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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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]]

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]

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 = 0.01, size = 37, normalized size = 0.49 \begin {gather*} \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 )} \end {gather*}

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))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.21, size = 197, normalized size = 2.63 \begin {gather*} \frac {1}{4} \sqrt {a^2+2 a b x^2+b^2 x^4}+\frac {1}{4} a \log \left (\sqrt {a^2+2 a b x^2+b^2 x^4}-a-\sqrt {b^2} x^2\right )-\frac {a \left (\sqrt {b^2}+b\right ) \log \left (\sqrt {a^2+2 a b x^2+b^2 x^4}+a-\sqrt {b^2} x^2\right )}{4 b}-\frac {a \sqrt {b^2} \log \left (b \sqrt {a^2+2 a b x^2+b^2 x^4}-a b-b \sqrt {b^2} x^2\right )}{4 b}-\frac {1}{4} \sqrt {b^2} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[b^2]*x^2) + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/4 + (a*Log[-a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4]])/4 - (a*(b + Sqrt[b^2])*Log[a - Sqrt[b^2]*x^2 + Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/(4*b) - (a*Sqrt[b^
2]*Log[-(a*b) - b*Sqrt[b^2]*x^2 + b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]])/(4*b)

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 11, normalized size = 0.15 \begin {gather*} \frac {1}{2} \, b x^{2} + a \log \relax (x) \end {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 30, normalized size = 0.40 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 34, normalized size = 0.45 \begin {gather*} \frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (b \,x^{2}+2 a \ln \relax (x )\right )}{2 b \,x^{2}+2 a} \end {gather*}

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)

________________________________________________________________________________________

maxima [A]  time = 1.39, size = 14, normalized size = 0.19 \begin {gather*} \frac {1}{2} \, b x^{2} + \frac {1}{2} \, a \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + 1/2*a*log(x^2)

________________________________________________________________________________________

mupad [B]  time = 4.39, size = 109, normalized size = 1.45 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{2}-\frac {\ln \left (a\,b+\frac {a^2}{x^2}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{x^2}\right )\,\sqrt {a^2}}{2}+\frac {a\,b\,\ln \left (a\,b+\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {b^2}+b^2\,x^2\right )}{2\,\sqrt {b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2)^(1/2)/x,x)

[Out]

(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2)/2 - (log(a*b + a^2/x^2 + ((a^2)^(1/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/x^2
)*(a^2)^(1/2))/2 + (a*b*log(a*b + ((a + b*x^2)^2)^(1/2)*(b^2)^(1/2) + b^2*x^2))/(2*(b^2)^(1/2))

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 10, normalized size = 0.13 \begin {gather*} a \log {\relax (x )} + \frac {b x^{2}}{2} \end {gather*}

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

________________________________________________________________________________________