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]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ \frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 c x^3} \]

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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fricas [A]  time = 0.54, size = 28, normalized size = 1.12 \[ \frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + b\right )}}{3 \, c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [A]  time = 0.15, size = 27, normalized size = 1.08 \[ \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{3 \, c} - \frac {b^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{3 \, c} \]

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

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maple [A]  time = 0.00, size = 29, normalized size = 1.16 \[ \frac {\left (c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 c x} \]

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 = 1.43, size = 14, normalized size = 0.56 \[ \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}}}{3 \, c} \]

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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mupad [B]  time = 4.14, size = 29, normalized size = 1.16 \[ \frac {\left (\frac {b}{3\,c}+\frac {x^2}{3}\right )\,\sqrt {c\,x^4+b\,x^2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{2} + c x^{4}}\, dx \]

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)

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