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

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

[Out]

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

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Rubi [A]  time = 0.0707354, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{x (2 A b-a B)}{a^2 \sqrt{a+b x^2}}-\frac{A}{a x \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.23623, size = 39, normalized size = 0.83 \[ - \frac{A}{a x \sqrt{a + b x^{2}}} - \frac{x \left (2 A b - B a\right )}{a^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0397755, size = 36, normalized size = 0.77 \[ \frac{-a A+a B x^2-2 A b x^2}{a^2 x \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \[ -{\frac{2\,Ab{x}^{2}-Ba{x}^{2}+Aa}{x{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224098, size = 58, normalized size = 1.23 \[ \frac{{\left ({\left (B a - 2 \, A b\right )} x^{2} - A a\right )} \sqrt{b x^{2} + a}}{a^{2} b x^{3} + a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 14.1452, size = 68, normalized size = 1.45 \[ A \left (- \frac{1}{a \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{2 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{2}} + 1}}\right ) + \frac{B x}{a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235366, size = 77, normalized size = 1.64 \[ \frac{2 \, A \sqrt{b}}{{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )} a} + \frac{{\left (B a - A b\right )} x}{\sqrt{b x^{2} + a} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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