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

Optimal. Leaf size=57 \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{c}}-\frac{A \sqrt{b x^2+c x^4}}{b x^2} \]

[Out]

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

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Rubi [A]  time = 0.29436, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{c}}-\frac{A \sqrt{b x^2+c x^4}}{b x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.3675, size = 49, normalized size = 0.86 \[ - \frac{A \sqrt{b x^{2} + c x^{4}}}{b x^{2}} + \frac{B \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0588401, size = 77, normalized size = 1.35 \[ \frac{b B x \sqrt{b+c x^2} \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )-A \sqrt{c} \left (b+c x^2\right )}{b \sqrt{c} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 67, normalized size = 1.2 \[ -{\frac{1}{b}\sqrt{c{x}^{2}+b} \left ( -B\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+b} \right ) xb+A\sqrt{c{x}^{2}+b}\sqrt{c} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231289, size = 1, normalized size = 0.02 \[ \left [\frac{B b \sqrt{c} x^{2} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) - 2 \, \sqrt{c x^{4} + b x^{2}} A c}{2 \, b c x^{2}}, -\frac{B b \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} A c}{b c x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224842, size = 54, normalized size = 0.95 \[ -\frac{B \arctan \left (\frac{\sqrt{c + \frac{b}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{A \sqrt{c + \frac{b}{x^{2}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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