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

Optimal. Leaf size=75 \[ -\frac{\sqrt{a+b x^2} (A-B x)}{x}+A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\sqrt{a} B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.205008, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\sqrt{a+b x^2} (A-B x)}{x}+A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\sqrt{a} B \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.6781, size = 65, normalized size = 0.87 \[ A \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - B \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} - \frac{\left (A - B x\right ) \sqrt{a + b x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0835402, size = 87, normalized size = 1.16 \[ \sqrt{a+b x^2} \left (B-\frac{A}{x}\right )+A \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )-\sqrt{a} B \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a} B \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 97, normalized size = 1.3 \[ -{\frac{A}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Axb}{a}\sqrt{b{x}^{2}+a}}+A\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) -B\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +B\sqrt{b{x}^{2}+a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/a/x*(b*x^2+a)^(3/2)+A*b/a*x*(b*x^2+a)^(1/2)+A*b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^
(1/2))-B*a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+B*(b*x^2+a)^(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(b*x^2 + a)*(B*x + A)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272651, size = 1, normalized size = 0.01 \[ \left [\frac{A \sqrt{b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + B \sqrt{a} x \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \, \sqrt{b x^{2} + a}{\left (B x - A\right )}}{2 \, x}, \frac{2 \, A \sqrt{-b} x \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) + B \sqrt{a} x \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \, \sqrt{b x^{2} + a}{\left (B x - A\right )}}{2 \, x}, -\frac{2 \, B \sqrt{-a} x \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) - A \sqrt{b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \, \sqrt{b x^{2} + a}{\left (B x - A\right )}}{2 \, x}, \frac{A \sqrt{-b} x \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) - B \sqrt{-a} x \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) + \sqrt{b x^{2} + a}{\left (B x - A\right )}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.8349, size = 124, normalized size = 1.65 \[ - \frac{A \sqrt{a}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + A \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )} - \frac{A b x}{\sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - B \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )} + \frac{B a}{\sqrt{b} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B \sqrt{b} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23325, size = 138, normalized size = 1.84 \[ \frac{2 \, B a \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - A \sqrt{b}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right ) + \sqrt{b x^{2} + a} B + \frac{2 \, A a \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*a*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - A*sqrt(b)*ln(ab
s(-sqrt(b)*x + sqrt(b*x^2 + a))) + sqrt(b*x^2 + a)*B + 2*A*a*sqrt(b)/((sqrt(b)*x
- sqrt(b*x^2 + a))^2 - a)