7*e^2*x^6 + 8*d^9*x^4 + 4*(d^6*e^2*x^6 - 2*d^8*x^4)*sqrt(e*x + d)*s
qrt(-e*x + d))

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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((c*x**4+b*x**2+a)/x**5/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.705993, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

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^5),x, algorithm="giac")

[Out]

sage0*x