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

Optimal. Leaf size=75 \[ \frac{b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{a \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]

[Out]

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

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Rubi [A]  time = 0.0219498, antiderivative size = 75, 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} \[ \frac{b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{a \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*(a + b*x^2)) + (a*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[x])/(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} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{a b+b^2 x^2}{x} \, dx}{a b+b^2 x^2}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (\frac{a b}{x}+b^2 x\right ) \, dx}{a b+b^2 x^2}\\ &=\frac{b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2}\\ \end{align*}

Mathematica [A]  time = 0.0121508, size = 37, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (2 a \log (x)+b x^2\right )}{2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.212, size = 34, normalized size = 0.5 \begin{align*}{\frac{b{x}^{2}+2\,a\ln \left ( x \right ) }{2\,b{x}^{2}+2\,a}\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,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48814, size = 30, normalized size = 0.4 \begin{align*} \frac{1}{2} \, b x^{2} + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*log(x)

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Sympy [A]  time = 0.106605, size = 10, normalized size = 0.13 \begin{align*} a \log{\left (x \right )} + \frac{b x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) + b*x**2/2

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Giac [A]  time = 1.12615, size = 41, normalized size = 0.55 \begin{align*} \frac{1}{2} \, b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{1}{2} \, a \log \left (x^{2}\right ) \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2*sgn(b*x^2 + a) + 1/2*a*log(x^2)*sgn(b*x^2 + a)