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

Optimal. Leaf size=281 \[ \frac{\sqrt{c} \left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{c} \left (-d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 2.87735, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{\sqrt{c} \left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{c} \left (-d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} a \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.19482, size = 269, normalized size = 0.96 \[ \frac{\frac{\sqrt{2} \left (\frac{c \left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{c \left (d \sqrt{b^2-4 a c}+2 a e-b d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}}-2 \sqrt{d} \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )+2 \sqrt{d} \log (x)}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.034, size = 294, normalized size = 1.1 \[ -{\frac{1}{a}\sqrt{d}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{e{x}^{2}+d} \right ) } \right ) }+{\frac{1}{2\,a}\sqrt{e{x}^{2}+d}}+{\frac{x}{2\,a}\sqrt{e}}-{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{6}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{4}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){{\it \_Z}}^{2}+c{d}^{4} \right ) }{\frac{{{\it \_R}}^{6}cd+ \left ( -4\,a{e}^{2}+4\,bde-3\,c{d}^{2} \right ){{\it \_R}}^{4}+d \left ( 4\,a{e}^{2}-4\,bde+3\,c{d}^{2} \right ){{\it \_R}}^{2}-c{d}^{4}}{{{\it \_R}}^{7}c+3\,{{\it \_R}}^{5}be-3\,{{\it \_R}}^{5}cd+8\,{{\it \_R}}^{3}a{e}^{2}-4\,{{\it \_R}}^{3}bde+3\,{{\it \_R}}^{3}c{d}^{2}+{\it \_R}\,b{d}^{2}e-{\it \_R}\,c{d}^{3}}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e}-{\it \_R} \right ) }}-{\frac{d}{2\,a} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a*d^(1/2)*ln((2*d+2*d^(1/2)*(e*x^2+d)^(1/2))/x)+1/2/a*(e*x^2+d)^(1/2)+1/2/a*x
*e^(1/2)-1/4/a*sum((_R^6*c*d+(-4*a*e^2+4*b*d*e-3*c*d^2)*_R^4+d*(4*a*e^2-4*b*d*e+
3*c*d^2)*_R^2-c*d^4)/(_R^7*c+3*_R^5*b*e-3*_R^5*c*d+8*_R^3*a*e^2-4*_R^3*b*d*e+3*_
R^3*c*d^2+_R*b*d^2*e-_R*c*d^3)*ln((e*x^2+d)^(1/2)-x*e^(1/2)-_R),_R=RootOf(c*_Z^8
+(4*b*e-4*c*d)*_Z^6+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^4+(4*b*d^2*e-4*c*d^3)*_Z^2+c*d
^4))-1/2/a*d/((e*x^2+d)^(1/2)-x*e^(1/2))

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

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 92.386, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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),x, algorithm="fricas")

[Out]

