Optimal. Leaf size=73 \[ \frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{(c+d x)^3}{3 b} \]
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Rubi [A] time = 0.0303147, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{d x (b c-a d)^2}{b^3}+\frac{(c+d x)^2 (b c-a d)}{2 b^2}+\frac{(b c-a d)^3 \log (a+b x)}{b^4}+\frac{(c+d x)^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{a+b x} \, dx &=\int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx\\ &=\frac{d (b c-a d)^2 x}{b^3}+\frac{(b c-a d) (c+d x)^2}{2 b^2}+\frac{(c+d x)^3}{3 b}+\frac{(b c-a d)^3 \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0271504, size = 74, normalized size = 1.01 \[ \frac{b d x \left (6 a^2 d^2-3 a b d (6 c+d x)+b^2 \left (18 c^2+9 c d x+2 d^2 x^2\right )\right )+6 (b c-a d)^3 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 133, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{3}}{3\,b}}-{\frac{{d}^{3}{x}^{2}a}{2\,{b}^{2}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,b}}+{\frac{{d}^{3}{a}^{2}x}{{b}^{3}}}-3\,{\frac{a{d}^{2}cx}{{b}^{2}}}+3\,{\frac{d{c}^{2}x}{b}}-{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{4}}}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{3}}}-3\,{\frac{a\ln \left ( bx+a \right ){c}^{2}d}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{3}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.34579, size = 154, normalized size = 2.11 \begin{align*} \frac{2 \, b^{2} d^{3} x^{3} + 3 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{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 (b x + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24773, size = 238, normalized size = 3.26 \begin{align*} \frac{2 \, b^{3} d^{3} x^{3} + 3 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 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 (b x + a\right )}{6 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.716172, size = 82, normalized size = 1.12 \begin{align*} \frac{d^{3} x^{3}}{3 b} - \frac{x^{2} \left (a d^{3} - 3 b c d^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19408, size = 155, normalized size = 2.12 \begin{align*} \frac{2 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} - 3 \, a b d^{3} x^{2} + 18 \, b^{2} c^{2} d x - 18 \, a b c d^{2} x + 6 \, a^{2} d^{3} x}{6 \, b^{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 | b x + a \right |}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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