3.925 \(\int \frac{\sqrt{a+b x^2+c x^4}}{x^3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{\sqrt{a+b x^2+c x^4}}{2 x^2}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{a}}+\frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.281461, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\sqrt{a+b x^2+c x^4}}{2 x^2}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{a}}+\frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2 + c*x^4]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 23.3574, size = 99, normalized size = 0.88 \[ \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2} - \frac{\sqrt{a + b x^{2} + c x^{4}}}{2 x^{2}} - \frac{b \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/2 - sqrt(a +
b*x**2 + c*x**4)/(2*x**2) - b*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 +
c*x**4)))/(4*sqrt(a))

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Mathematica [A]  time = 0.257574, size = 116, normalized size = 1.04 \[ \frac{1}{2} \left (-\frac{\sqrt{a+b x^2+c x^4}}{x^2}-\frac{b \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )}{2 \sqrt{a}}+\sqrt{c} \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )+\frac{b \log (x)}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2 + c*x^4]/x^3,x]

[Out]

(-(Sqrt[a + b*x^2 + c*x^4]/x^2) + (b*Log[x])/Sqrt[a] - (b*Log[2*a + b*x^2 + 2*Sq
rt[a]*Sqrt[a + b*x^2 + c*x^4]])/(2*Sqrt[a]) + Sqrt[c]*Log[b + 2*c*x^2 + 2*Sqrt[c
]*Sqrt[a + b*x^2 + c*x^4]])/2

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Maple [A]  time = 0.015, size = 140, normalized size = 1.3 \[ -{\frac{1}{2\,a{x}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b}{2\,a}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{b}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{c{x}^{2}}{2\,a}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{1}{2}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(sqrt(c*x^4 + b*x^2 + a)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.315703, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{a} \sqrt{c} x^{2} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) + b x^{2} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) - 4 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{8 \, \sqrt{a} x^{2}}, \frac{4 \, \sqrt{a} \sqrt{-c} x^{2} \arctan \left (\frac{2 \, c x^{2} + b}{2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c}}\right ) + b x^{2} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) - 4 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{8 \, \sqrt{a} x^{2}}, -\frac{b x^{2} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - \sqrt{-a} \sqrt{c} x^{2} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 4 \, a c\right ) + 2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{4 \, \sqrt{-a} x^{2}}, -\frac{b x^{2} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \, \sqrt{-a} \sqrt{-c} x^{2} \arctan \left (\frac{2 \, c x^{2} + b}{2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{4 \, \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)/x^3,x, algorithm="fricas")

[Out]

[1/8*(2*sqrt(a)*sqrt(c)*x^2*log(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x^4 + b*
x^2 + a)*(2*c*x^2 + b)*sqrt(c) - 4*a*c) + b*x^2*log((4*sqrt(c*x^4 + b*x^2 + a)*(
a*b*x^2 + 2*a^2) - ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqrt(a))/x^4) - 4*sqr
t(c*x^4 + b*x^2 + a)*sqrt(a))/(sqrt(a)*x^2), 1/8*(4*sqrt(a)*sqrt(-c)*x^2*arctan(
1/2*(2*c*x^2 + b)/(sqrt(c*x^4 + b*x^2 + a)*sqrt(-c))) + b*x^2*log((4*sqrt(c*x^4
+ b*x^2 + a)*(a*b*x^2 + 2*a^2) - ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqrt(a)
)/x^4) - 4*sqrt(c*x^4 + b*x^2 + a)*sqrt(a))/(sqrt(a)*x^2), -1/4*(b*x^2*arctan(1/
2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) - sqrt(-a)*sqrt(c)*x^2*log
(-8*c^2*x^4 - 8*b*c*x^2 - b^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c)
- 4*a*c) + 2*sqrt(c*x^4 + b*x^2 + a)*sqrt(-a))/(sqrt(-a)*x^2), -1/4*(b*x^2*arcta
n(1/2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) - 2*sqrt(-a)*sqrt(-c)*
x^2*arctan(1/2*(2*c*x^2 + b)/(sqrt(c*x^4 + b*x^2 + a)*sqrt(-c))) + 2*sqrt(c*x^4
+ b*x^2 + a)*sqrt(-a))/(sqrt(-a)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2} + c x^{4}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*x**2 + c*x**4)/x**3, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2} + a}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)/x^3,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)/x^3, x)