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

Optimal. Leaf size=32 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {660, 207} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/Sqrt[b]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 660

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 48, normalized size = 1.50 \[ -\frac {2 \sqrt {x} \sqrt {b+c x} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x]*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(Sqrt[b]*Sqrt[x*(b + c*x)])

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fricas [A]  time = 0.90, size = 71, normalized size = 2.22 \[ \left [\frac {\log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right )}{\sqrt {b}}, \frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right )}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2)/sqrt(b), 2*sqrt(-b)*arctan(sqrt(-b)*sqrt(x)/s
qrt(c*x^2 + b*x))/b]

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giac [A]  time = 0.21, size = 39, normalized size = 1.22 \[ \frac {2 \, \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {2 \, \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right )}{\sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) - 2*arctan(sqrt(b)/sqrt(-b))/sqrt(-b)

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maple [A]  time = 0.05, size = 37, normalized size = 1.16 \[ -\frac {2 \sqrt {\left (c x +b \right ) x}\, \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )}{\sqrt {c x +b}\, \sqrt {b}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x} \sqrt {x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {x}\,\sqrt {c\,x^2+b\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(x)*sqrt(x*(b + c*x))), x)

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