Optimal. Leaf size=77 \[ -\frac{a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac{x (a d+b c)}{b^2 d^2}+\frac{c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac{x^2}{2 b d} \]
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Rubi [A] time = 0.0562288, antiderivative size = 77, 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} \[ -\frac{a^3 \log (a+b x)}{b^3 (b c-a d)}-\frac{x (a d+b c)}{b^2 d^2}+\frac{c^3 \log (c+d x)}{d^3 (b c-a d)}+\frac{x^2}{2 b d} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{x^3}{(a+b x) (c+d x)} \, dx &=\int \left (\frac{-b c-a d}{b^2 d^2}+\frac{x}{b d}-\frac{a^3}{b^2 (b c-a d) (a+b x)}-\frac{c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx\\ &=-\frac{(b c+a d) x}{b^2 d^2}+\frac{x^2}{2 b d}-\frac{a^3 \log (a+b x)}{b^3 (b c-a d)}+\frac{c^3 \log (c+d x)}{d^3 (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0350276, size = 74, normalized size = 0.96 \[ \frac{-2 a^3 d^3 \log (a+b x)+b d x (b c-a d) (-2 a d-2 b c+b d x)+2 b^3 c^3 \log (c+d x)}{2 b^3 d^3 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 80, normalized size = 1. \begin{align*}{\frac{{x}^{2}}{2\,bd}}-{\frac{ax}{{b}^{2}d}}-{\frac{cx}{b{d}^{2}}}-{\frac{{c}^{3}\ln \left ( dx+c \right ) }{{d}^{3} \left ( ad-bc \right ) }}+{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{3} \left ( ad-bc \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12244, size = 104, normalized size = 1.35 \begin{align*} -\frac{a^{3} \log \left (b x + a\right )}{b^{4} c - a b^{3} d} + \frac{c^{3} \log \left (d x + c\right )}{b c d^{3} - a d^{4}} + \frac{b d x^{2} - 2 \,{\left (b c + a d\right )} x}{2 \, b^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31588, size = 189, normalized size = 2.45 \begin{align*} -\frac{2 \, a^{3} d^{3} \log \left (b x + a\right ) - 2 \, b^{3} c^{3} \log \left (d x + c\right ) -{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x}{2 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.6465, size = 219, normalized size = 2.84 \begin{align*} \frac{a^{3} \log{\left (x + \frac{\frac{a^{5} d^{4}}{b \left (a d - b c\right )} - \frac{2 a^{4} c d^{3}}{a d - b c} + \frac{a^{3} b c^{2} d^{2}}{a d - b c} + a^{3} c d^{2} + a b^{2} c^{3}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{b^{3} \left (a d - b c\right )} - \frac{c^{3} \log{\left (x + \frac{a^{3} c d^{2} - \frac{a^{2} b^{2} c^{3} d}{a d - b c} + \frac{2 a b^{3} c^{4}}{a d - b c} + a b^{2} c^{3} - \frac{b^{4} c^{5}}{d \left (a d - b c\right )}}{a^{3} d^{3} + b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )} + \frac{x^{2}}{2 b d} - \frac{x \left (a d + b c\right )}{b^{2} d^{2}} \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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