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

Optimal. Leaf size=71 \[ -\frac {a \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]

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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} \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x)) - (b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(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^6} \, dx &=\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*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.46 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (4 a+5 b x)}{20 x^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[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))

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IntegrateAlgebraic [B]  time = 0.61, size = 300, normalized size = 4.23 \begin {gather*} \frac {4 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-4 a^5 b-21 a^4 b^2 x-44 a^3 b^3 x^2-46 a^2 b^4 x^3-24 a b^5 x^4-5 b^6 x^5\right )+4 \sqrt {b^2} b^4 \left (4 a^6+25 a^5 b x+65 a^4 b^2 x^2+90 a^3 b^3 x^3+70 a^2 b^4 x^4+29 a b^5 x^5+5 b^6 x^6\right )}{5 \sqrt {b^2} x^5 \sqrt {a^2+2 a b x+b^2 x^2} \left (-16 a^4 b^4-64 a^3 b^5 x-96 a^2 b^6 x^2-64 a b^7 x^3-16 b^8 x^4\right )+5 x^5 \left (16 a^5 b^5+80 a^4 b^6 x+160 a^3 b^7 x^2+160 a^2 b^8 x^3+80 a b^9 x^4+16 b^{10} x^5\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-4*a^5*b - 21*a^4*b^2*x - 44*a^3*b^3*x^2 - 46*a^2*b^4*x^3 - 24*a*b^5*x^4
- 5*b^6*x^5) + 4*b^4*Sqrt[b^2]*(4*a^6 + 25*a^5*b*x + 65*a^4*b^2*x^2 + 90*a^3*b^3*x^3 + 70*a^2*b^4*x^4 + 29*a*
b^5*x^5 + 5*b^6*x^6))/(5*Sqrt[b^2]*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-16*a^4*b^4 - 64*a^3*b^5*x - 96*a^2*b^6*
x^2 - 64*a*b^7*x^3 - 16*b^8*x^4) + 5*x^5*(16*a^5*b^5 + 80*a^4*b^6*x + 160*a^3*b^7*x^2 + 160*a^2*b^8*x^3 + 80*a
*b^9*x^4 + 16*b^10*x^5))

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fricas [A]  time = 0.39, size = 13, normalized size = 0.18 \begin {gather*} -\frac {5 \, b x + 4 \, a}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(5*b*x + 4*a)/x^5

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giac [A]  time = 0.17, size = 40, normalized size = 0.56 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.04, size = 30, normalized size = 0.42 \begin {gather*} -\frac {\left (5 b x +4 a \right ) \sqrt {\left (b x +a \right )^{2}}}{20 \left (b x +a \right ) x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.46, size = 167, normalized size = 2.35 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.16, size = 29, normalized size = 0.41 \begin {gather*} -\frac {\left (4\,a+5\,b\,x\right )\,\sqrt {{\left (a+b\,x\right )}^2}}{20\,x^5\,\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^6,x)

[Out]

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

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sympy [A]  time = 0.16, size = 14, normalized size = 0.20 \begin {gather*} \frac {- 4 a - 5 b x}{20 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*a - 5*b*x)/(20*x**5)

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