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]

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

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )}}{c x} \]

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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fricas [A]  time = 0.63, size = 20, normalized size = 0.91 \[ \frac {\sqrt {c x^{4} + b x^{2}}}{c x} \]

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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giac [A]  time = 0.19, size = 31, normalized size = 1.41 \[ -\frac {2 \, \sqrt {b}}{{\left (\sqrt {c + \frac {b}{x^{2}}} - \frac {\sqrt {b}}{x}\right )}^{2} - c} \]

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)

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maple [A]  time = 0.00, size = 26, normalized size = 1.18 \[ \frac {\left (c \,x^{2}+b \right ) x}{\sqrt {c \,x^{4}+b \,x^{2}}\, c} \]

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 = 1.42, size = 13, normalized size = 0.59 \[ \frac {\sqrt {c x^{2} + b}}{c} \]

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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mupad [B]  time = 4.26, size = 20, normalized size = 0.91 \[ \frac {\sqrt {c\,x^4+b\,x^2}}{c\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^2 + c*x^4)^(1/2),x)

[Out]

(b*x^2 + c*x^4)^(1/2)/(c*x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]

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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