3.2.66 \(\int \frac {1}{x^3 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=145 \[ \frac {b (a+b x)}{a^2 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {-a-b x}{2 a x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 \log (x) (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.04, antiderivative size = 142, normalized size of antiderivative = 0.98, 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, 44} \begin {gather*} \frac {b (a+b x)}{a^2 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{2 a x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 \log (x) (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 {1}{x^3 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{x^3 \left (a b+b^2 x\right )} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {1}{a b x^3}-\frac {1}{a^2 x^2}+\frac {b}{a^3 x}-\frac {b^2}{a^3 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a+b x}{2 a x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{a^2 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) \log (x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x) \log (a+b x)}{a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 59, normalized size = 0.41 \begin {gather*} -\frac {(a+b x) \left (2 b^2 x^2 \log (a+b x)+a (a-2 b x)-2 b^2 x^2 \log (x)\right )}{2 a^3 x^2 \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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IntegrateAlgebraic [B]  time = 1.59, size = 739, normalized size = 5.10 \begin {gather*} \frac {2 b^2 \left (\sqrt {a^2+2 a b x+b^2 x^2}-\sqrt {b^2} x\right )^4 \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 a b x+b^2 x^2}}{a}\right )}{a^3 \left (a^4+4 a^3 b x+12 a^2 b^2 x^2-4 a^2 \sqrt {b^2} x \sqrt {a^2+2 a b x+b^2 x^2}-8 a b \sqrt {b^2} x^2 \sqrt {a^2+2 a b x+b^2 x^2}-8 \left (b^2\right )^{3/2} x^3 \sqrt {a^2+2 a b x+b^2 x^2}+16 a b^3 x^3+8 b^4 x^4\right )}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-a^{10} b^2-15 a^9 b^3 x-94 a^8 b^4 x^2-304 a^7 b^5 x^3-448 a^6 b^6 x^4+224 a^5 b^7 x^5+2240 a^4 b^8 x^6+4352 a^3 b^9 x^7+4352 a^2 b^{10} x^8+2304 a b^{11} x^9+512 b^{12} x^{10}\right )+\sqrt {b^2} \left (a^{11} b+16 a^{10} b^2 x+109 a^9 b^3 x^2+398 a^8 b^4 x^3+752 a^7 b^5 x^4+224 a^6 b^6 x^5-2464 a^5 b^7 x^6-6592 a^4 b^8 x^7-8704 a^3 b^9 x^8-6656 a^2 b^{10} x^9-2816 a b^{11} x^{10}-512 b^{12} x^{11}\right )}{a^2 \sqrt {b^2} x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (-2 a^9 b-34 a^8 b^2 x-256 a^7 b^3 x^2-1120 a^6 b^4 x^3-3136 a^5 b^5 x^4-5824 a^4 b^6 x^5-7168 a^3 b^7 x^6-5632 a^2 b^8 x^7-2560 a b^9 x^8-512 b^{10} x^9\right )+a^2 x^2 \left (2 a^{10} b^2+36 a^9 b^3 x+290 a^8 b^4 x^2+1376 a^7 b^5 x^3+4256 a^6 b^6 x^4+8960 a^5 b^7 x^5+12992 a^4 b^8 x^6+12800 a^3 b^9 x^7+8192 a^2 b^{10} x^8+3072 a b^{11} x^9+512 b^{12} x^{10}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-(a^10*b^2) - 15*a^9*b^3*x - 94*a^8*b^4*x^2 - 304*a^7*b^5*x^3 - 448*a^6*b^6*x^
4 + 224*a^5*b^7*x^5 + 2240*a^4*b^8*x^6 + 4352*a^3*b^9*x^7 + 4352*a^2*b^10*x^8 + 2304*a*b^11*x^9 + 512*b^12*x^1
0) + Sqrt[b^2]*(a^11*b + 16*a^10*b^2*x + 109*a^9*b^3*x^2 + 398*a^8*b^4*x^3 + 752*a^7*b^5*x^4 + 224*a^6*b^6*x^5
- 2464*a^5*b^7*x^6 - 6592*a^4*b^8*x^7 - 8704*a^3*b^9*x^8 - 6656*a^2*b^10*x^9 - 2816*a*b^11*x^10 - 512*b^12*x^
11))/(a^2*Sqrt[b^2]*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-2*a^9*b - 34*a^8*b^2*x - 256*a^7*b^3*x^2 - 1120*a^6*b^
4*x^3 - 3136*a^5*b^5*x^4 - 5824*a^4*b^6*x^5 - 7168*a^3*b^7*x^6 - 5632*a^2*b^8*x^7 - 2560*a*b^9*x^8 - 512*b^10*
x^9) + a^2*x^2*(2*a^10*b^2 + 36*a^9*b^3*x + 290*a^8*b^4*x^2 + 1376*a^7*b^5*x^3 + 4256*a^6*b^6*x^4 + 8960*a^5*b
^7*x^5 + 12992*a^4*b^8*x^6 + 12800*a^3*b^9*x^7 + 8192*a^2*b^10*x^8 + 3072*a*b^11*x^9 + 512*b^12*x^10)) + (2*b^
2*(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])^4*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b^2*x^2]/a
])/(a^3*(a^4 + 4*a^3*b*x + 12*a^2*b^2*x^2 + 16*a*b^3*x^3 + 8*b^4*x^4 - 4*a^2*Sqrt[b^2]*x*Sqrt[a^2 + 2*a*b*x +
b^2*x^2] - 8*a*b*Sqrt[b^2]*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2] - 8*(b^2)^(3/2)*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
))

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fricas [A]  time = 0.40, size = 41, normalized size = 0.28 \begin {gather*} -\frac {2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \relax (x) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 54, normalized size = 0.37 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} - \frac {2 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {2 \, a b x - a^{2}}{a^{3} x^{2}}\right )} \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 56, normalized size = 0.39 \begin {gather*} -\frac {\left (b x +a \right ) \left (-2 b^{2} x^{2} \ln \relax (x )+2 b^{2} x^{2} \ln \left (b x +a \right )-2 a b x +a^{2}\right )}{2 \sqrt {\left (b x +a \right )^{2}}\, a^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 95, normalized size = 0.66 \begin {gather*} -\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{3}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b}{2 \, a^{3} x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{2 \, a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^3\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*((a + b*x)^2)^(1/2)),x)

[Out]

int(1/(x^3*((a + b*x)^2)^(1/2)), x)

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sympy [A]  time = 0.24, size = 31, normalized size = 0.21 \begin {gather*} \frac {- a + 2 b x}{2 a^{2} x^{2}} + \frac {b^{2} \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a + 2*b*x)/(2*a**2*x**2) + b**2*(log(x) - log(a/b + x))/a**3

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