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

Optimal. Leaf size=25 \[ \frac{\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]

[Out]

(b*x^2 + c*x^4)^(3/2)/(3*c*x^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0159563, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4],x]

[Out]

(b*x^2 + c*x^4)^(3/2)/(3*c*x^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.1575, size = 19, normalized size = 0.76 \[ \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{3 c x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x**2 + c*x**4)**(3/2)/(3*c*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00956013, size = 25, normalized size = 1. \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(b + c*x^2))^(3/2)/(3*c*x^3)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 29, normalized size = 1.2 \[{\frac{c{x}^{2}+b}{3\,cx}\sqrt{c{x}^{4}+b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.701052, size = 19, normalized size = 0.76 \[ \frac{{\left (c x^{2} + b\right )}^{\frac{3}{2}}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c*x^2 + b)^(3/2)/c

_______________________________________________________________________________________

Fricas [A]  time = 0.259146, size = 38, normalized size = 1.52 \[ \frac{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}{3 \, c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(c*x^4 + b*x^2)*(c*x^2 + b)/(c*x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(b*x**2 + c*x**4), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270496, size = 36, normalized size = 1.44 \[ \frac{{\left (c x^{2} + b\right )}^{\frac{3}{2}}{\rm sign}\left (x\right )}{3 \, c} - \frac{b^{\frac{3}{2}}{\rm sign}\left (x\right )}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c*x^2 + b)^(3/2)*sign(x)/c - 1/3*b^(3/2)*sign(x)/c