Optimal. Leaf size=62 \[ \frac{b c-a d}{a^2 x}+\frac{b \log (x) (b c-a d)}{a^3}-\frac{b (b c-a d) \log (a+b x)}{a^3}-\frac{c}{2 a x^2} \]
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Rubi [A] time = 0.039749, antiderivative size = 62, 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, Rules used = {77} \[ \frac{b c-a d}{a^2 x}+\frac{b \log (x) (b c-a d)}{a^3}-\frac{b (b c-a d) \log (a+b x)}{a^3}-\frac{c}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{c+d x}{x^3 (a+b x)} \, dx &=\int \left (\frac{c}{a x^3}+\frac{-b c+a d}{a^2 x^2}-\frac{b (-b c+a d)}{a^3 x}+\frac{b^2 (-b c+a d)}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac{c}{2 a x^2}+\frac{b c-a d}{a^2 x}+\frac{b (b c-a d) \log (x)}{a^3}-\frac{b (b c-a d) \log (a+b x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0316975, size = 58, normalized size = 0.94 \[ \frac{-\frac{a (a c+2 a d x-2 b c x)}{x^2}+2 b \log (x) (b c-a d)+2 b (a d-b c) \log (a+b x)}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 75, normalized size = 1.2 \begin{align*} -{\frac{c}{2\,a{x}^{2}}}-{\frac{d}{ax}}+{\frac{bc}{{a}^{2}x}}-{\frac{b\ln \left ( x \right ) d}{{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) c}{{a}^{3}}}+{\frac{b\ln \left ( bx+a \right ) d}{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) c}{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.169, size = 85, normalized size = 1.37 \begin{align*} -\frac{{\left (b^{2} c - a b d\right )} \log \left (b x + a\right )}{a^{3}} + \frac{{\left (b^{2} c - a b d\right )} \log \left (x\right )}{a^{3}} - \frac{a c - 2 \,{\left (b c - a d\right )} x}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98901, size = 154, normalized size = 2.48 \begin{align*} -\frac{2 \,{\left (b^{2} c - a b d\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (b^{2} c - a b d\right )} x^{2} \log \left (x\right ) + a^{2} c - 2 \,{\left (a b c - a^{2} d\right )} x}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.718709, size = 131, normalized size = 2.11 \begin{align*} - \frac{a c + x \left (2 a d - 2 b c\right )}{2 a^{2} x^{2}} - \frac{b \left (a d - b c\right ) \log{\left (x + \frac{a^{2} b d - a b^{2} c - a b \left (a d - b c\right )}{2 a b^{2} d - 2 b^{3} c} \right )}}{a^{3}} + \frac{b \left (a d - b c\right ) \log{\left (x + \frac{a^{2} b d - a b^{2} c + a b \left (a d - b c\right )}{2 a b^{2} d - 2 b^{3} c} \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17258, size = 101, normalized size = 1.63 \begin{align*} \frac{{\left (b^{2} c - a b d\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c - a b^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{a^{2} c - 2 \,{\left (a b c - a^{2} d\right )} x}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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