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

Optimal. Leaf size=163 \[ \frac{b \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt{a+b x+c x^2}}-\frac{b (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{8 c^2 x}+\frac{\left (a+b x+c x^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3 c x} \]

[Out]

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

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Rubi [A]  time = 0.124519, antiderivative size = 163, 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 \[ \frac{b \left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt{a+b x+c x^2}}-\frac{b (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{8 c^2 x}+\frac{\left (a+b x+c x^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3 c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.5548, size = 146, normalized size = 0.9 \[ - \frac{b \left (b + 2 c x\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{8 c^{2} x} + \frac{b \left (- 4 a c + b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{5}{2}} x \sqrt{a + b x + c x^{2}}} + \frac{\left (a + b x + c x^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{3 c x} \]

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)

[Out]

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

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Mathematica [A]  time = 0.332574, size = 117, normalized size = 0.72 \[ \frac{3 x \left (b^3-4 a b c\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+2 \sqrt{c} x (a+x (b+c x)) \left (8 c \left (a+c x^2\right )-3 b^2+2 b c x\right )}{48 c^{5/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 167, normalized size = 1. \[{\frac{1}{48\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 16\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}-12\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}xb-6\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}{b}^{2}-12\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) ab{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(c*x^4+b*x^3+a*x^2)^(1/2)*(16*(c*x^2+b*x+a)^(3/2)*c^(5/2)-12*(c*x^2+b*x+a)^
(1/2)*c^(5/2)*x*b-6*(c*x^2+b*x+a)^(1/2)*c^(3/2)*b^2-12*ln(1/2*(2*(c*x^2+b*x+a)^(
1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a*b*c^2+3*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2
*c*x+b)/c^(1/2))*b^3*c)/x/(c*x^2+b*x+a)^(1/2)/c^(7/2)

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27975, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{c} x \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} -{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) - 4 \,{\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{96 \, c^{3} x}, -\frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) - 2 \,{\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{48 \, c^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a x^{2} + b x^{3} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a*x**2 + b*x**3 + c*x**4), x)

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GIAC/XCAS [A]  time = 0.300165, size = 224, normalized size = 1.37 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \, x{\rm sign}\left (x\right ) + \frac{b{\rm sign}\left (x\right )}{c}\right )} x - \frac{3 \, b^{2}{\rm sign}\left (x\right ) - 8 \, a c{\rm sign}\left (x\right )}{c^{2}}\right )} - \frac{{\left (b^{3}{\rm sign}\left (x\right ) - 4 \, a b c{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} + \frac{{\left (3 \, b^{3}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 12 \, a b c{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 6 \, \sqrt{a} b^{2} \sqrt{c} - 16 \, a^{\frac{3}{2}} c^{\frac{3}{2}}\right )}{\rm sign}\left (x\right )}{48 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + b*x + a)*(2*(4*x*sign(x) + b*sign(x)/c)*x - (3*b^2*sign(x) - 8
*a*c*sign(x))/c^2) - 1/16*(b^3*sign(x) - 4*a*b*c*sign(x))*ln(abs(-2*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2) + 1/48*(3*b^3*ln(abs(-b + 2*sqrt(a
)*sqrt(c))) - 12*a*b*c*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 6*sqrt(a)*b^2*sqrt(c) -
16*a^(3/2)*c^(3/2))*sign(x)/c^(5/2)