### 3.47 $$\int \frac{1}{x \sqrt{b x+c x^2}} \, dx$$

Optimal. Leaf size=21 $-\frac{2 \sqrt{b x+c x^2}}{b x}$

[Out]

(-2*Sqrt[b*x + c*x^2])/(b*x)

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Rubi [A]  time = 0.0073153, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {650} $-\frac{2 \sqrt{b x+c x^2}}{b x}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[b*x + c*x^2])/(b*x)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{b x+c x^2}} \, dx &=-\frac{2 \sqrt{b x+c x^2}}{b x}\\ \end{align*}

Mathematica [A]  time = 0.006572, size = 21, normalized size = 1. $-\frac{2 (b+c x)}{b \sqrt{x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*(b + c*x))/(b*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.047, size = 22, normalized size = 1.1 \begin{align*} -2\,{\frac{cx+b}{b\sqrt{c{x}^{2}+bx}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(c*x+b)/b/(c*x^2+b*x)^(1/2)

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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/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7453, size = 38, normalized size = 1.81 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}}{b x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(c*x^2 + b*x)/(b*x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(x*sqrt(x*(b + c*x))), x)

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Giac [A]  time = 1.19682, size = 31, normalized size = 1.48 \begin{align*} \frac{2}{\sqrt{c} x - \sqrt{c x^{2} + b x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(sqrt(c)*x - sqrt(c*x^2 + b*x))