Optimal. Leaf size=56 \[ \frac{a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac{c^2 \log (c+d x)}{d^2 (b c-a d)}+\frac{x}{b d} \]
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Rubi [A] time = 0.0466322, antiderivative size = 56, 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 = {72} \[ \frac{a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac{c^2 \log (c+d x)}{d^2 (b c-a d)}+\frac{x}{b d} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{x^2}{(a+b x) (c+d x)} \, dx &=\int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx\\ &=\frac{x}{b d}+\frac{a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac{c^2 \log (c+d x)}{d^2 (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0352156, size = 56, normalized size = 1. \[ \frac{a^2 \log (a+b x)}{b^2 (b c-a d)}-\frac{c^2 \log (c+d x)}{d^2 (b c-a d)}+\frac{x}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 1. \begin{align*}{\frac{x}{bd}}-{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{2} \left ( ad-bc \right ) }}+{\frac{{c}^{2}\ln \left ( dx+c \right ) }{{d}^{2} \left ( ad-bc \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23451, size = 81, normalized size = 1.45 \begin{align*} \frac{a^{2} \log \left (b x + a\right )}{b^{3} c - a b^{2} d} - \frac{c^{2} \log \left (d x + c\right )}{b c d^{2} - a d^{3}} + \frac{x}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45653, size = 128, normalized size = 2.29 \begin{align*} \frac{a^{2} d^{2} \log \left (b x + a\right ) - b^{2} c^{2} \log \left (d x + c\right ) +{\left (b^{2} c d - a b d^{2}\right )} x}{b^{3} c d^{2} - a b^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.04137, size = 190, normalized size = 3.39 \begin{align*} - \frac{a^{2} \log{\left (x + \frac{\frac{a^{4} d^{3}}{b \left (a d - b c\right )} - \frac{2 a^{3} c d^{2}}{a d - b c} + \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + a b c^{2}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{b^{2} \left (a d - b c\right )} + \frac{c^{2} \log{\left (x + \frac{- \frac{a^{2} b c^{2} d}{a d - b c} + a^{2} c d + \frac{2 a b^{2} c^{3}}{a d - b c} + a b c^{2} - \frac{b^{3} c^{4}}{d \left (a d - b c\right )}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{d^{2} \left (a d - b c\right )} + \frac{x}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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