### 3.2426 $$\int \frac{1}{x \sqrt{-a^2-2 a b x-b^2 x^2}} \, dx$$

Optimal. Leaf size=74 $\frac{\log (x) (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}$

[Out]

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

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Rubi [A]  time = 0.0264302, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.148, Rules used = {646, 36, 29, 31} $\frac{\log (x) (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{-a^2-2 a b x-b^2 x^2}} \, dx &=-\left (-\frac{\left (-a b-b^2 x\right ) \int \frac{1}{x \left (-a b-b^2 x\right )} \, dx}{\sqrt{-a^2-2 a b x-b^2 x^2}}\right )\\ &=-\frac{\left (-a b-b^2 x\right ) \int \frac{1}{x} \, dx}{a b \sqrt{-a^2-2 a b x-b^2 x^2}}+-\frac{\left (b \left (-a b-b^2 x\right )\right ) \int \frac{1}{-a b-b^2 x} \, dx}{a \sqrt{-a^2-2 a b x-b^2 x^2}}\\ &=\frac{(a+b x) \log (x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.01265, size = 33, normalized size = 0.45 $\frac{(a+b x) (\log (x)-\log (a+b x))}{a \sqrt{-(a+b x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.192, size = 32, normalized size = 0.4 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( \ln \left ( x \right ) -\ln \left ( bx+a \right ) \right ) }{a}{\frac{1}{\sqrt{- \left ( bx+a \right ) ^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.07598, size = 161, normalized size = 2.18 \begin{align*} -\sqrt{-\frac{1}{a^{2}}} \log \left (\frac{i \, a^{2} \sqrt{-\frac{1}{a^{2}}} + 2 \, b x + a}{2 \, b}\right ) + \sqrt{-\frac{1}{a^{2}}} \log \left (\frac{-i \, a^{2} \sqrt{-\frac{1}{a^{2}}} + 2 \, b x + a}{2 \, b}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- \left (a + b x\right )^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(-(a + b*x)**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef