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

Optimal. Leaf size=55 \[ \frac{B \sqrt{b x^2+c x^4}}{c x}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0601968, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{B \sqrt{b x^2+c x^4}}{c x}-\frac{A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.12021, size = 46, normalized size = 0.84 \[ - \frac{A \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{\sqrt{b}} + \frac{B \sqrt{b x^{2} + c x^{4}}}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.095474, size = 91, normalized size = 1.65 \[ \frac{x \left (A c \log (x) \sqrt{b+c x^2}-A c \sqrt{b+c x^2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+\sqrt{b} B \left (b+c x^2\right )\right )}{\sqrt{b} c \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 72, normalized size = 1.3 \[ -{\frac{x}{c}\sqrt{c{x}^{2}+b} \left ( A\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) c-B\sqrt{c{x}^{2}+b}\sqrt{b} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235248, size = 1, normalized size = 0.02 \[ \left [\frac{A \sqrt{b} c x \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}} B b}{2 \, b c x}, \frac{A \sqrt{-b} c x \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} B b}{b c x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238072, size = 81, normalized size = 1.47 \[ \frac{A{\rm ln}\left ({\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2}\right )}{2 \, \sqrt{b}} - \frac{2 \, B \sqrt{b}}{{\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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