3.239 \(\int \frac{1}{x \left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ \frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{3 a^{3/2}} \]

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) - ArcTan
h[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])]/(3*a^(3/2))

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Rubi [A]  time = 0.171916, antiderivative size = 92, 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 \[ \frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{3 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) - ArcTan
h[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])]/(3*a^(3/2))

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Rubi in Sympy [A]  time = 22.4225, size = 83, normalized size = 0.9 \[ \frac{2 \left (- 2 a c + b^{2} + b c x^{3}\right )}{3 a \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}} - \frac{\operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*a*c + b**2 + b*c*x**3)/(3*a*(-4*a*c + b**2)*sqrt(a + b*x**3 + c*x**6)) - a
tanh((2*a + b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/(3*a**(3/2))

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Mathematica [A]  time = 0.347083, size = 96, normalized size = 1.04 \[ \frac{\frac{2 \sqrt{a} \left (-2 a c+b^2+b c x^3\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )+\log \left (x^3\right )}{3 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*(b^2 - 2*a*c + b*c*x^3))/((b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) + L
og[x^3] - Log[2*a + b*x^3 + 2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6]])/(3*a^(3/2))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x/(c*x^6+b*x^3+a)^(3/2),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/((c*x^6 + b*x^3 + a)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293785, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b c x^{3} + b^{2} - 2 \, a c\right )} \sqrt{a} +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \log \left (\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right )}{6 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x^{6} + a^{2} b^{2} - 4 \, a^{3} c +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3}\right )} \sqrt{a}}, \frac{2 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b c x^{3} + b^{2} - 2 \, a c\right )} \sqrt{-a} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right )}{3 \,{\left ({\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} x^{6} + a^{2} b^{2} - 4 \, a^{3} c +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(a + b*x**3 + c*x**6)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x),x, algorithm="giac")

[Out]

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