Optimal. Leaf size=124 \[ \frac{b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}-\frac{b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac{b (3 A b-2 a B)}{2 a^4 x^2}+\frac{2 A b-a B}{4 a^3 x^4}-\frac{A}{6 a^2 x^6} \]
[Out]
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Rubi [A] time = 0.298546, antiderivative size = 124, 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 \[ \frac{b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}-\frac{b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac{b (3 A b-2 a B)}{2 a^4 x^2}+\frac{2 A b-a B}{4 a^3 x^4}-\frac{A}{6 a^2 x^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^7*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 34.9356, size = 119, normalized size = 0.96 \[ - \frac{A}{6 a^{2} x^{6}} + \frac{2 A b - B a}{4 a^{3} x^{4}} - \frac{b^{2} \left (A b - B a\right )}{2 a^{4} \left (a + b x^{2}\right )} - \frac{b \left (3 A b - 2 B a\right )}{2 a^{4} x^{2}} - \frac{b^{2} \left (4 A b - 3 B a\right ) \log{\left (x^{2} \right )}}{2 a^{5}} + \frac{b^{2} \left (4 A b - 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**7/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.186201, size = 110, normalized size = 0.89 \[ \frac{-\frac{2 a^3 A}{x^6}-\frac{3 a^2 (a B-2 A b)}{x^4}+\frac{6 a b^2 (a B-A b)}{a+b x^2}+6 b^2 (4 A b-3 a B) \log \left (a+b x^2\right )+12 b^2 \log (x) (3 a B-4 A b)+\frac{6 a b (2 a B-3 A b)}{x^2}}{12 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^7*(a + b*x^2)^2),x]
[Out]
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Maple [A] time = 0.023, size = 143, normalized size = 1.2 \[ -{\frac{A}{6\,{a}^{2}{x}^{6}}}+{\frac{Ab}{2\,{a}^{3}{x}^{4}}}-{\frac{B}{4\,{a}^{2}{x}^{4}}}-{\frac{3\,{b}^{2}A}{2\,{a}^{4}{x}^{2}}}+{\frac{Bb}{{a}^{3}{x}^{2}}}-4\,{\frac{{b}^{3}\ln \left ( x \right ) A}{{a}^{5}}}+3\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{4}}}+2\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) A}{{a}^{5}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) B}{2\,{a}^{4}}}-{\frac{A{b}^{3}}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{B{b}^{2}}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^7/(b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 1.35873, size = 184, normalized size = 1.48 \[ \frac{6 \,{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} x^{6} + 3 \,{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{4} - 2 \, A a^{3} -{\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} x^{2}}{12 \,{\left (a^{4} b x^{8} + a^{5} x^{6}\right )}} - \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{5}} + \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232668, size = 248, normalized size = 2. \[ \frac{6 \,{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} - 2 \, A a^{4} + 3 \,{\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{4} -{\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x^{2} - 6 \,{\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} +{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) + 12 \,{\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} +{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6}\right )} \log \left (x\right )}{12 \,{\left (a^{5} b x^{8} + a^{6} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.21963, size = 129, normalized size = 1.04 \[ \frac{- 2 A a^{3} + x^{6} \left (- 24 A b^{3} + 18 B a b^{2}\right ) + x^{4} \left (- 12 A a b^{2} + 9 B a^{2} b\right ) + x^{2} \left (4 A a^{2} b - 3 B a^{3}\right )}{12 a^{5} x^{6} + 12 a^{4} b x^{8}} + \frac{b^{2} \left (- 4 A b + 3 B a\right ) \log{\left (x \right )}}{a^{5}} - \frac{b^{2} \left (- 4 A b + 3 B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**7/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.233154, size = 240, normalized size = 1.94 \[ \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{5}} - \frac{{\left (3 \, B a b^{3} - 4 \, A b^{4}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5} b} + \frac{3 \, B a b^{3} x^{2} - 4 \, A b^{4} x^{2} + 4 \, B a^{2} b^{2} - 5 \, A a b^{3}}{2 \,{\left (b x^{2} + a\right )} a^{5}} - \frac{33 \, B a b^{2} x^{6} - 44 \, A b^{3} x^{6} - 12 \, B a^{2} b x^{4} + 18 \, A a b^{2} x^{4} + 3 \, B a^{3} x^{2} - 6 \, A a^{2} b x^{2} + 2 \, A a^{3}}{12 \, a^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^7),x, algorithm="giac")
[Out]