Optimal. Leaf size=86 \[ -\frac {b^2 \log (x) (A b-a B)}{a^4}+\frac {b^2 (A b-a B) \log (a+b x)}{a^4}-\frac {b (A b-a B)}{a^3 x}+\frac {A b-a B}{2 a^2 x^2}-\frac {A}{3 a x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 86, 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} \[ -\frac {b^2 \log (x) (A b-a B)}{a^4}+\frac {b^2 (A b-a B) \log (a+b x)}{a^4}+\frac {A b-a B}{2 a^2 x^2}-\frac {b (A b-a B)}{a^3 x}-\frac {A}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{x^4 (a+b x)} \, dx &=\int \left (\frac {A}{a x^4}+\frac {-A b+a B}{a^2 x^3}-\frac {b (-A b+a B)}{a^3 x^2}+\frac {b^2 (-A b+a B)}{a^4 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {A}{3 a x^3}+\frac {A b-a B}{2 a^2 x^2}-\frac {b (A b-a B)}{a^3 x}-\frac {b^2 (A b-a B) \log (x)}{a^4}+\frac {b^2 (A b-a B) \log (a+b x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 81, normalized size = 0.94 \[ \frac {\frac {a \left (-\left (a^2 (2 A+3 B x)\right )+3 a b x (A+2 B x)-6 A b^2 x^2\right )}{x^3}+6 b^2 \log (x) (a B-A b)+6 b^2 (A b-a B) \log (a+b x)}{6 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 94, normalized size = 1.09 \[ -\frac {6 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (b x + a\right ) - 6 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} \log \relax (x) + 2 \, A a^{3} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 99, normalized size = 1.15 \[ \frac {{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, A a^{3} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 101, normalized size = 1.17 \[ -\frac {A \,b^{3} \ln \relax (x )}{a^{4}}+\frac {A \,b^{3} \ln \left (b x +a \right )}{a^{4}}+\frac {B \,b^{2} \ln \relax (x )}{a^{3}}-\frac {B \,b^{2} \ln \left (b x +a \right )}{a^{3}}-\frac {A \,b^{2}}{a^{3} x}+\frac {B b}{a^{2} x}+\frac {A b}{2 a^{2} x^{2}}-\frac {B}{2 a \,x^{2}}-\frac {A}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 89, normalized size = 1.03 \[ -\frac {{\left (B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{a^{4}} + \frac {{\left (B a b^{2} - A b^{3}\right )} \log \relax (x)}{a^{4}} - \frac {2 \, A a^{2} - 6 \, {\left (B a b - A b^{2}\right )} x^{2} + 3 \, {\left (B a^{2} - A a b\right )} x}{6 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 96, normalized size = 1.12 \[ \frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,\left (A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (A\,b^3-B\,a\,b^2\right )}\right )\,\left (A\,b-B\,a\right )}{a^4}-\frac {\frac {A}{3\,a}-\frac {x\,\left (A\,b-B\,a\right )}{2\,a^2}+\frac {b\,x^2\,\left (A\,b-B\,a\right )}{a^3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.55, size = 165, normalized size = 1.92 \[ \frac {- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 6 B a b\right ) + x \left (3 A a b - 3 B a^{2}\right )}{6 a^{3} x^{3}} + \frac {b^{2} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{3} + B a^{2} b^{2} - a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} - \frac {b^{2} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{3} + B a^{2} b^{2} + a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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