3.131 \(\int \frac{1}{\sqrt{x} \sqrt{x \left (a+b x+c x^2\right )}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x} (2 a+b x)}{2 \sqrt{a} \sqrt{a x+b x^2+c x^3}}\right )}{\sqrt{a}} \]

[Out]

-(ArcTanh[(Sqrt[x]*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x + b*x^2 + c*x^3])]/Sqrt[a])

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Rubi [A]  time = 0.117825, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x} (2 a+b x)}{2 \sqrt{a} \sqrt{a x+b x^2+c x^3}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*Sqrt[x*(a + b*x + c*x^2)]),x]

[Out]

-(ArcTanh[(Sqrt[x]*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x + b*x^2 + c*x^3])]/Sqrt[a])

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Rubi in Sympy [A]  time = 19.1881, size = 70, normalized size = 1.49 \[ - \frac{\sqrt{x} \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{a} \sqrt{a x + b x^{2} + c x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x*
*2)))/(sqrt(a)*sqrt(a*x + b*x**2 + c*x**3))

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Mathematica [A]  time = 0.0682834, size = 72, normalized size = 1.53 \[ \frac{\sqrt{x} \sqrt{a+x (b+c x)} \left (\log (x)-\log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )\right )}{\sqrt{a} \sqrt{x (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 64, normalized size = 1.4 \[ -{1\sqrt{x}\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{x \left ( c{x}^{2}+bx+a \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(x*(c*x^2+b*x+a))^(1/2),x)

[Out]

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

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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 + a)*x)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.348107, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{4 \, \sqrt{c x^{3} + b x^{2} + a x}{\left (a b x + 2 \, a^{2}\right )} \sqrt{x} -{\left (8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x\right )} \sqrt{a}}{x^{3}}\right )}{2 \, \sqrt{a}}, \frac{\arctan \left (\frac{b x^{2} + 2 \, a x}{2 \, \sqrt{c x^{3} + b x^{2} + a x} \sqrt{-a} \sqrt{x}}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log(-(4*sqrt(c*x^3 + b*x^2 + a*x)*(a*b*x + 2*a^2)*sqrt(x) - (8*a*b*x^2 + (b
^2 + 4*a*c)*x^3 + 8*a^2*x)*sqrt(a))/x^3)/sqrt(a), arctan(1/2*(b*x^2 + 2*a*x)/(sq
rt(c*x^3 + b*x^2 + a*x)*sqrt(-a)*sqrt(x)))/sqrt(-a)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277979, size = 47, normalized size = 1. \[ \frac{2 \, \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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