Optimal. Leaf size=113 \[ -\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac{b^2 (4 A b-3 a B) \log (a+b x)}{a^5}-\frac{b^2 (A b-a B)}{a^4 (a+b x)}-\frac{b (3 A b-2 a B)}{a^4 x}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{A}{3 a^2 x^3} \]
[Out]
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Rubi [A] time = 0.211752, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac{b^2 (4 A b-3 a B) \log (a+b x)}{a^5}-\frac{b^2 (A b-a B)}{a^4 (a+b x)}-\frac{b (3 A b-2 a B)}{a^4 x}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{A}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^4*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 42.7386, size = 107, normalized size = 0.95 \[ - \frac{A}{3 a^{2} x^{3}} + \frac{2 A b - B a}{2 a^{3} x^{2}} - \frac{b^{2} \left (A b - B a\right )}{a^{4} \left (a + b x\right )} - \frac{b \left (3 A b - 2 B a\right )}{a^{4} x} - \frac{4 b^{2} \left (A b - \frac{3 B a}{4}\right ) \log{\left (x \right )}}{a^{5}} + \frac{b^{2} \left (4 A b - 3 B a\right ) \log{\left (a + b x \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**4/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.177921, size = 106, normalized size = 0.94 \[ \frac{-\frac{2 a^3 A}{x^3}-\frac{3 a^2 (a B-2 A b)}{x^2}+\frac{6 a b^2 (a B-A b)}{a+b x}+6 b^2 \log (x) (3 a B-4 A b)+6 b^2 (4 A b-3 a B) \log (a+b x)+\frac{6 a b (2 a B-3 A b)}{x}}{6 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^4*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
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Maple [A] time = 0.018, size = 134, normalized size = 1.2 \[ -{\frac{A}{3\,{a}^{2}{x}^{3}}}+{\frac{Ab}{{a}^{3}{x}^{2}}}-{\frac{B}{2\,{a}^{2}{x}^{2}}}-3\,{\frac{{b}^{2}A}{{a}^{4}x}}+2\,{\frac{Bb}{{a}^{3}x}}-4\,{\frac{A{b}^{3}\ln \left ( x \right ) }{{a}^{5}}}+3\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{4}}}-{\frac{A{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}+{\frac{{b}^{2}B}{{a}^{3} \left ( bx+a \right ) }}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{5}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^4/(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.686314, size = 173, normalized size = 1.53 \[ -\frac{2 \, A a^{3} - 6 \,{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} +{\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} x}{6 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} - \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{5}} + \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (x\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.293844, size = 242, normalized size = 2.14 \[ -\frac{2 \, A a^{4} - 6 \,{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x + 6 \,{\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} +{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} +{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.3944, size = 219, normalized size = 1.94 \[ \frac{- 2 A a^{3} + x^{3} \left (- 24 A b^{3} + 18 B a b^{2}\right ) + x^{2} \left (- 12 A a b^{2} + 9 B a^{2} b\right ) + x \left (4 A a^{2} b - 3 B a^{3}\right )}{6 a^{5} x^{3} + 6 a^{4} b x^{4}} + \frac{b^{2} \left (- 4 A b + 3 B a\right ) \log{\left (x + \frac{- 4 A a b^{3} + 3 B a^{2} b^{2} - a b^{2} \left (- 4 A b + 3 B a\right )}{- 8 A b^{4} + 6 B a b^{3}} \right )}}{a^{5}} - \frac{b^{2} \left (- 4 A b + 3 B a\right ) \log{\left (x + \frac{- 4 A a b^{3} + 3 B a^{2} b^{2} + a b^{2} \left (- 4 A b + 3 B a\right )}{- 8 A b^{4} + 6 B a b^{3}} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**4/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.273378, size = 180, normalized size = 1.59 \[ \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (3 \, B a b^{3} - 4 \, A b^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{2 \, A a^{4} - 6 \,{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x}{6 \,{\left (b x + a\right )} a^{5} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)*x^4),x, algorithm="giac")
[Out]