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

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Rubi [A]  time = 0.0408519, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}} \]

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]

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Rubi in Sympy [A]  time = 4.69134, size = 31, normalized size = 0.97 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(1/2)/(c*x**2+b*x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0270392, size = 48, normalized size = 1.5 \[ -\frac{2 \sqrt{x} \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{x (b+c x)}} \]

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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Maple [A]  time = 0.011, size = 37, normalized size = 1.2 \[ -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{\sqrt{x}\sqrt{cx+b}\sqrt{b}}{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) } \]

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)*(x*(c*x+b))^(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., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229004, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{\sqrt{b}}, -\frac{2 \, \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{\sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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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GIAC/XCAS [A]  time = 0.212129, size = 53, normalized size = 1.66 \[ \frac{2 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{2 \, \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right )}{\sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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