Optimal. Leaf size=73 \[ -\frac{(b c-a d)^2}{a^2 b (a+b x)}-\frac{2 c \log (x) (b c-a d)}{a^3}+\frac{2 c (b c-a d) \log (a+b x)}{a^3}-\frac{c^2}{a^2 x} \]
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Rubi [A] time = 0.0498078, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ -\frac{(b c-a d)^2}{a^2 b (a+b x)}-\frac{2 c \log (x) (b c-a d)}{a^3}+\frac{2 c (b c-a d) \log (a+b x)}{a^3}-\frac{c^2}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{x^2 (a+b x)^2} \, dx &=\int \left (\frac{c^2}{a^2 x^2}+\frac{2 c (-b c+a d)}{a^3 x}+\frac{(-b c+a d)^2}{a^2 (a+b x)^2}-\frac{2 b c (-b c+a d)}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac{c^2}{a^2 x}-\frac{(b c-a d)^2}{a^2 b (a+b x)}-\frac{2 c (b c-a d) \log (x)}{a^3}+\frac{2 c (b c-a d) \log (a+b x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0775493, size = 67, normalized size = 0.92 \[ \frac{-\frac{a (b c-a d)^2}{b (a+b x)}+2 c \log (x) (a d-b c)+2 c (b c-a d) \log (a+b x)-\frac{a c^2}{x}}{a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 106, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}}{{a}^{2}x}}+2\,{\frac{c\ln \left ( x \right ) d}{{a}^{2}}}-2\,{\frac{{c}^{2}\ln \left ( x \right ) b}{{a}^{3}}}-{\frac{{d}^{2}}{b \left ( bx+a \right ) }}+2\,{\frac{cd}{a \left ( bx+a \right ) }}-{\frac{{c}^{2}b}{{a}^{2} \left ( bx+a \right ) }}-2\,{\frac{c\ln \left ( bx+a \right ) d}{{a}^{2}}}+2\,{\frac{{c}^{2}\ln \left ( bx+a \right ) b}{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71824, size = 126, normalized size = 1.73 \begin{align*} -\frac{a b c^{2} +{\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{a^{2} b^{2} x^{2} + a^{3} b x} + \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left (b x + a\right )}{a^{3}} - \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left (x\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24454, size = 297, normalized size = 4.07 \begin{align*} -\frac{a^{2} b c^{2} +{\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x - 2 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{2} +{\left (a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \log \left (b x + a\right ) + 2 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{2} +{\left (a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \log \left (x\right )}{a^{3} b^{2} x^{2} + a^{4} b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.08552, size = 173, normalized size = 2.37 \begin{align*} - \frac{a b c^{2} + x \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{a^{3} b x + a^{2} b^{2} x^{2}} + \frac{2 c \left (a d - b c\right ) \log{\left (x + \frac{2 a^{2} c d - 2 a b c^{2} - 2 a c \left (a d - b c\right )}{4 a b c d - 4 b^{2} c^{2}} \right )}}{a^{3}} - \frac{2 c \left (a d - b c\right ) \log{\left (x + \frac{2 a^{2} c d - 2 a b c^{2} + 2 a c \left (a d - b c\right )}{4 a b c d - 4 b^{2} c^{2}} \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18589, size = 150, normalized size = 2.05 \begin{align*} \frac{b c^{2}}{a^{3}{\left (\frac{a}{b x + a} - 1\right )}} - \frac{2 \,{\left (b^{2} c^{2} - a b c d\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3} b} - \frac{\frac{b^{3} c^{2}}{b x + a} - \frac{2 \, a b^{2} c d}{b x + a} + \frac{a^{2} b d^{2}}{b x + a}}{a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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