3.1019 \(\int \frac{1}{x^3 \sqrt{a+b x^2+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=50 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{\sqrt{a+b x^2}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0733049, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{\sqrt{a+b x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.23566, size = 41, normalized size = 0.82 \[ - \frac{\sqrt{a + b x^{2}}}{2 a x^{2}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0364934, size = 64, normalized size = 1.28 \[ \frac{b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{2 a^{3/2}}-\frac{b \log (x)}{2 a^{3/2}}-\frac{\sqrt{a+b x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 48, normalized size = 1. \[ -{\frac{1}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{b}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^2+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286097, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{2} + a} \sqrt{a}}{4 \, a^{\frac{3}{2}} x^{2}}, \frac{b x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) - \sqrt{b x^{2} + a} \sqrt{-a}}{2 \, \sqrt{-a} a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.30155, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*a*x) + b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(3/2)
)

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GIAC/XCAS [A]  time = 0.26404, size = 65, normalized size = 1.3 \[ -\frac{1}{2} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x^{2} + a}}{a b x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*x^3),x, algorithm="giac")

[Out]

-1/2*b*(arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) + sqrt(b*x^2 + a)/(a*b*x^2
))