Optimal. Leaf size=69 \[ -\frac{b (A b-a B) \log \left (a+b x^2\right )}{2 a^3}+\frac{b \log (x) (A b-a B)}{a^3}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{4 a x^4} \]
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Rubi [A] time = 0.152458, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{b (A b-a B) \log \left (a+b x^2\right )}{2 a^3}+\frac{b \log (x) (A b-a B)}{a^3}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^5*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 20.0104, size = 63, normalized size = 0.91 \[ - \frac{A}{4 a x^{4}} + \frac{A b - B a}{2 a^{2} x^{2}} + \frac{b \left (A b - B a\right ) \log{\left (x^{2} \right )}}{2 a^{3}} - \frac{b \left (A b - B a\right ) \log{\left (a + b x^{2} \right )}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**5/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0478807, size = 70, normalized size = 1.01 \[ \frac{4 b x^4 \log (x) (A b-a B)-a \left (a A+2 a B x^2-2 A b x^2\right )+2 b x^4 (a B-A b) \log \left (a+b x^2\right )}{4 a^3 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^5*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 81, normalized size = 1.2 \[ -{\frac{A}{4\,a{x}^{4}}}+{\frac{Ab}{2\,{a}^{2}{x}^{2}}}-{\frac{B}{2\,a{x}^{2}}}+{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{3}}}-{\frac{bB\ln \left ( x \right ) }{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{3}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) B}{2\,{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^5/(b*x^2+a),x)
[Out]
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Maxima [A] time = 1.35597, size = 95, normalized size = 1.38 \[ \frac{{\left (B a b - A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} - \frac{{\left (B a b - A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac{2 \,{\left (B a - A b\right )} x^{2} + A a}{4 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236855, size = 99, normalized size = 1.43 \[ \frac{2 \,{\left (B a b - A b^{2}\right )} x^{4} \log \left (b x^{2} + a\right ) - 4 \,{\left (B a b - A b^{2}\right )} x^{4} \log \left (x\right ) - A a^{2} - 2 \,{\left (B a^{2} - A a b\right )} x^{2}}{4 \, a^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.80495, size = 61, normalized size = 0.88 \[ - \frac{A a + x^{2} \left (- 2 A b + 2 B a\right )}{4 a^{2} x^{4}} - \frac{b \left (- A b + B a\right ) \log{\left (x \right )}}{a^{3}} + \frac{b \left (- A b + B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**5/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.229093, size = 135, normalized size = 1.96 \[ -\frac{{\left (B a b - A b^{2}\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} + \frac{{\left (B a b^{2} - A b^{3}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b} + \frac{3 \, B a b x^{4} - 3 \, A b^{2} x^{4} - 2 \, B a^{2} x^{2} + 2 \, A a b x^{2} - A a^{2}}{4 \, a^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^5),x, algorithm="giac")
[Out]