3.235 \(\int \frac{\sqrt{b x^2+c x^4}}{x^4} \, dx\)

Optimal. Leaf size=56 \[ -\frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 \sqrt{b}}-\frac{\sqrt{b x^2+c x^4}}{2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0920732, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 \sqrt{b}}-\frac{\sqrt{b x^2+c x^4}}{2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4]/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.9372, size = 49, normalized size = 0.88 \[ - \frac{\sqrt{b x^{2} + c x^{4}}}{2 x^{3}} - \frac{c \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b*x**2 + c*x**4)/(2*x**3) - c*atanh(sqrt(b)*x/sqrt(b*x**2 + c*x**4))/(2*sq
rt(b))

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Mathematica [A]  time = 0.0651876, size = 89, normalized size = 1.59 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{b} \sqrt{b+c x^2}+c x^2 \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )-c x^2 \log (x)\right )}{2 \sqrt{b} x^3 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^2 + c*x^4]/x^4,x]

[Out]

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

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Maple [A]  time = 0.009, size = 85, normalized size = 1.5 \[ -{\frac{1}{2\,b{x}^{3}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( \ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) \sqrt{b}{x}^{2}c-\sqrt{c{x}^{2}+b}{x}^{2}c+ \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2)/x^4,x)

[Out]

-1/2*(c*x^4+b*x^2)^(1/2)*(ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*b^(1/2)*x^2*c-(c*x
^2+b)^(1/2)*x^2*c+(c*x^2+b)^(3/2))/x^3/(c*x^2+b)^(1/2)/b

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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(c*x^4 + b*x^2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274876, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} c x^{3} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}} b}{4 \, b x^{3}}, \frac{\sqrt{-b} c x^{3} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) - \sqrt{c x^{4} + b x^{2}} b}{2 \, b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**(1/2)/x**4,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))/x**4, x)

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GIAC/XCAS [A]  time = 0.289783, size = 61, normalized size = 1.09 \[ \frac{1}{2} \, c{\left (\frac{\arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{c x^{2} + b}}{c x^{2}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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