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

Optimal. Leaf size=76 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} \sqrt{2 c d-b e}} \]

[Out]

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

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Rubi [A]  time = 0.159295, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} \sqrt{2 c d-b e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.8589, size = 66, normalized size = 0.87 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x \sqrt{b e - 2 c d}}{\sqrt{d + e x^{2}} \sqrt{b e - c d}} \right )}}{\sqrt{e} \sqrt{b e - 2 c d} \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(sqrt(e)*x*sqrt(b*e - 2*c*d)/(sqrt(d + e*x**2)*sqrt(b*e - c*d)))/(sqrt(e)*s
qrt(b*e - 2*c*d)*sqrt(b*e - c*d))

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Mathematica [A]  time = 0.061687, size = 73, normalized size = 0.96 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{b e-2 c d}}{\sqrt{d+e x^2} \sqrt{b e-c d}}\right )}{\sqrt{e} \sqrt{b e-2 c d} \sqrt{b e-c d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.031, size = 2264, normalized size = 29.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*e*c/(-d*e)^(1/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(
b*e-c*d)*c*e)^(1/2))*((x-1/e*(-d*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-d*e)^(1/2
)))^(1/2)+1/2*e^(1/2)*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-
(-(b*e-c*d)*c*e)^(1/2))*ln(((x-1/e*(-d*e)^(1/2))*e+(-d*e)^(1/2))/e^(1/2)+((x-1/e
*(-d*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-d*e)^(1/2)))^(1/2))-1/2*e*c/(-d*e)^(1
/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/
2))*((x+1/e*(-d*e)^(1/2))^2*e-2*(-d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(1/2)+1/2*e^(
1/2)*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^
(1/2))*ln(((x+1/e*(-d*e)^(1/2))*e-(-d*e)^(1/2))/e^(1/2)+((x+1/e*(-d*e)^(1/2))^2*
e-2*(-d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(1/2))-1/2*e*c^2/((-d*e)^(1/2)*c+(-(b*e-c
*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)*(
(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e
)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2)-1/2*e^(1/2)*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)
^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))*ln(((-(b*e-c*d)*c*e)^(1/2)/c+(x-
(-(b*e-c*d)*c*e)^(1/2)/c/e)*e)/e^(1/2)+((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(
b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))-1/2*e
^2*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1
/2))/(-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e
-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(
-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1
/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x-(-(b*e-c*d)*c*e)^(1/2)/c/e))*b+e*c^2/((-d*e)^(
1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*
d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/
2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*e
)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-
2*c*d)/c)^(1/2))/(x-(-(b*e-c*d)*c*e)^(1/2)/c/e))*d+1/2*e*c^2/((-d*e)^(1/2)*c+(-(
b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1
/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d
)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2)-1/2*e^(1/2)*c/((-d*e)^(1/2)*c+(-(b*e-c*d)
*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))*ln((-(-(b*e-c*d)*c*e)^(1/2)
/c+(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)*e)/e^(1/2)+((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e
-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))
+1/2*e^2*c/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c
*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c-2*
(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)
*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c
*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x+(-(b*e-c*d)*c*e)^(1/2)/c/e))*b-e*c^2/((-
d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((-d*e)^(1/2)*c-(-(b*e-c*d)*c*e)^(1/2))/(-(
b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c-2*(-(b*e-c*d)*c*
e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x+(-(b*e-c*
d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)
-(b*e-2*c*d)/c)^(1/2))/(x+(-(b*e-c*d)*c*e)^(1/2)/c/e))*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x^{2} + d}}{c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(e*x^2 + d)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e), x)

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Fricas [A]  time = 0.367445, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (\frac{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} +{\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \,{\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2}\right )} \sqrt{2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} - 4 \,{\left ({\left (6 \, c^{3} d^{3} e^{2} - 13 \, b c^{2} d^{2} e^{3} + 9 \, b^{2} c d e^{4} - 2 \, b^{3} e^{5}\right )} x^{3} +{\left (2 \, c^{3} d^{4} e - 5 \, b c^{2} d^{3} e^{2} + 4 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \sqrt{e x^{2} + d}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \,{\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, \sqrt{2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}}}, \frac{\arctan \left (-\frac{\sqrt{-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}}{\left (c d^{2} - b d e +{\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )}}{2 \,{\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt{e x^{2} + d} x}\right )}{2 \, \sqrt{-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*log(((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2*d^2*e^2 - 24*b*c*d*e^3
+ 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e - 11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2)*sqrt(2*c^
2*d^2*e - 3*b*c*d*e^2 + b^2*e^3) - 4*((6*c^3*d^3*e^2 - 13*b*c^2*d^2*e^3 + 9*b^2*
c*d*e^4 - 2*b^3*e^5)*x^3 + (2*c^3*d^4*e - 5*b*c^2*d^3*e^2 + 4*b^2*c*d^2*e^3 - b^
3*d*e^4)*x)*sqrt(e*x^2 + d))/(c^2*e^2*x^4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c
^2*d*e - b*c*e^2)*x^2))/sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3), 1/2*arctan(-1
/2*sqrt(-2*c^2*d^2*e + 3*b*c*d*e^2 - b^2*e^3)*(c*d^2 - b*d*e + (3*c*d*e - 2*b*e^
2)*x^2)/((2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*sqrt(e*x^2 + d)*x))/sqrt(-2*c^2*d
^2*e + 3*b*c*d*e^2 - b^2*e^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d + e x^{2}} \left (b e - c d + c e x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(d + e*x**2)*(b*e - c*d + c*e*x**2)), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out