3.136 \(\int \frac{\sqrt{x}}{\sqrt{x^3 \left (a+b x^2+c x^4\right )}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{\tanh ^{-1}\left (\frac{x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^3+b x^5+c x^7}}\right )}{2 \sqrt{a}} \]

[Out]

-ArcTanh[(x^(3/2)*(2*a + b*x^2))/(2*Sqrt[a]*Sqrt[a*x^3 + b*x^5 + c*x^7])]/(2*Sqr
t[a])

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Rubi [A]  time = 0.120266, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{\tanh ^{-1}\left (\frac{x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^3+b x^5+c x^7}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/Sqrt[x^3*(a + b*x^2 + c*x^4)],x]

[Out]

-ArcTanh[(x^(3/2)*(2*a + b*x^2))/(2*Sqrt[a]*Sqrt[a*x^3 + b*x^5 + c*x^7])]/(2*Sqr
t[a])

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Rubi in Sympy [A]  time = 22.7174, size = 78, normalized size = 1.47 \[ - \frac{x^{\frac{3}{2}} \sqrt{a + b x^{2} + c x^{4}} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 \sqrt{a} \sqrt{a x^{3} + b x^{5} + c x^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(x**3*(c*x**4+b*x**2+a))**(1/2),x)

[Out]

-x**(3/2)*sqrt(a + b*x**2 + c*x**4)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x
**2 + c*x**4)))/(2*sqrt(a)*sqrt(a*x**3 + b*x**5 + c*x**7))

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Mathematica [A]  time = 0.0846057, size = 91, normalized size = 1.72 \[ \frac{x^{3/2} \sqrt{a+b x^2+c x^4} \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )\right )}{2 \sqrt{a} \sqrt{x^3 \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/Sqrt[x^3*(a + b*x^2 + c*x^4)],x]

[Out]

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

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Maple [A]  time = 0.012, size = 74, normalized size = 1.4 \[ -{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{c{x}^{4}+b{x}^{2}+a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{{x}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(x^3*(c*x^4+b*x^2+a))^(1/2),x)

[Out]

-1/2/(x^3*(c*x^4+b*x^2+a))^(1/2)*x^(3/2)*(c*x^4+b*x^2+a)^(1/2)/a^(1/2)*ln((2*a+b
*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^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(sqrt(x)/sqrt((c*x^4 + b*x^2 + a)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282487, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{4 \, \sqrt{c x^{7} + b x^{5} + a x^{3}}{\left (a b x^{2} + 2 \, a^{2}\right )} \sqrt{x} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{4} + 8 \, a^{2} x^{2}\right )} \sqrt{a}}{x^{6}}\right )}{4 \, \sqrt{a}}, \frac{\sqrt{-a} \arctan \left (\frac{{\left (b x^{4} + 2 \, a x^{2}\right )} \sqrt{-a}}{2 \, \sqrt{c x^{7} + b x^{5} + a x^{3}} a \sqrt{x}}\right )}{2 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/sqrt((c*x^4 + b*x^2 + a)*x^3),x, algorithm="fricas")

[Out]

[1/4*log((4*sqrt(c*x^7 + b*x^5 + a*x^3)*(a*b*x^2 + 2*a^2)*sqrt(x) - ((b^2 + 4*a*
c)*x^6 + 8*a*b*x^4 + 8*a^2*x^2)*sqrt(a))/x^6)/sqrt(a), 1/2*sqrt(-a)*arctan(1/2*(
b*x^4 + 2*a*x^2)*sqrt(-a)/(sqrt(c*x^7 + b*x^5 + a*x^3)*a*sqrt(x)))/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(x**(1/2)/(x**3*(c*x**4+b*x**2+a))**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{{\left (c x^{4} + b x^{2} + a\right )} x^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/sqrt((c*x^4 + b*x^2 + a)*x^3),x, algorithm="giac")

[Out]

integrate(sqrt(x)/sqrt((c*x^4 + b*x^2 + a)*x^3), x)