3.926 \(\int \frac{\sqrt{a+b x^2+c x^4}}{x^5} \, dx\)

Optimal. Leaf size=88 \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{3/2}}-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4} \]

[Out]

-((2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(8*a*x^4) + ((b^2 - 4*a*c)*ArcTanh[(2*a
+ b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(16*a^(3/2))

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Rubi [A]  time = 0.172143, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{3/2}}-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2 + c*x^4]/x^5,x]

[Out]

-((2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(8*a*x^4) + ((b^2 - 4*a*c)*ArcTanh[(2*a
+ b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(16*a^(3/2))

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Rubi in Sympy [A]  time = 16.0541, size = 76, normalized size = 0.86 \[ - \frac{\left (2 a + b x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{8 a x^{4}} + \frac{\left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a + b*x**2)*sqrt(a + b*x**2 + c*x**4)/(8*a*x**4) + (-4*a*c + b**2)*atanh((2*
a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 + c*x**4)))/(16*a**(3/2))

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Mathematica [A]  time = 0.234385, size = 93, normalized size = 1.06 \[ -\frac{\left (b^2-4 a c\right ) \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{16 a^{3/2}}-\frac{\left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{8 a x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2 + c*x^4]/x^5,x]

[Out]

-((2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(8*a*x^4) - ((b^2 - 4*a*c)*(Log[x^2] -
Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]]))/(16*a^(3/2))

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Maple [B]  time = 0.015, size = 193, normalized size = 2.2 \[ -{\frac{1}{4\,a{x}^{4}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{8\,{a}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{bc{x}^{2}}{8\,{a}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{c}{4\,a}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{c}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/a/x^4*(c*x^4+b*x^2+a)^(3/2)+1/8*b/a^2/x^2*(c*x^4+b*x^2+a)^(3/2)-1/8*b^2/a^2
*(c*x^4+b*x^2+a)^(1/2)+1/16*b^2/a^(3/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^
(1/2))/x^2)-1/8*b/a^2*c*(c*x^4+b*x^2+a)^(1/2)*x^2+1/4*c/a*(c*x^4+b*x^2+a)^(1/2)-
1/4*c/a^(1/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295778, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 4 \, a c\right )} x^{4} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{a}}{32 \, a^{\frac{3}{2}} x^{4}}, \frac{{\left (b^{2} - 4 \, a c\right )} x^{4} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{16 \, \sqrt{-a} a x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)/x^5,x, algorithm="fricas")

[Out]

[-1/32*((b^2 - 4*a*c)*x^4*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)*(
b*x^2 + 2*a)*sqrt(a))/(a^(3/2)*x^4), 1/16*((b^2 - 4*a*c)*x^4*arctan(1/2*(b*x^2 +
2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) - 2*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 +
2*a)*sqrt(-a))/(sqrt(-a)*a*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2} + c x^{4}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**(1/2)/x**5,x)

[Out]

Integral(sqrt(a + b*x**2 + c*x**4)/x**5, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2} + a}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2 + a)/x^5,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)/x^5, x)