\(\int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx\) [147]

Optimal result

Integrand size = 24, antiderivative size = 71 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]

[Out]

-1/5*a*((b*x+a)^2)^(1/2)/x^5/(b*x+a)-1/4*b*((b*x+a)^2)^(1/2)/x^4/(b*x+a)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]

[In]

Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^6,x]

[Out]

-1/5*(a*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x^5*(a + b*x)) - (b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x))

Rule 45

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 660

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*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{x^6} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a b}{x^6}+\frac {b^2}{x^5}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {\sqrt {(a+b x)^2} (4 a+5 b x)}{20 x^5 (a+b x)} \]

[In]

Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^6,x]

[Out]

-1/20*(Sqrt[(a + b*x)^2]*(4*a + 5*b*x))/(x^5*(a + b*x))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.28

method	result	size
default	\(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (5 b x +4 a \right )}{20 x^{5}}\)	\(20\)
risch	\(\frac {\left (-\frac {b x}{4}-\frac {a}{5}\right ) \sqrt {\left (b x +a \right )^{2}}}{x^{5} \left (b x +a \right )}\)	\(29\)
gosper	\(-\frac {\left (5 b x +4 a \right ) \sqrt {\left (b x +a \right )^{2}}}{20 x^{5} \left (b x +a \right )}\)	\(30\)

[In]

int(((b*x+a)^2)^(1/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/20*csgn(b*x+a)*(5*b*x+4*a)/x^5

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {5 \, b x + 4 \, a}{20 \, x^{5}} \]

[In]

integrate(((b*x+a)^2)^(1/2)/x^6,x, algorithm="fricas")

[Out]

-1/20*(5*b*x + 4*a)/x^5

Sympy [F]

\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=\int \frac {\sqrt {\left (a + b x\right )^{2}}}{x^{6}}\, dx \]

[In]

integrate(((b*x+a)**2)**(1/2)/x**6,x)

[Out]

Integral(sqrt((a + b*x)**2)/x**6, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (45) = 90\).

Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}}{2 \, a^{5}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}}{2 \, a^{4} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{3}}{2 \, a^{5} x^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}}{20 \, a^{4} x^{3}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b}{20 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{5 \, a^{2} x^{5}} \]

[In]

integrate(((b*x+a)^2)^(1/2)/x^6,x, algorithm="maxima")

[Out]

-1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^5/a^5 - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4/(a^4*x) + 1/2*(b^2*x^2 + 2*
a*b*x + a^2)^(3/2)*b^3/(a^5*x^2) - 9/20*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2/(a^4*x^3) + 7/20*(b^2*x^2 + 2*a*b*
x + a^2)^(3/2)*b/(a^3*x^4) - 1/5*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)/(a^2*x^5)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=\frac {b^{5} \mathrm {sgn}\left (b x + a\right )}{20 \, a^{4}} - \frac {5 \, b x \mathrm {sgn}\left (b x + a\right ) + 4 \, a \mathrm {sgn}\left (b x + a\right )}{20 \, x^{5}} \]

[In]

integrate(((b*x+a)^2)^(1/2)/x^6,x, algorithm="giac")

[Out]

1/20*b^5*sgn(b*x + a)/a^4 - 1/20*(5*b*x*sgn(b*x + a) + 4*a*sgn(b*x + a))/x^5

Mupad [B] (verification not implemented)

Time = 9.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx=-\frac {\left (4\,a+5\,b\,x\right )\,\sqrt {{\left (a+b\,x\right )}^2}}{20\,x^5\,\left (a+b\,x\right )} \]

[In]

int(((a + b*x)^2)^(1/2)/x^6,x)

[Out]

-((4*a + 5*b*x)*((a + b*x)^2)^(1/2))/(20*x^5*(a + b*x))