3.132 \(\int \frac{\sqrt{x}}{\sqrt{x^3 \left (a+b x+c x^2\right )}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\tanh ^{-1}\left (\frac{x^{3/2} (2 a+b x)}{2 \sqrt{a} \sqrt{a x^3+b x^4+c x^5}}\right )}{\sqrt{a}} \]

[Out]

-(ArcTanh[(x^(3/2)*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^3 + b*x^4 + c*x^5])]/Sqrt[a]
)

_______________________________________________________________________________________

Rubi [A]  time = 0.146853, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tanh ^{-1}\left (\frac{x^{3/2} (2 a+b x)}{2 \sqrt{a} \sqrt{a x^3+b x^4+c x^5}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/Sqrt[x^3*(a + b*x + c*x^2)],x]

[Out]

-(ArcTanh[(x^(3/2)*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^3 + b*x^4 + c*x^5])]/Sqrt[a]
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.595, size = 71, normalized size = 1.45 \[ - \frac{x^{\frac{3}{2}} \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{a} \sqrt{a x^{3} + b x^{4} + c x^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(x**3*(c*x**2+b*x+a))**(1/2),x)

[Out]

-x**(3/2)*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**3 + b*x**4 + c*x**5))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0608748, size = 74, normalized size = 1.51 \[ \frac{x^{3/2} \sqrt{a+x (b+c x)} \left (\log (x)-\log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )\right )}{\sqrt{a} \sqrt{x^3 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/Sqrt[x^3*(a + b*x + c*x^2)],x]

[Out]

(x^(3/2)*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^3*(a + x*(b + c*x))])

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 66, normalized size = 1.4 \[ -{1{x}^{{\frac{3}{2}}}\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{{x}^{3} \left ( c{x}^{2}+bx+a \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(x^3*(c*x^2+b*x+a))^(1/2),x)

[Out]

-1/(x^3*(c*x^2+b*x+a))^(1/2)*x^(3/2)*(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)

_______________________________________________________________________________________

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(sqrt(x)/sqrt((c*x^2 + b*x + a)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.33202, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{4 \, \sqrt{c x^{5} + b x^{4} + a x^{3}}{\left (a b x + 2 \, a^{2}\right )} \sqrt{x} -{\left (8 \, a b x^{3} +{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a^{2} x^{2}\right )} \sqrt{a}}{x^{4}}\right )}{2 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{b x^{3} + 2 \, a x^{2}}{2 \, \sqrt{c x^{5} + b x^{4} + a x^{3}} \sqrt{-a} \sqrt{x}}\right )}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/sqrt((c*x^2 + b*x + a)*x^3),x, algorithm="fricas")

[Out]

[1/2*log(-(4*sqrt(c*x^5 + b*x^4 + a*x^3)*(a*b*x + 2*a^2)*sqrt(x) - (8*a*b*x^3 +
(b^2 + 4*a*c)*x^4 + 8*a^2*x^2)*sqrt(a))/x^4)/sqrt(a), -sqrt(-a)*arctan(1/2*(b*x^
3 + 2*a*x^2)/(sqrt(c*x^5 + b*x^4 + a*x^3)*sqrt(-a)*sqrt(x)))/a]

_______________________________________________________________________________________

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(x**(1/2)/(x**3*(c*x**2+b*x+a))**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.28359, size = 47, normalized size = 0.96 \[ \frac{2 \, \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/sqrt((c*x^2 + b*x + a)*x^3),x, algorithm="giac")

[Out]

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