3.961 \(\int \frac{1}{x^5 \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{5/2}}+\frac{3 b \sqrt{a+b x^2+c x^4}}{8 a^2 x^2}-\frac{\sqrt{a+b x^2+c x^4}}{4 a x^4} \]

[Out]

-Sqrt[a + b*x^2 + c*x^4]/(4*a*x^4) + (3*b*Sqrt[a + b*x^2 + c*x^4])/(8*a^2*x^2) -
((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(1
6*a^(5/2))

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Rubi [A]  time = 0.239125, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{5/2}}+\frac{3 b \sqrt{a+b x^2+c x^4}}{8 a^2 x^2}-\frac{\sqrt{a+b x^2+c x^4}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*x^2 + c*x^4]/(4*a*x^4) + (3*b*Sqrt[a + b*x^2 + c*x^4])/(8*a^2*x^2) -
((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(1
6*a^(5/2))

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Rubi in Sympy [A]  time = 22.2489, size = 97, normalized size = 0.9 \[ - \frac{\sqrt{a + b x^{2} + c x^{4}}}{4 a x^{4}} + \frac{3 b \sqrt{a + b x^{2} + c x^{4}}}{8 a^{2} x^{2}} - \frac{\left (- 4 a c + 3 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x**2 + c*x**4)/(4*a*x**4) + 3*b*sqrt(a + b*x**2 + c*x**4)/(8*a**2*x*
*2) - (-4*a*c + 3*b**2)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 + c*x**4
)))/(16*a**(5/2))

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Mathematica [A]  time = 0.252742, size = 96, normalized size = 0.89 \[ \frac{\left (3 b^2-4 a c\right ) \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{16 a^{5/2}}+\frac{\left (3 b x^2-2 a\right ) \sqrt{a+b x^2+c x^4}}{8 a^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

((-2*a + 3*b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(8*a^2*x^4) + ((3*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^(5/2))

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Maple [A]  time = 0.018, size = 127, normalized size = 1.2 \[ -{\frac{1}{4\,a{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,b}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{c}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302769, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b^{2} - 4 \, a c\right )} x^{4} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) - 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (3 \, b x^{2} - 2 \, a\right )} \sqrt{a}}{32 \, a^{\frac{5}{2}} x^{4}}, -\frac{{\left (3 \, b^{2} - 4 \, a c\right )} x^{4} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (3 \, b x^{2} - 2 \, a\right )} \sqrt{-a}}{16 \, \sqrt{-a} a^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 + a)*x^5),x, algorithm="fricas")

[Out]

[-1/32*((3*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
)*(3*b*x^2 - 2*a)*sqrt(a))/(a^(5/2)*x^4), -1/16*((3*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)*
(3*b*x^2 - 2*a)*sqrt(-a))/(sqrt(-a)*a^2*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**5*sqrt(a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2} + a} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 + a)*x^5),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2 + a)*x^5), x)