Optimal. Leaf size=109 \[ \frac{x \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}+\frac{a^4 \log (a+b x)}{b^4 (b c-a d)}-\frac{x^2 (a d+b c)}{2 b^2 d^2}-\frac{c^4 \log (c+d x)}{d^4 (b c-a d)}+\frac{x^3}{3 b d} \]
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Rubi [A] time = 0.0802101, antiderivative size = 109, 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} \[ \frac{x \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}+\frac{a^4 \log (a+b x)}{b^4 (b c-a d)}-\frac{x^2 (a d+b c)}{2 b^2 d^2}-\frac{c^4 \log (c+d x)}{d^4 (b c-a d)}+\frac{x^3}{3 b d} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{x^4}{(a+b x) (c+d x)} \, dx &=\int \left (\frac{b^2 c^2+a b c d+a^2 d^2}{b^3 d^3}-\frac{(b c+a d) x}{b^2 d^2}+\frac{x^2}{b d}+\frac{a^4}{b^3 (b c-a d) (a+b x)}+\frac{c^4}{d^3 (-b c+a d) (c+d x)}\right ) \, dx\\ &=\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) x}{b^3 d^3}-\frac{(b c+a d) x^2}{2 b^2 d^2}+\frac{x^3}{3 b d}+\frac{a^4 \log (a+b x)}{b^4 (b c-a d)}-\frac{c^4 \log (c+d x)}{d^4 (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0505118, size = 105, normalized size = 0.96 \[ \frac{b d x (b c-a d) \left (6 a^2 d^2-3 a b d (d x-2 c)+b^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )+6 a^4 d^4 \log (a+b x)-6 b^4 c^4 \log (c+d x)}{6 b^4 d^4 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 116, normalized size = 1.1 \begin{align*}{\frac{{x}^{3}}{3\,bd}}-{\frac{a{x}^{2}}{2\,{b}^{2}d}}-{\frac{c{x}^{2}}{2\,b{d}^{2}}}+{\frac{{a}^{2}x}{{b}^{3}d}}+{\frac{acx}{{b}^{2}{d}^{2}}}+{\frac{{c}^{2}x}{b{d}^{3}}}+{\frac{{c}^{4}\ln \left ( dx+c \right ) }{{d}^{4} \left ( ad-bc \right ) }}-{\frac{{a}^{4}\ln \left ( bx+a \right ) }{{b}^{4} \left ( ad-bc \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07471, size = 153, normalized size = 1.4 \begin{align*} \frac{a^{4} \log \left (b x + a\right )}{b^{5} c - a b^{4} d} - \frac{c^{4} \log \left (d x + c\right )}{b c d^{4} - a d^{5}} + \frac{2 \, b^{2} d^{2} x^{3} - 3 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} + 6 \,{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16297, size = 239, normalized size = 2.19 \begin{align*} \frac{6 \, a^{4} d^{4} \log \left (b x + a\right ) - 6 \, b^{4} c^{4} \log \left (d x + c\right ) + 2 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} - 3 \,{\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{2} + 6 \,{\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x}{6 \,{\left (b^{5} c d^{4} - a b^{4} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.92049, size = 252, normalized size = 2.31 \begin{align*} - \frac{a^{4} \log{\left (x + \frac{\frac{a^{6} d^{5}}{b \left (a d - b c\right )} - \frac{2 a^{5} c d^{4}}{a d - b c} + \frac{a^{4} b c^{2} d^{3}}{a d - b c} + a^{4} c d^{3} + a b^{3} c^{4}}{a^{4} d^{4} + b^{4} c^{4}} \right )}}{b^{4} \left (a d - b c\right )} + \frac{c^{4} \log{\left (x + \frac{a^{4} c d^{3} - \frac{a^{2} b^{3} c^{4} d}{a d - b c} + \frac{2 a b^{4} c^{5}}{a d - b c} + a b^{3} c^{4} - \frac{b^{5} c^{6}}{d \left (a d - b c\right )}}{a^{4} d^{4} + b^{4} c^{4}} \right )}}{d^{4} \left (a d - b c\right )} + \frac{x^{3}}{3 b d} - \frac{x^{2} \left (a d + b c\right )}{2 b^{2} d^{2}} + \frac{x \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right )}{b^{3} d^{3}} \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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