3.27 \(\int \frac{A+B x}{x^2 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{A \sqrt{a+b x^2}}{a x}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.121638, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{A \sqrt{a+b x^2}}{a x}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.29236, size = 39, normalized size = 0.83 \[ - \frac{A \sqrt{a + b x^{2}}}{a x} - \frac{B \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0678633, size = 58, normalized size = 1.23 \[ -\frac{A \sqrt{a+b x^2}}{a x}-\frac{B \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}}+\frac{B \log (x)}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 49, normalized size = 1. \[ -{\frac{A}{ax}\sqrt{b{x}^{2}+a}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255294, size = 1, normalized size = 0.02 \[ \left [\frac{B a x \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{2} + a} A \sqrt{a}}{2 \, a^{\frac{3}{2}} x}, -\frac{B a x \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + \sqrt{b x^{2} + a} A \sqrt{-a}}{\sqrt{-a} a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(B*a*x*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2) - 2*sqrt(b*x
^2 + a)*A*sqrt(a))/(a^(3/2)*x), -(B*a*x*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + sqrt(
b*x^2 + a)*A*sqrt(-a))/(sqrt(-a)*a*x)]

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Sympy [A]  time = 3.01679, size = 41, normalized size = 0.87 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{a} - \frac{B \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217755, size = 88, normalized size = 1.87 \[ \frac{2 \, B \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2 \, A \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) + 2*A*sqrt(b)/((sqr
t(b)*x - sqrt(b*x^2 + a))^2 - a)