3.226 \(\int \frac{1}{x^7 \sqrt{a+b x^3+c x^6}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{24 a^{5/2}}+\frac{b \sqrt{a+b x^3+c x^6}}{4 a^2 x^3}-\frac{\sqrt{a+b x^3+c x^6}}{6 a x^6} \]

[Out]

-Sqrt[a + b*x^3 + c*x^6]/(6*a*x^6) + (b*Sqrt[a + b*x^3 + c*x^6])/(4*a^2*x^3) - (
(3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(24*
a^(5/2))

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Rubi [A]  time = 0.24279, 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 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{24 a^{5/2}}+\frac{b \sqrt{a+b x^3+c x^6}}{4 a^2 x^3}-\frac{\sqrt{a+b x^3+c x^6}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

-Sqrt[a + b*x^3 + c*x^6]/(6*a*x^6) + (b*Sqrt[a + b*x^3 + c*x^6])/(4*a^2*x^3) - (
(3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(24*
a^(5/2))

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Rubi in Sympy [A]  time = 27.3505, size = 95, normalized size = 0.88 \[ - \frac{\sqrt{a + b x^{3} + c x^{6}}}{6 a x^{6}} + \frac{b \sqrt{a + b x^{3} + c x^{6}}}{4 a^{2} x^{3}} - \frac{\left (- 4 a c + 3 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{24 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

-sqrt(a + b*x**3 + c*x**6)/(6*a*x**6) + b*sqrt(a + b*x**3 + c*x**6)/(4*a**2*x**3
) - (-4*a*c + 3*b**2)*atanh((2*a + b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6))
)/(24*a**(5/2))

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Mathematica [A]  time = 0.305705, size = 97, normalized size = 0.9 \[ \frac{\left (3 b^2-4 a c\right ) \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )-\frac{2 \sqrt{a} \left (2 a-3 b x^3\right ) \sqrt{a+b x^3+c x^6}}{x^6}}{24 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

((-2*Sqrt[a]*(2*a - 3*b*x^3)*Sqrt[a + b*x^3 + c*x^6])/x^6 + (3*b^2 - 4*a*c)*(Log
[x^3] - Log[2*a + b*x^3 + 2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6]]))/(24*a^(5/2))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(1/x^7/(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^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280626, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b^{2} - 4 \, a c\right )} x^{6} \log \left (-\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (3 \, b x^{3} - 2 \, a\right )} \sqrt{a}}{48 \, a^{\frac{5}{2}} x^{6}}, -\frac{{\left (3 \, b^{2} - 4 \, a c\right )} x^{6} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \, \sqrt{c x^{6} + b x^{3} + a}{\left (3 \, b x^{3} - 2 \, a\right )} \sqrt{-a}}{24 \, \sqrt{-a} a^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^7),x, algorithm="fricas")

[Out]

[-1/48*((3*b^2 - 4*a*c)*x^6*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
)*(3*b*x^3 - 2*a)*sqrt(a))/(a^(5/2)*x^6), -1/24*((3*b^2 - 4*a*c)*x^6*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)*
(3*b*x^3 - 2*a)*sqrt(-a))/(sqrt(-a)*a^2*x^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{7} \sqrt{a + b x^{3} + c x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(1/(x**7*sqrt(a + b*x**3 + c*x**6)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{6} + b x^{3} + a} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^7),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^7), x)