3.564 \(\int \frac{A+B x^2}{x^4 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{\sqrt{a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac{A \sqrt{a+b x^2}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0850896, antiderivative size = 53, 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 \[ \frac{\sqrt{a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac{A \sqrt{a+b x^2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^4*Sqrt[a + b*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.52643, size = 44, normalized size = 0.83 \[ - \frac{A \sqrt{a + b x^{2}}}{3 a x^{3}} + \frac{\sqrt{a + b x^{2}} \left (2 A b - 3 B a\right )}{3 a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**4/(b*x**2+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.042519, size = 39, normalized size = 0.74 \[ -\frac{\sqrt{a+b x^2} \left (a \left (A+3 B x^2\right )-2 A b x^2\right )}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^4*Sqrt[a + b*x^2]),x]

[Out]

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

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Maple [A]  time = 0.006, size = 36, normalized size = 0.7 \[ -{\frac{-2\,Ab{x}^{2}+3\,Ba{x}^{2}+Aa}{3\,{x}^{3}{a}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^4/(b*x^2+a)^(1/2),x)

[Out]

-1/3*(b*x^2+a)^(1/2)*(-2*A*b*x^2+3*B*a*x^2+A*a)/x^3/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)/(sqrt(b*x^2 + a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225003, size = 46, normalized size = 0.87 \[ -\frac{{\left ({\left (3 \, B a - 2 \, A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.64634, size = 70, normalized size = 1.32 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a x^{2}} + \frac{2 A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{2}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**4/(b*x**2+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.246316, size = 162, normalized size = 3.06 \[ \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B \sqrt{b} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a \sqrt{b} + 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A b^{\frac{3}{2}} + 3 \, B a^{2} \sqrt{b} - 2 \, A a b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*x^4),x, algorithm="giac")

[Out]

2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*sqrt(b) - 6*(sqrt(b)*x - sqrt(b*x^2 + a
))^2*B*a*sqrt(b) + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*b^(3/2) + 3*B*a^2*sqrt(b)
- 2*A*a*b^(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3