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

Optimal. Leaf size=240 \[ \frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

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

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Rubi [A]  time = 0.925522, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 103.247, size = 219, normalized size = 0.91 \[ \frac{\sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}} \operatorname{atanh}{\left (\frac{x \sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{d + e x^{2}}} \right )}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} \operatorname{atanh}{\left (\frac{x \sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{d + e x^{2}}} \right )}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*e - 2*c*d + e*sqrt(-4*a*c + b**2))*atanh(x*sqrt(b*e - 2*c*d + e*sqrt(-4*a
*c + b**2))/(sqrt(b + sqrt(-4*a*c + b**2))*sqrt(d + e*x**2)))/(sqrt(b + sqrt(-4*
a*c + b**2))*sqrt(-4*a*c + b**2)) - sqrt(b*e - 2*c*d - e*sqrt(-4*a*c + b**2))*at
anh(x*sqrt(b*e - 2*c*d - e*sqrt(-4*a*c + b**2))/(sqrt(b - sqrt(-4*a*c + b**2))*s
qrt(d + e*x**2)))/(sqrt(b - sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.125985, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.02, size = 161, normalized size = 0.7 \[ -{\frac{1}{2}{e}^{{\frac{3}{2}}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{{{\it \_R}}^{2}+2\,{\it \_R}\,d+{d}^{2}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*e^(3/2)*sum((_R^2+2*_R*d+d^2)/(_R^3*c+3*_R^2*b*e-3*_R^2*c*d+8*_R*a*e^2-4*_R
*b*d*e+3*_R*c*d^2+b*d^2*e-c*d^3)*ln(((e*x^2+d)^(1/2)-x*e^(1/2))^2-_R),_R=RootOf(
c*_Z^4+(4*b*e-4*c*d)*_Z^3+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z
+c*d^4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.04514, size = 1330, normalized size = 5.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(d + e*x**2)/(a + b*x**2 + c*x**4), 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*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

Timed out