### 3.142 $$\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x} \, dx$$

Optimal. Leaf size=62 $\frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}$

[Out]

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

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Rubi [A]  time = 0.0166675, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {646, 43} $\frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{a \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 646

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0095251, size = 27, normalized size = 0.44 $\frac{\sqrt{(a+b x)^2} (a \log (x)+b x)}{a+b x}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [C]  time = 0.328, size = 19, normalized size = 0.3 \begin{align*}{\it csgn} \left ( bx+a \right ) \left ( bx+a+a\ln \left ( bx \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

csgn(b*x+a)*(b*x+a+a*ln(b*x))

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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+a)^2)^(1/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.86431, size = 22, normalized size = 0.35 \begin{align*} b x + a \log \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(x)

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Sympy [A]  time = 0.264238, size = 7, normalized size = 0.11 \begin{align*} a \log{\left (x \right )} + b x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) + b*x

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Giac [A]  time = 1.21377, size = 28, normalized size = 0.45 \begin{align*} b x \mathrm{sgn}\left (b x + a\right ) + a \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*sgn(b*x + a) + a*log(abs(x))*sgn(b*x + a)