Optimal. Leaf size=113 \[ -\frac {b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac {b^2 (4 A b-3 a B) \log (a+b x)}{a^5}-\frac {b^2 (A b-a B)}{a^4 (a+b x)}-\frac {b (3 A b-2 a B)}{a^4 x}+\frac {2 A b-a B}{2 a^3 x^2}-\frac {A}{3 a^2 x^3} \]
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Rubi [A] time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ -\frac {b^2 (A b-a B)}{a^4 (a+b x)}-\frac {b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac {b^2 (4 A b-3 a B) \log (a+b x)}{a^5}+\frac {2 A b-a B}{2 a^3 x^2}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {A}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^4 (a+b x)^2} \, dx &=\int \left (\frac {A}{a^2 x^4}+\frac {-2 A b+a B}{a^3 x^3}-\frac {b (-3 A b+2 a B)}{a^4 x^2}+\frac {b^2 (-4 A b+3 a B)}{a^5 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)^2}-\frac {b^3 (-4 A b+3 a B)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac {A}{3 a^2 x^3}+\frac {2 A b-a B}{2 a^3 x^2}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {b^2 (A b-a B)}{a^4 (a+b x)}-\frac {b^2 (4 A b-3 a B) \log (x)}{a^5}+\frac {b^2 (4 A b-3 a B) \log (a+b x)}{a^5}\\ \end {align*}
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Mathematica [A] time = 0.11, 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.
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fricas [A] time = 1.00, size = 179, normalized size = 1.58 \[ -\frac {2 \, A a^{4} - 6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 3 \, {\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} + {\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x + 6 \, {\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} + {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} + {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \relax (x)}{6 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 161, normalized size = 1.42 \[ \frac {{\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{5} b} + \frac {\frac {B a b^{6}}{b x + a} - \frac {A b^{7}}{b x + a}}{a^{4} b^{4}} - \frac {15 \, B a b^{2} - 26 \, A b^{3} - \frac {3 \, {\left (11 \, B a^{2} b^{3} - 20 \, A a b^{4}\right )}}{{\left (b x + a\right )} b} + \frac {18 \, {\left (B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \, a^{5} {\left (\frac {a}{b x + a} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 134, normalized size = 1.19 \[ -\frac {A \,b^{3}}{\left (b x +a \right ) a^{4}}-\frac {4 A \,b^{3} \ln \relax (x )}{a^{5}}+\frac {4 A \,b^{3} \ln \left (b x +a \right )}{a^{5}}+\frac {B \,b^{2}}{\left (b x +a \right ) a^{3}}+\frac {3 B \,b^{2} \ln \relax (x )}{a^{4}}-\frac {3 B \,b^{2} \ln \left (b x +a \right )}{a^{4}}-\frac {3 A \,b^{2}}{a^{4} x}+\frac {2 B b}{a^{3} x}+\frac {A b}{a^{3} x^{2}}-\frac {B}{2 a^{2} x^{2}}-\frac {A}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 128, normalized size = 1.13 \[ -\frac {2 \, A a^{3} - 6 \, {\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \, {\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} + {\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} x}{6 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} - \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{5}} + \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \relax (x)}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 131, normalized size = 1.16 \[ \frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,\left (4\,A\,b-3\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (4\,A\,b^3-3\,B\,a\,b^2\right )}\right )\,\left (4\,A\,b-3\,B\,a\right )}{a^5}-\frac {\frac {A}{3\,a}-\frac {x\,\left (4\,A\,b-3\,B\,a\right )}{6\,a^2}+\frac {b^2\,x^3\,\left (4\,A\,b-3\,B\,a\right )}{a^4}+\frac {b\,x^2\,\left (4\,A\,b-3\,B\,a\right )}{2\,a^3}}{b\,x^4+a\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.84, 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.
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