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

Optimal. Leaf size=22 \[ \frac{\sqrt{b x^2+c x^4}}{c x} \]

[Out]

Sqrt[b*x^2 + c*x^4]/(c*x)

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Rubi [A]  time = 0.0168739, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {1588} \[ \frac{\sqrt{b x^2+c x^4}}{c x} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[b*x^2 + c*x^4],x]

[Out]

Sqrt[b*x^2 + c*x^4]/(c*x)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{b x^2+c x^4}} \, dx &=\frac{\sqrt{b x^2+c x^4}}{c x}\\ \end{align*}

Mathematica [A]  time = 0.0052852, size = 22, normalized size = 1. \[ \frac{\sqrt{x^2 \left (b+c x^2\right )}}{c x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[b*x^2 + c*x^4],x]

[Out]

Sqrt[x^2*(b + c*x^2)]/(c*x)

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Maple [A]  time = 0.043, size = 26, normalized size = 1.2 \begin{align*}{\frac{x \left ( c{x}^{2}+b \right ) }{c}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01831, size = 18, normalized size = 0.82 \begin{align*} \frac{\sqrt{c x^{2} + b}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2 + b)/c

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Fricas [A]  time = 1.0958, size = 36, normalized size = 1.64 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}}{c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^4 + b*x^2)/(c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/sqrt(x**2*(b + c*x**2)), x)

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Giac [A]  time = 1.13133, size = 42, normalized size = 1.91 \begin{align*} -\frac{2 \, \sqrt{b}}{{\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b)/((sqrt(c + b/x^2) - sqrt(b)/x)^2 - c)