Optimal. Leaf size=74 \[ \frac{b x (b c-a d)^2}{d^3}-\frac{(a+b x)^2 (b c-a d)}{2 d^2}-\frac{(b c-a d)^3 \log (c+d x)}{d^4}+\frac{(a+b x)^3}{3 d} \]
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Rubi [A] time = 0.0361576, antiderivative size = 74, 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} \[ \frac{b x (b c-a d)^2}{d^3}-\frac{(a+b x)^2 (b c-a d)}{2 d^2}-\frac{(b c-a d)^3 \log (c+d x)}{d^4}+\frac{(a+b x)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^4}{a c+(b c+a d) x+b d x^2} \, dx &=\int \frac{(a+b x)^3}{c+d x} \, dx\\ &=\int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx\\ &=\frac{b (b c-a d)^2 x}{d^3}-\frac{(b c-a d) (a+b x)^2}{2 d^2}+\frac{(a+b x)^3}{3 d}-\frac{(b c-a d)^3 \log (c+d x)}{d^4}\\ \end{align*}
Mathematica [A] time = 0.026245, size = 74, normalized size = 1. \[ \frac{b d x \left (18 a^2 d^2+9 a b d (d x-2 c)+b^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )-6 (b c-a d)^3 \log (c+d x)}{6 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 133, normalized size = 1.8 \begin{align*}{\frac{{b}^{3}{x}^{3}}{3\,d}}+{\frac{3\,a{b}^{2}{x}^{2}}{2\,d}}-{\frac{{b}^{3}{x}^{2}c}{2\,{d}^{2}}}+3\,{\frac{b{a}^{2}x}{d}}-3\,{\frac{ac{b}^{2}x}{{d}^{2}}}+{\frac{{b}^{3}{c}^{2}x}{{d}^{3}}}+{\frac{\ln \left ( dx+c \right ){a}^{3}}{d}}-3\,{\frac{\ln \left ( dx+c \right ) cb{a}^{2}}{{d}^{2}}}+3\,{\frac{\ln \left ( dx+c \right ) a{c}^{2}{b}^{2}}{{d}^{3}}}-{\frac{\ln \left ( dx+c \right ){b}^{3}{c}^{3}}{{d}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06319, size = 154, normalized size = 2.08 \begin{align*} \frac{2 \, b^{3} d^{2} x^{3} - 3 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x}{6 \, d^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (d x + c\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57452, size = 238, normalized size = 3.22 \begin{align*} \frac{2 \, b^{3} d^{3} x^{3} - 3 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x - 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (d x + c\right )}{6 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.536191, size = 82, normalized size = 1.11 \begin{align*} \frac{b^{3} x^{3}}{3 d} + \frac{x^{2} \left (3 a b^{2} d - b^{3} c\right )}{2 d^{2}} + \frac{x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} + \frac{\left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15981, size = 157, normalized size = 2.12 \begin{align*} \frac{2 \, b^{3} d^{2} x^{3} - 3 \, b^{3} c d x^{2} + 9 \, a b^{2} d^{2} x^{2} + 6 \, b^{3} c^{2} x - 18 \, a b^{2} c d x + 18 \, a^{2} b d^{2} x}{6 \, d^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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