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

Optimal. Leaf size=84 \[ \frac{x \sqrt{a+b x^2} (a B+2 A b)}{2 a}+\frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b}}-\frac{A \left (a+b x^2\right )^{3/2}}{a x} \]

[Out]

((2*A*b + a*B)*x*Sqrt[a + b*x^2])/(2*a) - (A*(a + b*x^2)^(3/2))/(a*x) + ((2*A*b
+ a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])

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Rubi [A]  time = 0.10249, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \sqrt{a+b x^2} (a B+2 A b)}{2 a}+\frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b}}-\frac{A \left (a+b x^2\right )^{3/2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

((2*A*b + a*B)*x*Sqrt[a + b*x^2])/(2*a) - (A*(a + b*x^2)^(3/2))/(a*x) + ((2*A*b
+ a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])

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Rubi in Sympy [A]  time = 10.8135, size = 71, normalized size = 0.85 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{a x} + \frac{\left (2 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 \sqrt{b}} + \frac{x \sqrt{a + b x^{2}} \left (2 A b + B a\right )}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(a + b*x**2)**(3/2)/(a*x) + (2*A*b + B*a)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(
2*sqrt(b)) + x*sqrt(a + b*x**2)*(2*A*b + B*a)/(2*a)

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Mathematica [A]  time = 0.0637051, size = 65, normalized size = 0.77 \[ \sqrt{a+b x^2} \left (\frac{B x}{2}-\frac{A}{x}\right )+\frac{(a B+2 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 93, normalized size = 1.1 \[{\frac{Bx}{2}\sqrt{b{x}^{2}+a}}+{\frac{Ba}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}-{\frac{A}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Axb}{a}\sqrt{b{x}^{2}+a}}+A\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*B*(b*x^2+a)^(1/2)+1/2*B*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-A*(b*x^2+a
)^(3/2)/a/x+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))

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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(b*x^2 + a)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221886, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B a + 2 \, A b\right )} x \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (B x^{2} - 2 \, A\right )} \sqrt{b x^{2} + a} \sqrt{b}}{4 \, \sqrt{b} x}, \frac{{\left (B a + 2 \, A b\right )} x \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (B x^{2} - 2 \, A\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{2 \, \sqrt{-b} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.9012, size = 107, normalized size = 1.27 \[ - \frac{A \sqrt{a}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + A \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )} - \frac{A b x}{\sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B \sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+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)*x*sqrt(1 + b*x**2/a)/2 + B*a*asinh(sqrt(
b)*x/sqrt(a))/(2*sqrt(b))

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GIAC/XCAS [A]  time = 0.241461, size = 113, normalized size = 1.35 \[ \frac{1}{2} \, \sqrt{b x^{2} + a} B x + \frac{2 \, A a \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} - \frac{{\left (B a \sqrt{b} + 2 \, A b^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(b*x^2 + a)*B*x + 2*A*a*sqrt(b)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a) -
1/4*(B*a*sqrt(b) + 2*A*b^(3/2))*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2)/b