[-1/4*(sqrt(1/2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^
2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-(2*
b^2*d^2 - 4*a*b*d*e + 2*a^2*e^2 + (b^2*d*e - a*b*e^2)*x^2 + 4*sqrt(1/2)*(a^3*b^2
- 4*a^4*c)*sqrt(e*x^2 + d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^
5*c))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b
*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - ((a^2*b^2 - 4*a^3*c
)*e*x^2 + 2*(a^2*b^2 - 4*a^3*c)*d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2
- 4*a^5*c)))/x^2) - sqrt(1/2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a
^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^
3*c))*log(-(2*b^2*d^2 - 4*a*b*d*e + 2*a^2*e^2 + (b^2*d*e - a*b*e^2)*x^2 - 4*sqrt
(1/2)*(a^3*b^2 - 4*a^4*c)*sqrt(e*x^2 + d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(
a^4*b^2 - 4*a^5*c))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b
^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - ((a^2
*b^2 - 4*a^3*c)*e*x^2 + 2*(a^2*b^2 - 4*a^3*c)*d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2
*e^2)/(a^4*b^2 - 4*a^5*c)))/x^2) - sqrt(1/2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d -
(a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(
a^2*b^2 - 4*a^3*c))*log(-(2*b^2*d^2 - 4*a*b*d*e + 2*a^2*e^2 + (b^2*d*e - a*b*e^2
)*x^2 + 4*sqrt(1/2)*(a^3*b^2 - 4*a^4*c)*sqrt(e*x^2 + d)*sqrt((b^2*d^2 - 2*a*b*d*
e + a^2*e^2)/(a^4*b^2 - 4*a^5*c))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*
a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a
^3*c)) + ((a^2*b^2 - 4*a^3*c)*e*x^2 + 2*(a^2*b^2 - 4*a^3*c)*d)*sqrt((b^2*d^2 - 2
*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/x^2) + sqrt(1/2)*a*sqrt(-(a*b*e - (b^2
- 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2
- 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-(2*b^2*d^2 - 4*a*b*d*e + 2*a^2*e^2 + (b^2
*d*e - a*b*e^2)*x^2 - 4*sqrt(1/2)*(a^3*b^2 - 4*a^4*c)*sqrt(e*x^2 + d)*sqrt((b^2*
d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d -
(a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/
(a^2*b^2 - 4*a^3*c)) + ((a^2*b^2 - 4*a^3*c)*e*x^2 + 2*(a^2*b^2 - 4*a^3*c)*d)*sqr
t((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/x^2) - 2*sqrt(d)*log(-(e
*x^2 - 2*sqrt(e*x^2 + d)*sqrt(d) + 2*d)/x^2))/a, -1/4*(sqrt(1/2)*a*sqrt(-(a*b*e
- (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^
4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-(2*b^2*d^2 - 4*a*b*d*e + 2*a^2*e^2
+ (b^2*d*e - a*b*e^2)*x^2 + 4*sqrt(1/2)*(a^3*b^2 - 4*a^4*c)*sqrt(e*x^2 + d)*sqrt
((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c))*sqrt(-(a*b*e - (b^2 - 2*a*
c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5
*c)))/(a^2*b^2 - 4*a^3*c)) - ((a^2*b^2 - 4*a^3*c)*e*x^2 + 2*(a^2*b^2 - 4*a^3*c)*
d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/x^2) - sqrt(1/2)*a
*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e
+ a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-(2*b^2*d^2 - 4*a*b*d*
e + 2*a^2*e^2 + (b^2*d*e - a*b*e^2)*x^2 - 4*sqrt(1/2)*(a^3*b^2 - 4*a^4*c)*sqrt(e
*x^2 + d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c))*sqrt(-(a*b*e
- (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a
^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - ((a^2*b^2 - 4*a^3*c)*e*x^2 + 2*(a^2*b
^2 - 4*a^3*c)*d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/x^2)
- sqrt(1/2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^
2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-(2*b^2*
d^2 - 4*a*b*d*e + 2*a^2*e^2 + (b^2*d*e - a*b*e^2)*x^2 + 4*sqrt(1/2)*(a^3*b^2 - 4
*a^4*c)*sqrt(e*x^2 + d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)
)*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e
+ a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) + ((a^2*b^2 - 4*a^3*c)*e*
x^2 + 2*(a^2*b^2 - 4*a^3*c)*d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4
*a^5*c)))/x^2) + sqrt(1/2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c
)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)
)*log(-(2*b^2*d^2 - 4*a*b*d*e + 2*a^2*e^2 + (b^2*d*e - a*b*e^2)*x^2 - 4*sqrt(1/2
)*(a^3*b^2 - 4*a^4*c)*sqrt(e*x^2 + d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*
b^2 - 4*a^5*c))*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d
^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) + ((a^2*b^2
- 4*a^3*c)*e*x^2 + 2*(a^2*b^2 - 4*a^3*c)*d)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2
)/(a^4*b^2 - 4*a^5*c)))/x^2) + 4*sqrt(-d)*arctan(d/(sqrt(e*x^2 + d)*sqrt(-d))))/
a]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(d + e*x**2)/(x*(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),x, algorithm="giac")

[Out]

Timed out