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

Optimal. Leaf size=112 \[ -\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{6 \sqrt{a}}+\frac{1}{3} \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.251808, 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 \[ -\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{6 \sqrt{a}}+\frac{1}{3} \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.7965, size = 99, normalized size = 0.88 \[ \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3} - \frac{\sqrt{a + b x^{3} + c x^{6}}}{3 x^{3}} - \frac{b \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{6 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.21062, size = 112, normalized size = 1. \[ \frac{1}{6} \left (-\frac{2 \sqrt{a+b x^3+c x^6}}{x^3}+\frac{b \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )}{\sqrt{a}}+2 \sqrt{c} \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}\sqrt{c{x}^{6}+b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((c*x^6+b*x^3+a)^(1/2)/x^4,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(sqrt(c*x^6 + b*x^3 + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29594, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{a} \sqrt{c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{c} - 4 \, a c\right ) + b x^{3} \log \left (\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) - 4 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{a}}{12 \, \sqrt{a} x^{3}}, \frac{4 \, \sqrt{a} \sqrt{-c} x^{3} \arctan \left (\frac{2 \, c x^{3} + b}{2 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c}}\right ) + b x^{3} \log \left (\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) - 4 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{a}}{12 \, \sqrt{a} x^{3}}, -\frac{b x^{3} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - \sqrt{-a} \sqrt{c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{c} - 4 \, a c\right ) + 2 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a}}{6 \, \sqrt{-a} x^{3}}, -\frac{b x^{3} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \, \sqrt{-a} \sqrt{-c} x^{3} \arctan \left (\frac{2 \, c x^{3} + b}{2 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c}}\right ) + 2 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a}}{6 \, \sqrt{-a} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(2*sqrt(a)*sqrt(c)*x^3*log(-8*c^2*x^6 - 8*b*c*x^3 - b^2 - 4*sqrt(c*x^6 + b
*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*a*c) + 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*sq
rt(c*x^6 + b*x^3 + a)*sqrt(a))/(sqrt(a)*x^3), 1/12*(4*sqrt(a)*sqrt(-c)*x^3*arcta
n(1/2*(2*c*x^3 + b)/(sqrt(c*x^6 + b*x^3 + a)*sqrt(-c))) + 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))/(sqrt(a)*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)) - sqrt(-a)*sqrt(c)*x^3*l
og(-8*c^2*x^6 - 8*b*c*x^3 - b^2 - 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(c
) - 4*a*c) + 2*sqrt(c*x^6 + b*x^3 + a)*sqrt(-a))/(sqrt(-a)*x^3), -1/6*(b*x^3*arc
tan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a)*a)) - 2*sqrt(-a)*sqrt(-c
)*x^3*arctan(1/2*(2*c*x^3 + b)/(sqrt(c*x^6 + b*x^3 + a)*sqrt(-c))) + 2*sqrt(c*x^
6 + b*x^3 + a)*sqrt(-a))/(sqrt(-a)*x^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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