Optimal. Leaf size=145 \[ \frac{x^2 \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 b^3 d^3}-\frac{x (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{b^4 d^4}-\frac{a^5 \log (a+b x)}{b^5 (b c-a d)}-\frac{x^3 (a d+b c)}{3 b^2 d^2}+\frac{c^5 \log (c+d x)}{d^5 (b c-a d)}+\frac{x^4}{4 b d} \]
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Rubi [A] time = 0.144992, antiderivative size = 145, 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} \[ \frac{x^2 \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 b^3 d^3}-\frac{x (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{b^4 d^4}-\frac{a^5 \log (a+b x)}{b^5 (b c-a d)}-\frac{x^3 (a d+b c)}{3 b^2 d^2}+\frac{c^5 \log (c+d x)}{d^5 (b c-a d)}+\frac{x^4}{4 b d} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{x^5}{(a+b x) (c+d x)} \, dx &=\int \left (\frac{(b c+a d) \left (-b^2 c^2-a^2 d^2\right )}{b^4 d^4}+\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}{b^2 d^2}+\frac{x^3}{b d}-\frac{a^5}{b^4 (b c-a d) (a+b x)}-\frac{c^5}{d^4 (-b c+a d) (c+d x)}\right ) \, dx\\ &=-\frac{(b c+a d) \left (b^2 c^2+a^2 d^2\right ) x}{b^4 d^4}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) x^2}{2 b^3 d^3}-\frac{(b c+a d) x^3}{3 b^2 d^2}+\frac{x^4}{4 b d}-\frac{a^5 \log (a+b x)}{b^5 (b c-a d)}+\frac{c^5 \log (c+d x)}{d^5 (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0610115, size = 133, normalized size = 0.92 \[ \frac{b d x \left (4 a^2 b^2 d^4 x^2-6 a^3 b d^4 x+12 a^4 d^4-3 a b^3 d^4 x^3+b^4 c \left (6 c^2 d x-12 c^3-4 c d^2 x^2+3 d^3 x^3\right )\right )-12 a^5 d^5 \log (a+b x)+12 b^5 c^5 \log (c+d x)}{12 b^5 d^5 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 175, normalized size = 1.2 \begin{align*}{\frac{{x}^{4}}{4\,bd}}-{\frac{{x}^{3}a}{3\,{b}^{2}d}}-{\frac{c{x}^{3}}{3\,b{d}^{2}}}+{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}d}}+{\frac{a{x}^{2}c}{2\,{b}^{2}{d}^{2}}}+{\frac{{x}^{2}{c}^{2}}{2\,b{d}^{3}}}-{\frac{{a}^{3}x}{{b}^{4}d}}-{\frac{{a}^{2}cx}{{b}^{3}{d}^{2}}}-{\frac{a{c}^{2}x}{{b}^{2}{d}^{3}}}-{\frac{{c}^{3}x}{b{d}^{4}}}-{\frac{{c}^{5}\ln \left ( dx+c \right ) }{{d}^{5} \left ( ad-bc \right ) }}+{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{5} \left ( ad-bc \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.20029, size = 217, normalized size = 1.5 \begin{align*} -\frac{a^{5} \log \left (b x + a\right )}{b^{6} c - a b^{5} d} + \frac{c^{5} \log \left (d x + c\right )}{b c d^{5} - a d^{6}} + \frac{3 \, b^{3} d^{3} x^{4} - 4 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 6 \,{\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 12 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} x}{12 \, b^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40593, size = 294, normalized size = 2.03 \begin{align*} -\frac{12 \, a^{5} d^{5} \log \left (b x + a\right ) - 12 \, b^{5} c^{5} \log \left (d x + c\right ) - 3 \,{\left (b^{5} c d^{4} - a b^{4} d^{5}\right )} x^{4} + 4 \,{\left (b^{5} c^{2} d^{3} - a^{2} b^{3} d^{5}\right )} x^{3} - 6 \,{\left (b^{5} c^{3} d^{2} - a^{3} b^{2} d^{5}\right )} x^{2} + 12 \,{\left (b^{5} c^{4} d - a^{4} b d^{5}\right )} x}{12 \,{\left (b^{6} c d^{5} - a b^{5} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.59195, size = 298, normalized size = 2.06 \begin{align*} \frac{a^{5} \log{\left (x + \frac{\frac{a^{7} d^{6}}{b \left (a d - b c\right )} - \frac{2 a^{6} c d^{5}}{a d - b c} + \frac{a^{5} b c^{2} d^{4}}{a d - b c} + a^{5} c d^{4} + a b^{4} c^{5}}{a^{5} d^{5} + b^{5} c^{5}} \right )}}{b^{5} \left (a d - b c\right )} - \frac{c^{5} \log{\left (x + \frac{a^{5} c d^{4} - \frac{a^{2} b^{4} c^{5} d}{a d - b c} + \frac{2 a b^{5} c^{6}}{a d - b c} + a b^{4} c^{5} - \frac{b^{6} c^{7}}{d \left (a d - b c\right )}}{a^{5} d^{5} + b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )} + \frac{x^{4}}{4 b d} - \frac{x^{3} \left (a d + b c\right )}{3 b^{2} d^{2}} + \frac{x^{2} \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right )}{2 b^{3} d^{3}} - \frac{x \left (a^{3} d^{3} + a^{2} b c d^{2} + a b^{2} c^{2} d + b^{3} c^{3}\right )}{b^{4} d^{4}} \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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