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)

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Rubi [A]  time = 0.0058113, 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, Rules used = {2000} \[ \frac{\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]

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)

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin{align*} \int \sqrt{b x^2+c x^4} \, dx &=\frac{\left (b x^2+c x^4\right )^{3/2}}{3 c x^3}\\ \end{align*}

Mathematica [A]  time = 0.0053323, 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)

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Maple [A]  time = 0.044, size = 29, normalized size = 1.2 \begin{align*}{\frac{c{x}^{2}+b}{3\,cx}\sqrt{c{x}^{4}+b{x}^{2}}} \end{align*}

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)

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Maxima [A]  time = 0.984779, size = 19, normalized size = 0.76 \begin{align*} \frac{{\left (c x^{2} + b\right )}^{\frac{3}{2}}}{3 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*(c*x^2 + b)^(3/2)/c

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Fricas [A]  time = 1.51192, size = 58, normalized size = 2.32 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}{3 \, c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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

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)

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Giac [A]  time = 1.30426, size = 36, normalized size = 1.44 \begin{align*} \frac{{\left (c x^{2} + b\right )}^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, c} - \frac{b^{\frac{3}{2}} \mathrm{sgn}\left (x\right )}{3 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

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