Optimal. Leaf size=73 \[ \frac{e x (b d-a e)^2}{b^3}+\frac{(d+e x)^2 (b d-a e)}{2 b^2}+\frac{(b d-a e)^3 \log (a+b x)}{b^4}+\frac{(d+e x)^3}{3 b} \]
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Rubi [A] time = 0.0286476, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{e x (b d-a e)^2}{b^3}+\frac{(d+e x)^2 (b d-a e)}{2 b^2}+\frac{(b d-a e)^3 \log (a+b x)}{b^4}+\frac{(d+e x)^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^3}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^3}{a+b x} \, dx\\ &=\int \left (\frac{e (b d-a e)^2}{b^3}+\frac{(b d-a e)^3}{b^3 (a+b x)}+\frac{e (b d-a e) (d+e x)}{b^2}+\frac{e (d+e x)^2}{b}\right ) \, dx\\ &=\frac{e (b d-a e)^2 x}{b^3}+\frac{(b d-a e) (d+e x)^2}{2 b^2}+\frac{(d+e x)^3}{3 b}+\frac{(b d-a e)^3 \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0293133, size = 74, normalized size = 1.01 \[ \frac{b e x \left (6 a^2 e^2-3 a b e (6 d+e x)+b^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (b d-a e)^3 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 133, normalized size = 1.8 \begin{align*}{\frac{{e}^{3}{x}^{3}}{3\,b}}-{\frac{{e}^{3}{x}^{2}a}{2\,{b}^{2}}}+{\frac{3\,{e}^{2}{x}^{2}d}{2\,b}}+{\frac{{e}^{3}{a}^{2}x}{{b}^{3}}}-3\,{\frac{a{e}^{2}dx}{{b}^{2}}}+3\,{\frac{e{d}^{2}x}{b}}-{\frac{\ln \left ( bx+a \right ){a}^{3}{e}^{3}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx+a \right ){a}^{2}d{e}^{2}}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) a{d}^{2}e}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{3}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959333, size = 154, normalized size = 2.11 \begin{align*} \frac{2 \, b^{2} e^{3} x^{3} + 3 \,{\left (3 \, b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x}{6 \, b^{3}} + \frac{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{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 = 1.49214, size = 238, normalized size = 3.26 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} + 3 \,{\left (3 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{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 = 0.486319, size = 82, normalized size = 1.12 \begin{align*} \frac{e^{3} x^{3}}{3 b} - \frac{x^{2} \left (a e^{3} - 3 b d e^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} e^{3} - 3 a b d e^{2} + 3 b^{2} d^{2} e\right )}{b^{3}} - \frac{\left (a e - b d\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 = 1.08954, size = 149, normalized size = 2.04 \begin{align*} \frac{2 \, b^{2} x^{3} e^{3} + 9 \, b^{2} d x^{2} e^{2} + 18 \, b^{2} d^{2} x e - 3 \, a b x^{2} e^{3} - 18 \, a b d x e^{2} + 6 \, a^{2} x e^{3}}{6 \, b^{3}} + \frac{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{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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