3.225 \(\int \frac{\sqrt{b x^2+c x^4}}{x^3} \, dx\)

Optimal. Leaf size=52 \[ \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{\sqrt{b x^2+c x^4}}{x^2} \]

[Out]

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

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Rubi [A]  time = 0.146377, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{\sqrt{b x^2+c x^4}}{x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.6633, size = 44, normalized size = 0.85 \[ \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )} - \frac{\sqrt{b x^{2} + c x^{4}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0490217, size = 66, normalized size = 1.27 \[ \frac{\sqrt{c} x \sqrt{b+c x^2} \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )-b-c x^2}{\sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 84, normalized size = 1.6 \[{\frac{1}{b{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( \sqrt{c{x}^{2}+b}{c}^{{\frac{3}{2}}}{x}^{2}+\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) xbc- \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}\sqrt{c} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2)/x^3,x)

[Out]

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

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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(c*x^4 + b*x^2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276618, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c} x^{2} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}}}{2 \, x^{2}}, \frac{\sqrt{-c} x^{2} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}\right ) - \sqrt{c x^{4} + b x^{2}}}{x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x**2*(b + c*x**2))/x**3, x)

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GIAC/XCAS [A]  time = 0.293318, size = 82, normalized size = 1.58 \[ -\frac{1}{2} \, \sqrt{c}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \, b \sqrt{c}{\rm sign}\left (x\right )}{{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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