Optimal. Leaf size=140 \[ -\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}+\frac{2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}-\frac{b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 (A b-a B)}{2 a^4 (a+b x)^2}+\frac{3 A b-a B}{2 a^4 x^2}-\frac{A}{3 a^3 x^3} \]
[Out]
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Rubi [A] time = 0.287136, antiderivative size = 140, 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 \[ -\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}+\frac{2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}-\frac{b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 (A b-a B)}{2 a^4 (a+b x)^2}+\frac{3 A b-a B}{2 a^4 x^2}-\frac{A}{3 a^3 x^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^4*(a + b*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 53.5509, size = 134, normalized size = 0.96 \[ - \frac{A}{3 a^{3} x^{3}} - \frac{b^{2} \left (A b - B a\right )}{2 a^{4} \left (a + b x\right )^{2}} + \frac{3 A b - B a}{2 a^{4} x^{2}} - \frac{b^{2} \left (4 A b - 3 B a\right )}{a^{5} \left (a + b x\right )} - \frac{3 b \left (2 A b - B a\right )}{a^{5} x} - \frac{2 b^{2} \left (5 A b - 3 B a\right ) \log{\left (x \right )}}{a^{6}} + \frac{2 b^{2} \left (5 A b - 3 B a\right ) \log{\left (a + b x \right )}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**4/(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.238561, size = 129, normalized size = 0.92 \[ \frac{\frac{a \left (a^4 (-(2 A+3 B x))+a^3 b x (5 A+12 B x)+2 a^2 b^2 x^2 (27 B x-10 A)+18 a b^3 x^3 (2 B x-5 A)-60 A b^4 x^4\right )}{x^3 (a+b x)^2}+12 b^2 \log (x) (3 a B-5 A b)+12 b^2 (5 A b-3 a B) \log (a+b x)}{6 a^6} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^4*(a + b*x)^3),x]
[Out]
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Maple [A] time = 0.017, size = 168, normalized size = 1.2 \[ -{\frac{A}{3\,{a}^{3}{x}^{3}}}+{\frac{3\,Ab}{2\,{a}^{4}{x}^{2}}}-{\frac{B}{2\,{a}^{3}{x}^{2}}}-6\,{\frac{A{b}^{2}}{{a}^{5}x}}+3\,{\frac{Bb}{{a}^{4}x}}-10\,{\frac{A\ln \left ( x \right ){b}^{3}}{{a}^{6}}}+6\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{5}}}-4\,{\frac{A{b}^{3}}{{a}^{5} \left ( bx+a \right ) }}+3\,{\frac{B{b}^{2}}{{a}^{4} \left ( bx+a \right ) }}-{\frac{A{b}^{3}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{B{b}^{2}}{2\,{a}^{3} \left ( bx+a \right ) ^{2}}}+10\,{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{6}}}-6\,{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^4/(b*x+a)^3,x)
[Out]
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Maxima [A] time = 1.36404, size = 221, normalized size = 1.58 \[ -\frac{2 \, A a^{4} - 12 \,{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} +{\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x}{6 \,{\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} - \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{6}} + \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x\right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215402, size = 354, normalized size = 2.53 \[ -\frac{2 \, A a^{5} - 12 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x + 12 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 12 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.19622, size = 262, normalized size = 1.87 \[ \frac{- 2 A a^{4} + x^{4} \left (- 60 A b^{4} + 36 B a b^{3}\right ) + x^{3} \left (- 90 A a b^{3} + 54 B a^{2} b^{2}\right ) + x^{2} \left (- 20 A a^{2} b^{2} + 12 B a^{3} b\right ) + x \left (5 A a^{3} b - 3 B a^{4}\right )}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac{2 b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (x + \frac{- 10 A a b^{3} + 6 B a^{2} b^{2} - 2 a b^{2} \left (- 5 A b + 3 B a\right )}{- 20 A b^{4} + 12 B a b^{3}} \right )}}{a^{6}} - \frac{2 b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (x + \frac{- 10 A a b^{3} + 6 B a^{2} b^{2} + 2 a b^{2} \left (- 5 A b + 3 B a\right )}{- 20 A b^{4} + 12 B a b^{3}} \right )}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**4/(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.361478, size = 213, normalized size = 1.52 \[ \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{6}} - \frac{2 \,{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac{2 \, A a^{5} - 12 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x}{6 \,{\left (b x + a\right )}^{2} a^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^3*x^4),x, algorithm="giac")
[Out]