Optimal. Leaf size=55 \[ -\frac {A b-2 a B}{b^3 (a+b x)}+\frac {a (A b-a B)}{2 b^3 (a+b x)^2}+\frac {B \log (a+b x)}{b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {77} \[ -\frac {A b-2 a B}{b^3 (a+b x)}+\frac {a (A b-a B)}{2 b^3 (a+b x)^2}+\frac {B \log (a+b x)}{b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{(a+b x)^3} \, dx &=\int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^3}+\frac {A b-2 a B}{b^2 (a+b x)^2}+\frac {B}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {a (A b-a B)}{2 b^3 (a+b x)^2}-\frac {A b-2 a B}{b^3 (a+b x)}+\frac {B \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 0.98 \[ \frac {3 a^2 B-a b (A-4 B x)+2 B (a+b x)^2 \log (a+b x)-2 A b^2 x}{2 b^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 81, normalized size = 1.47 \[ \frac {3 \, B a^{2} - A a b + 2 \, {\left (2 \, B a b - A b^{2}\right )} x + 2 \, {\left (B b^{2} x^{2} + 2 \, B a b x + B a^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 54, normalized size = 0.98 \[ \frac {B \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac {2 \, {\left (2 \, B a - A b\right )} x + \frac {3 \, B a^{2} - A a b}{b}}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 1.27 \[ \frac {A a}{2 \left (b x +a \right )^{2} b^{2}}-\frac {B \,a^{2}}{2 \left (b x +a \right )^{2} b^{3}}-\frac {A}{\left (b x +a \right ) b^{2}}+\frac {2 B a}{\left (b x +a \right ) b^{3}}+\frac {B \ln \left (b x +a \right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 65, normalized size = 1.18 \[ \frac {3 \, B a^{2} - A a b + 2 \, {\left (2 \, B a b - A b^{2}\right )} x}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {B \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 63, normalized size = 1.15 \[ \frac {\frac {3\,B\,a^2-A\,a\,b}{2\,b^3}-\frac {x\,\left (A\,b-2\,B\,a\right )}{b^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {B\,\ln \left (a+b\,x\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 63, normalized size = 1.15 \[ \frac {B \log {\left (a + b x \right )}}{b^{3}} + \frac {- A a b + 3 B a^{2} + x \left (- 2 A b^{2} + 4 B a b\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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