3.66 \(\int \frac{A+B x^2}{x^4 \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=59 \[ \frac{\sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3} \]

[Out]

-A/(3*a*x^3) + (A*b - a*B)/(a^2*x) + (Sqrt[b]*(A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqr
t[a]])/a^(5/2)

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Rubi [A]  time = 0.104534, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-A/(3*a*x^3) + (A*b - a*B)/(a^2*x) + (Sqrt[b]*(A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqr
t[a]])/a^(5/2)

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Rubi in Sympy [A]  time = 14.6906, size = 49, normalized size = 0.83 \[ - \frac{A}{3 a x^{3}} + \frac{A b - B a}{a^{2} x} + \frac{\sqrt{b} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(3*a*x**3) + (A*b - B*a)/(a**2*x) + sqrt(b)*(A*b - B*a)*atan(sqrt(b)*x/sqrt(a
))/a**(5/2)

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Mathematica [A]  time = 0.0912105, size = 60, normalized size = 1.02 \[ -\frac{\sqrt{b} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-A/(3*a*x^3) + (A*b - a*B)/(a^2*x) - (Sqrt[b]*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/
Sqrt[a]])/a^(5/2)

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Maple [A]  time = 0.009, size = 72, normalized size = 1.2 \[ -{\frac{A}{3\,a{x}^{3}}}+{\frac{Ab}{{a}^{2}x}}-{\frac{B}{ax}}+{\frac{{b}^{2}A}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{Bb}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^4/(b*x^2+a),x)

[Out]

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

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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)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23766, size = 1, normalized size = 0.02 \[ \left [-\frac{3 \,{\left (B a - A b\right )} x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 6 \,{\left (B a - A b\right )} x^{2} + 2 \, A a}{6 \, a^{2} x^{3}}, -\frac{3 \,{\left (B a - A b\right )} x^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 3 \,{\left (B a - A b\right )} x^{2} + A a}{3 \, a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(3*(B*a - A*b)*x^3*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 +
a)) + 6*(B*a - A*b)*x^2 + 2*A*a)/(a^2*x^3), -1/3*(3*(B*a - A*b)*x^3*sqrt(b/a)*ar
ctan(b*x/(a*sqrt(b/a))) + 3*(B*a - A*b)*x^2 + A*a)/(a^2*x^3)]

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Sympy [A]  time = 2.26094, size = 129, normalized size = 2.19 \[ \frac{\sqrt{- \frac{b}{a^{5}}} \left (- A b + B a\right ) \log{\left (- \frac{a^{3} \sqrt{- \frac{b}{a^{5}}} \left (- A b + B a\right )}{- A b^{2} + B a b} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{5}}} \left (- A b + B a\right ) \log{\left (\frac{a^{3} \sqrt{- \frac{b}{a^{5}}} \left (- A b + B a\right )}{- A b^{2} + B a b} + x \right )}}{2} - \frac{A a + x^{2} \left (- 3 A b + 3 B a\right )}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-b/a**5)*(-A*b + B*a)*log(-a**3*sqrt(-b/a**5)*(-A*b + B*a)/(-A*b**2 + B*a*b
) + x)/2 - sqrt(-b/a**5)*(-A*b + B*a)*log(a**3*sqrt(-b/a**5)*(-A*b + B*a)/(-A*b*
*2 + B*a*b) + x)/2 - (A*a + x**2*(-3*A*b + 3*B*a))/(3*a**2*x**3)

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GIAC/XCAS [A]  time = 0.225285, size = 77, normalized size = 1.31 \[ -\frac{{\left (B a b - A b^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} - \frac{3 \, B a x^{2} - 3 \, A b x^{2} + A a}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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