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

Optimal. Leaf size=173 \[ \frac{\sqrt{a x^2+b x^3+c x^4}}{x}-\frac{\sqrt{a} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}+\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}} \]

[Out]

Sqrt[a*x^2 + b*x^3 + c*x^4]/x - (Sqrt[a]*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a +
b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/Sqrt[a*x^2 + b*x^3 + c*x^4] + (b*x*Sqrt
[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqr
t[c]*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi [A]  time = 0.260683, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{a x^2+b x^3+c x^4}}{x}-\frac{\sqrt{a} x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}+\frac{b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a*x^2 + b*x^3 + c*x^4]/x - (Sqrt[a]*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a +
b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/Sqrt[a*x^2 + b*x^3 + c*x^4] + (b*x*Sqrt
[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqr
t[c]*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi in Sympy [A]  time = 37.0255, size = 158, normalized size = 0.91 \[ - \frac{\sqrt{a} x \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{a x^{2} + b x^{3} + c x^{4}}} + \frac{b x \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{c} \sqrt{a x^{2} + b x^{3} + c x^{4}}} + \frac{\sqrt{a x^{2} + b x^{3} + c x^{4}}}{x} \]

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**2,x)

[Out]

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

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Mathematica [A]  time = 0.164966, size = 144, normalized size = 0.83 \[ \frac{x \sqrt{a+x (b+c x)} \left (2 \sqrt{c} \sqrt{a+x (b+c x)}-2 \sqrt{a} \sqrt{c} \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+b \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+2 \sqrt{a} \sqrt{c} \log (x)\right )}{2 \sqrt{c} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)] + 2*Sqrt[a]*Sqrt[c]*Lo
g[x] - 2*Sqrt[a]*Sqrt[c]*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]] + b*Lo
g[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(2*Sqrt[c]*Sqrt[x^2*(a + x*(b +
c*x))])

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Maple [A]  time = 0.007, size = 126, normalized size = 0.7 \[ -{\frac{1}{2\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 2\,\sqrt{a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ) \sqrt{c}-2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}-b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b \right ){\frac{1}{\sqrt{c}}}} \right ) \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312332, size = 1, normalized size = 0.01 \[ \left [\frac{b \sqrt{c} x \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} +{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) + 2 \, \sqrt{a} c x \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}{4 \, c x}, -\frac{b \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) - \sqrt{a} c x \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}{2 \, c x}, -\frac{4 \, \sqrt{-a} c x \arctan \left (\frac{b x^{2} + 2 \, a x}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} \sqrt{-a}}\right ) - b \sqrt{c} x \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} +{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}{4 \, c x}, -\frac{2 \, \sqrt{-a} c x \arctan \left (\frac{b x^{2} + 2 \, a x}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} \sqrt{-a}}\right ) + b \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) - 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}{2 \, c x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(b*sqrt(c)*x*log(-(4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c^2*x + b*c) + (8*c^2*x
^3 + 8*b*c*x^2 + (b^2 + 4*a*c)*x)*sqrt(c))/x) + 2*sqrt(a)*c*x*log(-(8*a*b*x^2 +
(b^2 + 4*a*c)*x^3 + 8*a^2*x - 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*sqrt(a))
/x^3) + 4*sqrt(c*x^4 + b*x^3 + a*x^2)*c)/(c*x), -1/2*(b*sqrt(-c)*x*arctan(1/2*(2
*c*x^2 + b*x)*sqrt(-c)/(sqrt(c*x^4 + b*x^3 + a*x^2)*c)) - sqrt(a)*c*x*log(-(8*a*
b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x - 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*
sqrt(a))/x^3) - 2*sqrt(c*x^4 + b*x^3 + a*x^2)*c)/(c*x), -1/4*(4*sqrt(-a)*c*x*arc
tan(1/2*(b*x^2 + 2*a*x)/(sqrt(c*x^4 + b*x^3 + a*x^2)*sqrt(-a))) - b*sqrt(c)*x*lo
g(-(4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c^2*x + b*c) + (8*c^2*x^3 + 8*b*c*x^2 + (b^
2 + 4*a*c)*x)*sqrt(c))/x) - 4*sqrt(c*x^4 + b*x^3 + a*x^2)*c)/(c*x), -1/2*(2*sqrt
(-a)*c*x*arctan(1/2*(b*x^2 + 2*a*x)/(sqrt(c*x^4 + b*x^3 + a*x^2)*sqrt(-a))) + b*
sqrt(-c)*x*arctan(1/2*(2*c*x^2 + b*x)*sqrt(-c)/(sqrt(c*x^4 + b*x^3 + a*x^2)*c))
- 2*sqrt(c*x^4 + b*x^3 + a*x^2)*c)/(c*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError