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

Optimal. Leaf size=83 \[ \frac{\left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 c}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 c^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.121603, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 c}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.08482, size = 73, normalized size = 0.88 \[ \frac{\left (b + 2 c x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{8 c} - \frac{\left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0726342, size = 81, normalized size = 0.98 \[ \frac{\left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 c}-\frac{\left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{16 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 101, normalized size = 1.2 \[{\frac{2\,c{x}^{2}+b}{8\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{a}{4}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{b}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^4+b*x^2+a)^(1/2),x)

[Out]

1/8*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/c+1/4/c^(1/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*
x^4+b*x^2+a)^(1/2))*a-1/16/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2
))*b^2

_______________________________________________________________________________________

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^2 + a)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.291955, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{c} -{\left (b^{2} - 4 \, a c\right )} \log \left (-4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{32 \, c^{\frac{3}{2}}}, \frac{2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} + b\right )} \sqrt{-c} -{\left (b^{2} - 4 \, a c\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{16 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/32*(4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) - (b^2 - 4*a*c)*log(-4*sq
rt(c*x^4 + b*x^2 + a)*(2*c^2*x^2 + b*c) - (8*c^2*x^4 + 8*b*c*x^2 + b^2 + 4*a*c)*
sqrt(c)))/c^(3/2), 1/16*(2*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(-c) - (b^2
- 4*a*c)*arctan(1/2*(2*c*x^2 + b)*sqrt(-c)/(sqrt(c*x^4 + b*x^2 + a)*c)))/(sqrt(
-c)*c)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284176, size = 103, normalized size = 1.24 \[ \frac{1}{8} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, x^{2} + \frac{b}{c}\right )} + \frac{{\left (b^{2} - 4 \, a c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(c*x^4 + b*x^2 + a)*(2*x^2 + b/c) + 1/16*(b^2 - 4*a*c)*ln(abs(-2*(sqrt(c
)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) - b))/c^(3/2)