3.225 \(\int \frac{1}{x^4 \sqrt{a+b x^3+c x^6}} \, dx\)

Optimal. Leaf size=72 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{6 a^{3/2}}-\frac{\sqrt{a+b x^3+c x^6}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.135255, antiderivative size = 72, 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 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{6 a^{3/2}}-\frac{\sqrt{a+b x^3+c x^6}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.8378, size = 61, normalized size = 0.85 \[ - \frac{\sqrt{a + b x^{3} + c x^{6}}}{3 a x^{3}} + \frac{b \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{6 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

-sqrt(a + b*x**3 + c*x**6)/(3*a*x**3) + b*atanh((2*a + b*x**3)/(2*sqrt(a)*sqrt(a
+ b*x**3 + c*x**6)))/(6*a**(3/2))

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Mathematica [A]  time = 0.133368, size = 78, normalized size = 1.08 \[ -\frac{b \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+x^3 \left (b+c x^3\right )}+2 a+b x^3\right )\right )}{6 a^{3/2}}-\frac{\sqrt{a+b x^3+c x^6}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

-Sqrt[a + b*x^3 + c*x^6]/(3*a*x^3) - (b*(Log[x^3] - Log[2*a + b*x^3 + 2*Sqrt[a]*
Sqrt[a + x^3*(b + c*x^3)]]))/(6*a^(3/2))

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(1/x^4/(c*x^6+b*x^3+a)^(1/2),x)

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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^6 + b*x^3 + a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286317, size = 1, normalized size = 0.01 \[ \left [\frac{b x^{3} \log \left (-\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) - 4 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{a}}{12 \, a^{\frac{3}{2}} x^{3}}, \frac{b x^{3} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a}}{6 \, \sqrt{-a} a x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^4),x, algorithm="fricas")

[Out]

[1/12*(b*x^3*log(-(4*sqrt(c*x^6 + b*x^3 + a)*(a*b*x^3 + 2*a^2) + ((b^2 + 4*a*c)*
x^6 + 8*a*b*x^3 + 8*a^2)*sqrt(a))/x^6) - 4*sqrt(c*x^6 + b*x^3 + a)*sqrt(a))/(a^(
3/2)*x^3), 1/6*(b*x^3*arctan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a)
*a)) - 2*sqrt(c*x^6 + b*x^3 + a)*sqrt(-a))/(sqrt(-a)*a*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{a + b x^{3} + c x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(1/(x**4*sqrt(a + b*x**3 + c*x**6)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{6} + b x^{3} + a} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^4),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^4), x)