Optimal. Leaf size=106 \[ \frac{b^3 \log (x) (A b-a B)}{a^5}-\frac{b^3 (A b-a B) \log (a+b x)}{a^5}+\frac{b^2 (A b-a B)}{a^4 x}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{4 a x^4} \]
[Out]
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Rubi [A] time = 0.153017, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{b^3 \log (x) (A b-a B)}{a^5}-\frac{b^3 (A b-a B) \log (a+b x)}{a^5}+\frac{b^2 (A b-a B)}{a^4 x}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^5*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 31.3941, size = 92, normalized size = 0.87 \[ - \frac{A}{4 a x^{4}} + \frac{A b - B a}{3 a^{2} x^{3}} - \frac{b \left (A b - B a\right )}{2 a^{3} x^{2}} + \frac{b^{2} \left (A b - B a\right )}{a^{4} x} + \frac{b^{3} \left (A b - B a\right ) \log{\left (x \right )}}{a^{5}} - \frac{b^{3} \left (A b - B a\right ) \log{\left (a + b x \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**5/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.11372, size = 100, normalized size = 0.94 \[ \frac{\frac{a \left (a^3 (-(3 A+4 B x))+2 a^2 b x (2 A+3 B x)-6 a b^2 x^2 (A+2 B x)+12 A b^3 x^3\right )}{x^4}+12 b^3 \log (x) (A b-a B)-12 b^3 (A b-a B) \log (a+b x)}{12 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^5*(a + b*x)),x]
[Out]
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Maple [A] time = 0.014, size = 125, normalized size = 1.2 \[ -{\frac{A}{4\,a{x}^{4}}}+{\frac{Ab}{3\,{a}^{2}{x}^{3}}}-{\frac{B}{3\,a{x}^{3}}}+{\frac{{b}^{4}\ln \left ( x \right ) A}{{a}^{5}}}-{\frac{{b}^{3}B\ln \left ( x \right ) }{{a}^{4}}}-{\frac{{b}^{2}A}{2\,{a}^{3}{x}^{2}}}+{\frac{bB}{2\,{a}^{2}{x}^{2}}}+{\frac{{b}^{3}A}{{a}^{4}x}}-{\frac{{b}^{2}B}{{a}^{3}x}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) A}{{a}^{5}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) B}{{a}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^5/(b*x+a),x)
[Out]
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Maxima [A] time = 1.37354, size = 151, normalized size = 1.42 \[ \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (x\right )}{a^{5}} - \frac{3 \, A a^{3} + 12 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 4 \,{\left (B a^{3} - A a^{2} b\right )} x}{12 \, a^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209221, size = 158, normalized size = 1.49 \[ \frac{12 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (x\right ) - 3 \, A a^{4} - 12 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \,{\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.64012, size = 189, normalized size = 1.78 \[ - \frac{3 A a^{3} + x^{3} \left (- 12 A b^{3} + 12 B a b^{2}\right ) + x^{2} \left (6 A a b^{2} - 6 B a^{2} b\right ) + x \left (- 4 A a^{2} b + 4 B a^{3}\right )}{12 a^{4} x^{4}} - \frac{b^{3} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{4} + B a^{2} b^{3} - a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} + \frac{b^{3} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{4} + B a^{2} b^{3} + a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**5/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.302465, size = 165, normalized size = 1.56 \[ -\frac{{\left (B a b^{3} - A b^{4}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} + \frac{{\left (B a b^{4} - A b^{5}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{3 \, A a^{4} + 12 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 6 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 4 \,{\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)*x^5),x, algorithm="giac")
[Out]