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

Optimal. Leaf size=88 \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{24 a^{3/2}}-\frac{\left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{12 a x^6} \]

[Out]

-((2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(12*a*x^6) + ((b^2 - 4*a*c)*ArcTanh[(2*
a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(24*a^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.154004, antiderivative size = 88, 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 \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{24 a^{3/2}}-\frac{\left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{12 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

-((2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(12*a*x^6) + ((b^2 - 4*a*c)*ArcTanh[(2*
a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(24*a^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.8226, size = 76, normalized size = 0.86 \[ - \frac{\left (2 a + b x^{3}\right ) \sqrt{a + b x^{3} + c x^{6}}}{12 a x^{6}} + \frac{\left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{24 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a + b*x**3)*sqrt(a + b*x**3 + c*x**6)/(12*a*x**6) + (-4*a*c + b**2)*atanh((2
*a + b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/(24*a**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.224819, size = 93, normalized size = 1.06 \[ -\frac{\left (b^2-4 a c\right ) \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )}{24 a^{3/2}}-\frac{\left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{12 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

-((2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(12*a*x^6) - ((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^(3/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.278603, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 4 \, a c\right )} x^{6} \log \left (\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) + 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{a}}{48 \, a^{\frac{3}{2}} x^{6}}, \frac{{\left (b^{2} - 4 \, a c\right )} x^{6} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{24 \, \sqrt{-a} a x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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