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

Optimal. Leaf size=72 \[ \frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]

[Out]

-((a*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2))) + (b*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(a + b*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0203706, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1112, 14} \[ \frac{b x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^2,x]

[Out]

-((a*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2))) + (b*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(a + b*x^2)

Rule 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0074029, size = 35, normalized size = 0.49 \[ \frac{\left (b x^2-a\right ) \sqrt{\left (a+b x^2\right )^2}}{x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^2,x]

[Out]

((-a + b*x^2)*Sqrt[(a + b*x^2)^2])/(x*(a + b*x^2))

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 34, normalized size = 0.5 \begin{align*} -{\frac{-b{x}^{2}+a}{x \left ( b{x}^{2}+a \right ) }\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-b*x^2+a)*((b*x^2+a)^2)^(1/2)/x/(b*x^2+a)

________________________________________________________________________________________

Maxima [A]  time = 1.00983, size = 18, normalized size = 0.25 \begin{align*} \frac{b x^{2} - a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^2 - a)/x

________________________________________________________________________________________

Fricas [A]  time = 1.49991, size = 20, normalized size = 0.28 \begin{align*} \frac{b x^{2} - a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^2 - a)/x

________________________________________________________________________________________

Sympy [A]  time = 0.257612, size = 5, normalized size = 0.07 \begin{align*} - \frac{a}{x} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**2+a)**2)**(1/2)/x**2,x)

[Out]

-a/x + b*x

________________________________________________________________________________________

Giac [A]  time = 1.16587, size = 35, normalized size = 0.49 \begin{align*} b x \mathrm{sgn}\left (b x^{2} + a\right ) - \frac{a \mathrm{sgn}\left (b x^{2} + a\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*sgn(b*x^2 + a) - a*sgn(b*x^2 + a)/x