3.512 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.180363, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x^2} (2 a B+A b)}{2 a}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{3/2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.0302, size = 68, normalized size = 0.81 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{2 a x^{2}} + \frac{\sqrt{a + b x^{2}} \left (\frac{A b}{2} + B a\right )}{a} - \frac{\left (\frac{A b}{2} + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.116005, size = 81, normalized size = 0.96 \[ \sqrt{a+b x^2} \left (B-\frac{A}{2 x^2}\right )+\frac{(-2 a B-A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{2 \sqrt{a}}-\frac{\log (x) (-2 a B-A b)}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^3,x]

[Out]

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

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Maple [A]  time = 0.012, size = 106, normalized size = 1.3 \[ -{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Ab}{2\,a}\sqrt{b{x}^{2}+a}}-B\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +B\sqrt{b{x}^{2}+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(b*x^2+a)^(1/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23009, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, B a + A b\right )} x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (2 \, B x^{2} - A\right )} \sqrt{b x^{2} + a} \sqrt{a}}{4 \, \sqrt{a} x^{2}}, -\frac{{\left (2 \, B a + A b\right )} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (2 \, B x^{2} - A\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{2 \, \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 31.7916, size = 107, normalized size = 1.27 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 \sqrt{a}} - B \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )} + \frac{B a}{\sqrt{b} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B \sqrt{b} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**3,x)

[Out]

-A*sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*x) - A*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*sqrt(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 = 0.242215, size = 92, normalized size = 1.1 \[ \frac{2 \, \sqrt{b x^{2} + a} B b + \frac{{\left (2 \, B a b + A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x^{2} + a} A b}{x^{2}}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^3,x, algorithm="giac")

[Out]

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