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

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

________________________________________________________________________________________

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}}{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 {gather*}

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*}

________________________________________________________________________________________

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

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))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 3.41, size = 741, normalized size = 10.44 \begin {gather*} \frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \left (-2 a^{15} b^3-55 a^{14} b^4 x-704 a^{13} b^5 x^2-5563 a^{12} b^6 x^3-30344 a^{11} b^7 x^4-121000 a^{10} b^8 x^5-364320 a^9 b^9 x^6-843216 a^8 b^{10} x^7-1512192 a^7 b^{11} x^8-2100736 a^6 b^{12} x^9-2241536 a^5 b^{13} x^{10}-1803520 a^4 b^{14} x^{11}-1058816 a^3 b^{15} x^{12}-428032 a^2 b^{16} x^{13}-106496 a b^{17} x^{14}-12288 b^{18} x^{15}\right )+2 \sqrt {b^2} \left (2 a^{16} b^2+57 a^{15} b^3 x+759 a^{14} b^4 x^2+6267 a^{13} b^5 x^3+35907 a^{12} b^6 x^4+151344 a^{11} b^7 x^5+485320 a^{10} b^8 x^6+1207536 a^9 b^9 x^7+2355408 a^8 b^{10} x^8+3612928 a^7 b^{11} x^9+4342272 a^6 b^{12} x^{10}+4045056 a^5 b^{13} x^{11}+2862336 a^4 b^{14} x^{12}+1486848 a^3 b^{15} x^{13}+534528 a^2 b^{16} x^{14}+118784 a b^{17} x^{15}+12288 b^{18} x^{16}\right )}{3 \sqrt {b^2} x^3 \sqrt {a^2+2 a b x+b^2 x^2} \left (-4 a^{14} b^2-104 a^{13} b^3 x-1252 a^{12} b^4 x^2-9248 a^{11} b^5 x^3-46816 a^{10} b^6 x^4-171776 a^9 b^7 x^5-470976 a^8 b^8 x^6-979968 a^7 b^9 x^7-1554432 a^6 b^{10} x^8-1869824 a^5 b^{11} x^9-1678336 a^4 b^{12} x^{10}-1089536 a^3 b^{13} x^{11}-483328 a^2 b^{14} x^{12}-131072 a b^{15} x^{13}-16384 b^{16} x^{14}\right )+3 x^3 \left (4 a^{15} b^3+108 a^{14} b^4 x+1356 a^{13} b^5 x^2+10500 a^{12} b^6 x^3+56064 a^{11} b^7 x^4+218592 a^{10} b^8 x^5+642752 a^9 b^9 x^6+1450944 a^8 b^{10} x^7+2534400 a^7 b^{11} x^8+3424256 a^6 b^{12} x^9+3548160 a^5 b^{13} x^{10}+2767872 a^4 b^{14} x^{11}+1572864 a^3 b^{15} x^{12}+614400 a^2 b^{16} x^{13}+147456 a b^{17} x^{14}+16384 b^{18} x^{15}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-2*a^15*b^3 - 55*a^14*b^4*x - 704*a^13*b^5*x^2 - 5563*a^12*b^6*x^3 - 30344*a
^11*b^7*x^4 - 121000*a^10*b^8*x^5 - 364320*a^9*b^9*x^6 - 843216*a^8*b^10*x^7 - 1512192*a^7*b^11*x^8 - 2100736*
a^6*b^12*x^9 - 2241536*a^5*b^13*x^10 - 1803520*a^4*b^14*x^11 - 1058816*a^3*b^15*x^12 - 428032*a^2*b^16*x^13 -
106496*a*b^17*x^14 - 12288*b^18*x^15) + 2*Sqrt[b^2]*(2*a^16*b^2 + 57*a^15*b^3*x + 759*a^14*b^4*x^2 + 6267*a^13
*b^5*x^3 + 35907*a^12*b^6*x^4 + 151344*a^11*b^7*x^5 + 485320*a^10*b^8*x^6 + 1207536*a^9*b^9*x^7 + 2355408*a^8*
b^10*x^8 + 3612928*a^7*b^11*x^9 + 4342272*a^6*b^12*x^10 + 4045056*a^5*b^13*x^11 + 2862336*a^4*b^14*x^12 + 1486
848*a^3*b^15*x^13 + 534528*a^2*b^16*x^14 + 118784*a*b^17*x^15 + 12288*b^18*x^16))/(3*Sqrt[b^2]*x^3*Sqrt[a^2 +
2*a*b*x + b^2*x^2]*(-4*a^14*b^2 - 104*a^13*b^3*x - 1252*a^12*b^4*x^2 - 9248*a^11*b^5*x^3 - 46816*a^10*b^6*x^4
- 171776*a^9*b^7*x^5 - 470976*a^8*b^8*x^6 - 979968*a^7*b^9*x^7 - 1554432*a^6*b^10*x^8 - 1869824*a^5*b^11*x^9 -
1678336*a^4*b^12*x^10 - 1089536*a^3*b^13*x^11 - 483328*a^2*b^14*x^12 - 131072*a*b^15*x^13 - 16384*b^16*x^14)
+ 3*x^3*(4*a^15*b^3 + 108*a^14*b^4*x + 1356*a^13*b^5*x^2 + 10500*a^12*b^6*x^3 + 56064*a^11*b^7*x^4 + 218592*a^
10*b^8*x^5 + 642752*a^9*b^9*x^6 + 1450944*a^8*b^10*x^7 + 2534400*a^7*b^11*x^8 + 3424256*a^6*b^12*x^9 + 3548160
*a^5*b^13*x^10 + 2767872*a^4*b^14*x^11 + 1572864*a^3*b^15*x^12 + 614400*a^2*b^16*x^13 + 147456*a*b^17*x^14 + 1
6384*b^18*x^15))

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 13, normalized size = 0.18 \begin {gather*} -\frac {3 \, b x + 2 \, a}{6 \, x^{3}} \end {gather*}

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maxima [B]  time = 1.43, size = 109, normalized size = 1.54 \begin {gather*} -\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}} \end {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.16, size = 29, normalized size = 0.41 \begin {gather*} -\frac {\left (2\,a+3\,b\,x\right )\,\sqrt {{\left (a+b\,x\right )}^2}}{6\,x^3\,\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^4,x)

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________