3.1.47 \(\int \frac {1}{x \sqrt {b x+c x^2}} \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 \sqrt {b x+c x^2}}{b x} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2 (b+c x)}{b \sqrt {x (b+c x)}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.13, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {b x+c x^2}}{b x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[b*x + c*x^2])/(b*x)

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fricas [A]  time = 0.39, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x}}{b x} \end {gather*}

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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giac [A]  time = 0.27, size = 23, normalized size = 1.10 \begin {gather*} \frac {2}{\sqrt {c} x - \sqrt {c x^{2} + b x}} \end {gather*}

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))

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maple [A]  time = 0.04, size = 22, normalized size = 1.05 \begin {gather*} -\frac {2 \left (c x +b \right )}{\sqrt {c \,x^{2}+b x}\, b} \end {gather*}

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 [A]  time = 1.29, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x}}{b x} \end {gather*}

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]

-2*sqrt(c*x^2 + b*x)/(b*x)

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mupad [B]  time = 0.17, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}}{b\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x + c*x^2)^(1/2)),x)

[Out]

-(2*(b*x + c*x^2)^(1/2))/(b*x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}

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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