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

Optimal. Leaf size=58 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x^2}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.142435, antiderivative size = 58, 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 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.108, size = 48, normalized size = 0.83 \[ - \frac{A \sqrt{a + b x^{2}}}{2 a x^{2}} + \frac{\left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0721462, size = 81, normalized size = 1.4 \[ -\frac{(2 a B-A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{2 a^{3/2}}+\frac{\log (x) (2 a B-A b)}{2 a^{3/2}}-\frac{A \sqrt{a+b x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 79, normalized size = 1.4 \[ -{\frac{A}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23573, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (2 \, B a - A b\right )} x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \, \sqrt{b x^{2} + a} A \sqrt{a}}{4 \, a^{\frac{3}{2}} x^{2}}, -\frac{{\left (2 \, B a - A b\right )} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + \sqrt{b x^{2} + a} A \sqrt{-a}}{2 \, \sqrt{-a} a x^{2}}\right ] \]

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

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Sympy [A]  time = 23.598, size = 66, normalized size = 1.14 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} + \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} - \frac{B \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*a*x) + A*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(
3/2)) - B*asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a)

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GIAC/XCAS [A]  time = 0.244565, size = 84, normalized size = 1.45 \[ \frac{\frac{{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{b x^{2} + a} A b}{a x^{2}}}{2 \, b} \]

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