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

Optimal. Leaf size=71 \[ -\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, 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} \[ -\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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^4} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{x^4} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.46 \[ -\frac {\sqrt {(a+b x)^2} (2 a+3 b x)}{6 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(Sqrt[(a + b*x)^2]*(2*a + 3*b*x))/(x^3*(a + b*x))

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fricas [A]  time = 0.84, size = 13, normalized size = 0.18 \[ -\frac {3 \, b x + 2 \, a}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

-1/6*(3*b*x + 2*a)/x^3

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giac [A]  time = 0.17, size = 40, normalized size = 0.56 \[ \frac {b^{3} \mathrm {sgn}\left (b x + a\right )}{6 \, a^{2}} - \frac {3 \, b x \mathrm {sgn}\left (b x + a\right ) + 2 \, a \mathrm {sgn}\left (b x + a\right )}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/x^4,x, algorithm="giac")

[Out]

1/6*b^3*sgn(b*x + a)/a^2 - 1/6*(3*b*x*sgn(b*x + a) + 2*a*sgn(b*x + a))/x^3

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maple [A]  time = 0.04, size = 30, normalized size = 0.42 \[ -\frac {\left (3 b x +2 a \right ) \sqrt {\left (b x +a \right )^{2}}}{6 \left (b x +a \right ) x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x+a)^2)^(1/2)/x^4,x)

[Out]

-1/6*(3*b*x+2*a)*((b*x+a)^2)^(1/2)/x^3/(b*x+a)

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maxima [B]  time = 1.43, size = 109, normalized size = 1.54 \[ -\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}}{2 \, a^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}}{2 \, a^{2} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b}{2 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.16, size = 29, normalized size = 0.41 \[ -\frac {\left (2\,a+3\,b\,x\right )\,\sqrt {{\left (a+b\,x\right )}^2}}{6\,x^3\,\left (a+b\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^2)^(1/2)/x^4,x)

[Out]

-((2*a + 3*b*x)*((a + b*x)^2)^(1/2))/(6*x^3*(a + b*x))

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sympy [A]  time = 0.14, size = 14, normalized size = 0.20 \[ \frac {- 2 a - 3 b x}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)**2)**(1/2)/x**4,x)

[Out]

(-2*a - 3*b*x)/(6*x**3)

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