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

Optimal. Leaf size=119 \[ \frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]

[Out]

((b + 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*c*x) - ((b^2 - 4*a*c)*x*Sqrt[a + b*
x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2)*Sq
rt[a*x^2 + b*x^3 + c*x^4])

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Rubi [A]  time = 0.131767, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 c x}-\frac{x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

((b + 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*c*x) - ((b^2 - 4*a*c)*x*Sqrt[a + b*
x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2)*Sq
rt[a*x^2 + b*x^3 + c*x^4])

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Rubi in Sympy [A]  time = 19.825, size = 107, normalized size = 0.9 \[ \frac{\left (b + 2 c x\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{4 c x} - \frac{x \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

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,x)

[Out]

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

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Mathematica [A]  time = 0.236237, size = 100, normalized size = 0.84 \[ \frac{x \left (2 \sqrt{c} (b+2 c x) (a+x (b+c x))-\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{8 c^{3/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 146, normalized size = 1.2 \[{\frac{1}{8\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 4\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x+2\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}b+4\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{c}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b \right ){\frac{1}{\sqrt{c}}}} \right ){b}^{2}c \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(c*x^4+b*x^3+a*x^2)^(1/2)*(4*(c*x^2+b*x+a)^(1/2)*c^(5/2)*x+2*(c*x^2+b*x+a)^(
1/2)*c^(3/2)*b+4*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a*c^2-l
n(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*b^2*c)/(c*x^2+b*x+a)^(1/2
)/c^(5/2)/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(sqrt(c*x^4 + b*x^3 + a*x^2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285538, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{c} x \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} +{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )}}{16 \, c^{2} x}, \frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )}}{8 \, c^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*((b^2 - 4*a*c)*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) - 4*sqrt(c*x^4 + b*x
^3 + a*x^2)*(2*c^2*x + b*c))/(c^2*x), 1/8*((b^2 - 4*a*c)*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)*(2*c^2*x + b*c))/(c^2*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.294393, size = 169, normalized size = 1.42 \[ \frac{1}{8} \,{\left (2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, x + \frac{b}{c}\right )} + \frac{{\left (b^{2} - 4 \, a c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}}\right )}{\rm sign}\left (x\right ) - \frac{{\left (b^{2}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 4 \, a c{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 2 \, \sqrt{a} b \sqrt{c}\right )}{\rm sign}\left (x\right )}{8 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(2*sqrt(c*x^2 + b*x + a)*(2*x + b/c) + (b^2 - 4*a*c)*ln(abs(-2*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(3/2))*sign(x) - 1/8*(b^2*ln(abs(-b + 2*s
qrt(a)*sqrt(c))) - 4*a*c*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 2*sqrt(a)*b*sqrt(c))*
sign(x)/c^(3/2)