Optimal. Leaf size=98 \[ -\frac{b x (b c-a d)^3}{d^4}+\frac{(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac{(a+b x)^3 (b c-a d)}{3 d^2}+\frac{(b c-a d)^4 \log (c+d x)}{d^5}+\frac{(a+b x)^4}{4 d} \]
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Rubi [A] time = 0.046089, antiderivative size = 98, 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} \[ -\frac{b x (b c-a d)^3}{d^4}+\frac{(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac{(a+b x)^3 (b c-a d)}{3 d^2}+\frac{(b c-a d)^4 \log (c+d x)}{d^5}+\frac{(a+b x)^4}{4 d} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{a c+(b c+a d) x+b d x^2} \, dx &=\int \frac{(a+b x)^4}{c+d x} \, dx\\ &=\int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx\\ &=-\frac{b (b c-a d)^3 x}{d^4}+\frac{(b c-a d)^2 (a+b x)^2}{2 d^3}-\frac{(b c-a d) (a+b x)^3}{3 d^2}+\frac{(a+b x)^4}{4 d}+\frac{(b c-a d)^4 \log (c+d x)}{d^5}\\ \end{align*}
Mathematica [A] time = 0.0403413, size = 115, normalized size = 1.17 \[ \frac{b d x \left (36 a^2 b d^2 (d x-2 c)+48 a^3 d^3+8 a b^2 d \left (6 c^2-3 c d x+2 d^2 x^2\right )+b^3 \left (6 c^2 d x-12 c^3-4 c d^2 x^2+3 d^3 x^3\right )\right )+12 (b c-a d)^4 \log (c+d x)}{12 d^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 209, normalized size = 2.1 \begin{align*}{\frac{{b}^{4}{x}^{4}}{4\,d}}+{\frac{4\,a{b}^{3}{x}^{3}}{3\,d}}-{\frac{{b}^{4}{x}^{3}c}{3\,{d}^{2}}}+3\,{\frac{{b}^{2}{x}^{2}{a}^{2}}{d}}-2\,{\frac{{b}^{3}{x}^{2}ac}{{d}^{2}}}+{\frac{{x}^{2}{b}^{4}{c}^{2}}{2\,{d}^{3}}}+4\,{\frac{x{a}^{3}b}{d}}-6\,{\frac{{b}^{2}c{a}^{2}x}{{d}^{2}}}+4\,{\frac{xa{b}^{3}{c}^{2}}{{d}^{3}}}-{\frac{{b}^{4}{c}^{3}x}{{d}^{4}}}+{\frac{\ln \left ( dx+c \right ){a}^{4}}{d}}-4\,{\frac{\ln \left ( dx+c \right ){a}^{3}bc}{{d}^{2}}}+6\,{\frac{\ln \left ( dx+c \right ){a}^{2}{b}^{2}{c}^{2}}{{d}^{3}}}-4\,{\frac{\ln \left ( dx+c \right ) a{b}^{3}{c}^{3}}{{d}^{4}}}+{\frac{\ln \left ( dx+c \right ){b}^{4}{c}^{4}}{{d}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09025, size = 239, normalized size = 2.44 \begin{align*} \frac{3 \, b^{4} d^{3} x^{4} - 4 \,{\left (b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d - 4 \, a b^{3} c d^{2} + 6 \, a^{2} b^{2} d^{3}\right )} x^{2} - 12 \,{\left (b^{4} c^{3} - 4 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{12 \, d^{4}} + \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (d x + c\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6222, size = 369, normalized size = 3.77 \begin{align*} \frac{3 \, b^{4} d^{4} x^{4} - 4 \,{\left (b^{4} c d^{3} - 4 \, a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} - 12 \,{\left (b^{4} c^{3} d - 4 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (d x + c\right )}{12 \, d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.891777, size = 134, normalized size = 1.37 \begin{align*} \frac{b^{4} x^{4}}{4 d} + \frac{x^{3} \left (4 a b^{3} d - b^{4} c\right )}{3 d^{2}} + \frac{x^{2} \left (6 a^{2} b^{2} d^{2} - 4 a b^{3} c d + b^{4} c^{2}\right )}{2 d^{3}} + \frac{x \left (4 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 4 a b^{3} c^{2} d - b^{4} c^{3}\right )}{d^{4}} + \frac{\left (a d - b c\right )^{4} \log{\left (c + d x \right )}}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26526, size = 248, normalized size = 2.53 \begin{align*} \frac{3 \, b^{4} d^{3} x^{4} - 4 \, b^{4} c d^{2} x^{3} + 16 \, a b^{3} d^{3} x^{3} + 6 \, b^{4} c^{2} d x^{2} - 24 \, a b^{3} c d^{2} x^{2} + 36 \, a^{2} b^{2} d^{3} x^{2} - 12 \, b^{4} c^{3} x + 48 \, a b^{3} c^{2} d x - 72 \, a^{2} b^{2} c d^{2} x + 48 \, a^{3} b d^{3} x}{12 \, d^{4}} + \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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