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

Optimal. Leaf size=59 \[ \frac{B x \sqrt{b x^2+c x^4}}{3 c}-\frac{\sqrt{b x^2+c x^4} (2 b B-3 A c)}{3 c^2 x} \]

[Out]

-((2*b*B - 3*A*c)*Sqrt[b*x^2 + c*x^4])/(3*c^2*x) + (B*x*Sqrt[b*x^2 + c*x^4])/(3*c)

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Rubi [A]  time = 0.135175, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2039, 1588} \[ \frac{B x \sqrt{b x^2+c x^4}}{3 c}-\frac{\sqrt{b x^2+c x^4} (2 b B-3 A c)}{3 c^2 x} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

-((2*b*B - 3*A*c)*Sqrt[b*x^2 + c*x^4])/(3*c^2*x) + (B*x*Sqrt[b*x^2 + c*x^4])/(3*c)

Rule 2039

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[(d*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(b*(m + n + p*(j + n) + 1)), x] - Dist[(a*d*(m
+ j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)), Int[(e*x)^m*(a*x^j + b*x^(j + n))^p, x
], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[
m + n + p*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])

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

Mathematica [A]  time = 0.0276201, size = 40, normalized size = 0.68 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (3 A c-2 b B+B c x^2\right )}{3 c^2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(-2*b*B + 3*A*c + B*c*x^2))/(3*c^2*x)

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Maple [A]  time = 0.004, size = 42, normalized size = 0.7 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ) \left ( Bc{x}^{2}+3\,Ac-2\,Bb \right ) x}{3\,{c}^{2}}{\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*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.15904, size = 68, normalized size = 1.15 \begin{align*} \frac{\sqrt{c x^{2} + b} A}{c} + \frac{{\left (c^{2} x^{4} - b c x^{2} - 2 \, b^{2}\right )} B}{3 \, \sqrt{c x^{2} + b} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.00099, size = 80, normalized size = 1.36 \begin{align*} \frac{\sqrt{c x^{4} + b x^{2}}{\left (B c x^{2} - 2 \, B b + 3 \, A c\right )}}{3 \, c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (A + B x^{2}\right )}{\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*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x^2 + A)*x^2/sqrt(c*x^4 + b*x^2), x)