Optimal. Leaf size=51 \[ -\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}+\frac{d^2 x}{b^2} \]
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Rubi [A] time = 0.0442666, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 43} \[ -\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}+\frac{d^2 x}{b^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^4} \, dx &=\int \frac{(c+d x)^2}{(a+b x)^2} \, dx\\ &=\int \left (\frac{d^2}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)^2}+\frac{2 d (b c-a d)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{d^2 x}{b^2}-\frac{(b c-a d)^2}{b^3 (a+b x)}+\frac{2 d (b c-a d) \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0364032, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{a+b x}+2 d (b c-a d) \log (a+b x)+b d^2 x}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 86, normalized size = 1.7 \begin{align*}{\frac{{d}^{2}x}{{b}^{2}}}-2\,{\frac{{d}^{2}\ln \left ( bx+a \right ) a}{{b}^{3}}}+2\,{\frac{d\ln \left ( bx+a \right ) c}{{b}^{2}}}-{\frac{{a}^{2}{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}+2\,{\frac{acd}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{c}^{2}}{b \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0366, size = 90, normalized size = 1.76 \begin{align*} \frac{d^{2} x}{b^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{b^{4} x + a b^{3}} + \frac{2 \,{\left (b c d - a d^{2}\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52461, size = 184, normalized size = 3.61 \begin{align*} \frac{b^{2} d^{2} x^{2} + a b d^{2} x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} + 2 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.765751, size = 60, normalized size = 1.18 \begin{align*} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{a b^{3} + b^{4} x} + \frac{d^{2} x}{b^{2}} - \frac{2 d \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22063, size = 88, normalized size = 1.73 \begin{align*} \frac{d^{2} x}{b^{2}} + \frac{2 \,{\left (b c d - a d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{{\left (b x + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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