3.2.22 \(\int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx\)

Optimal. Leaf size=35 \[ -\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 a x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 37} \begin {gather*} -\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^3,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{x^3} \, dx}{a b+b^2 x}\\ &=-\frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 a x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (a+2 b x)}{2 x^2 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^3,x]

[Out]

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

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IntegrateAlgebraic [B]  time = 0.36, size = 108, normalized size = 3.09 \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-a b-2 b^2 x\right )+\sqrt {b^2} \left (a^2+3 a b x+2 b^2 x^2\right )}{2 x^2 \left (a b+b^2 x\right )-2 \sqrt {b^2} x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^3,x]

[Out]

((-(a*b) - 2*b^2*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + Sqrt[b^2]*(a^2 + 3*a*b*x + 2*b^2*x^2))/(2*x^2*(a*b + b^2*x
) - 2*Sqrt[b^2]*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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fricas [A]  time = 0.41, size = 11, normalized size = 0.31 \begin {gather*} -\frac {2 \, b x + a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/x^3,x, algorithm="fricas")

[Out]

-1/2*(2*b*x + a)/x^2

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giac [A]  time = 0.15, size = 39, normalized size = 1.11 \begin {gather*} -\frac {b^{2} \mathrm {sgn}\left (b x + a\right )}{2 \, a} - \frac {2 \, b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 28, normalized size = 0.80 \begin {gather*} -\frac {\left (2 b x +a \right ) \sqrt {\left (b x +a \right )^{2}}}{2 \left (b x +a \right ) x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x+a)^2)^(1/2)/x^3,x)

[Out]

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

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maxima [B]  time = 1.41, size = 80, normalized size = 2.29 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}}{2 \, a^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b}{2 \, a x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{2 \, a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/x^3,x, algorithm="maxima")

[Out]

1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2/a^2 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b/(a*x) - 1/2*(b^2*x^2 + 2*a*b*x
+ a^2)^(3/2)/(a^2*x^2)

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mupad [B]  time = 0.16, size = 27, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+2\,b\,x\right )}{2\,x^2\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^2)^(1/2)/x^3,x)

[Out]

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

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sympy [A]  time = 0.13, size = 12, normalized size = 0.34 \begin {gather*} \frac {- a - 2 b x}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)**2)**(1/2)/x**3,x)

[Out]

(-a - 2*b*x)/(2*x**2)

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