3.36 \(\int \frac{\sqrt{a x^2+b x^3+c x^4}}{x^5} \, dx\)

Optimal. Leaf size=155 \[ -\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{16 a^{5/2}}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4} \]

[Out]

-Sqrt[a*x^2 + b*x^3 + c*x^4]/(3*x^4) - (b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(12*a*x^3
) + ((3*b^2 - 8*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(24*a^2*x^2) - (b*(b^2 - 4*a*c
)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(16*a^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.412583, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{16 a^{5/2}}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^5,x]

[Out]

-Sqrt[a*x^2 + b*x^3 + c*x^4]/(3*x^4) - (b*Sqrt[a*x^2 + b*x^3 + c*x^4])/(12*a*x^3
) + ((3*b^2 - 8*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(24*a^2*x^2) - (b*(b^2 - 4*a*c
)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^3 + c*x^4])])/(16*a^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 64.5217, size = 167, normalized size = 1.08 \[ - \frac{\sqrt{a x^{2} + b x^{3} + c x^{4}}}{3 x^{4}} - \frac{b \sqrt{a x^{2} + b x^{3} + c x^{4}}}{12 a x^{3}} + \frac{\left (- 8 a c + 3 b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{24 a^{2} x^{2}} - \frac{b x \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{16 a^{\frac{5}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**3+a*x**2)**(1/2)/x**5,x)

[Out]

-sqrt(a*x**2 + b*x**3 + c*x**4)/(3*x**4) - b*sqrt(a*x**2 + b*x**3 + c*x**4)/(12*
a*x**3) + (-8*a*c + 3*b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(24*a**2*x**2) - b*x*
(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x
+ c*x**2)))/(16*a**(5/2)*sqrt(a*x**2 + b*x**3 + c*x**4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.38975, size = 145, normalized size = 0.94 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (8 a^2+2 a x (b+4 c x)-3 b^2 x^2\right )+3 b x^3 \log (x) \left (b^2-4 a c\right )-3 b x^3 \left (b^2-4 a c\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )\right )}{48 a^{5/2} x^4 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^2 + b*x^3 + c*x^4]/x^5,x]

[Out]

(Sqrt[x^2*(a + x*(b + c*x))]*(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(8*a^2 - 3*b^2*x^
2 + 2*a*x*(b + 4*c*x)) + 3*b*(b^2 - 4*a*c)*x^3*Log[x] - 3*b*(b^2 - 4*a*c)*x^3*Lo
g[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]]))/(48*a^(5/2)*x^4*Sqrt[a + x*(b +
c*x)])

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 234, normalized size = 1.5 \[ -{\frac{1}{48\,{x}^{4}{a}^{3}}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( -12\,{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) c{x}^{3}b-6\,\sqrt{c{x}^{2}+bx+a}c{x}^{4}{b}^{2}+12\,\sqrt{c{x}^{2}+bx+a}c{x}^{3}ab+3\,\sqrt{a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{3}{b}^{3}+6\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}{b}^{2}-6\,\sqrt{c{x}^{2}+bx+a}{x}^{3}{b}^{3}-12\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}xab+16\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{a}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^3+a*x^2)^(1/2)/x^5,x)

[Out]

-1/48*(c*x^4+b*x^3+a*x^2)^(1/2)*(-12*a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)
^(1/2))/x)*c*x^3*b-6*(c*x^2+b*x+a)^(1/2)*c*x^4*b^2+12*(c*x^2+b*x+a)^(1/2)*c*x^3*
a*b+3*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*x^3*b^3+6*(c*x^2+b*x
+a)^(3/2)*x^2*b^2-6*(c*x^2+b*x+a)^(1/2)*x^3*b^3-12*(c*x^2+b*x+a)^(3/2)*x*a*b+16*
(c*x^2+b*x+a)^(3/2)*a^2)/x^4/(c*x^2+b*x+a)^(1/2)/a^3

_______________________________________________________________________________________

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^4 + b*x^3 + a*x^2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.292678, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{a} x^{4} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )} +{\left (8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{3}}\right ) + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, a^{2} b x + 8 \, a^{3} -{\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{2}\right )}}{96 \, a^{3} x^{4}}, \frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{-a} x^{4} \arctan \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}\right ) - 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, a^{2} b x + 8 \, a^{3} -{\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{2}\right )}}{48 \, a^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^3 + a*x^2)/x^5,x, algorithm="fricas")

[Out]

[-1/96*(3*(b^3 - 4*a*b*c)*sqrt(a)*x^4*log(-(4*sqrt(c*x^4 + b*x^3 + a*x^2)*(a*b*x
+ 2*a^2) + (8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x)*sqrt(a))/x^3) + 4*sqrt(c*x
^4 + b*x^3 + a*x^2)*(2*a^2*b*x + 8*a^3 - (3*a*b^2 - 8*a^2*c)*x^2))/(a^3*x^4), 1/
48*(3*(b^3 - 4*a*b*c)*sqrt(-a)*x^4*arctan(1/2*(b*x^2 + 2*a*x)*sqrt(-a)/(sqrt(c*x
^4 + b*x^3 + a*x^2)*a)) - 2*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*a^2*b*x + 8*a^3 - (3*
a*b^2 - 8*a^2*c)*x^2))/(a^3*x^4)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (a + b x + c x^{2}\right )}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**3+a*x**2)**(1/2)/x**5,x)

[Out]

Integral(sqrt(x**2*(a + b*x + c*x**2))/x**5, x)

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^3 + a*x^2)/x^5,x, algorithm="giac")

[Out]

Timed out