Optimal. Leaf size=92 \[ \frac{e^2 (a+b x)^6 (b d-a e)}{2 b^4}+\frac{3 e (a+b x)^5 (b d-a e)^2}{5 b^4}+\frac{(a+b x)^4 (b d-a e)^3}{4 b^4}+\frac{e^3 (a+b x)^7}{7 b^4} \]
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Rubi [A] time = 0.0954058, antiderivative size = 92, 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 = {27, 43} \[ \frac{e^2 (a+b x)^6 (b d-a e)}{2 b^4}+\frac{3 e (a+b x)^5 (b d-a e)^2}{5 b^4}+\frac{(a+b x)^4 (b d-a e)^3}{4 b^4}+\frac{e^3 (a+b x)^7}{7 b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^3 \, dx\\ &=\int \left (\frac{(b d-a e)^3 (a+b x)^3}{b^3}+\frac{3 e (b d-a e)^2 (a+b x)^4}{b^3}+\frac{3 e^2 (b d-a e) (a+b x)^5}{b^3}+\frac{e^3 (a+b x)^6}{b^3}\right ) \, dx\\ &=\frac{(b d-a e)^3 (a+b x)^4}{4 b^4}+\frac{3 e (b d-a e)^2 (a+b x)^5}{5 b^4}+\frac{e^2 (b d-a e) (a+b x)^6}{2 b^4}+\frac{e^3 (a+b x)^7}{7 b^4}\\ \end{align*}
Mathematica [A] time = 0.022843, size = 161, normalized size = 1.75 \[ \frac{3}{5} b e x^5 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac{1}{4} x^4 \left (9 a^2 b d e^2+a^3 e^3+9 a b^2 d^2 e+b^3 d^3\right )+a d x^3 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac{3}{2} a^2 d^2 x^2 (a e+b d)+a^3 d^3 x+\frac{1}{2} b^2 e^2 x^6 (a e+b d)+\frac{1}{7} b^3 e^3 x^7 \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 244, normalized size = 2.7 \begin{align*}{\frac{{b}^{3}{e}^{3}{x}^{7}}{7}}+{\frac{ \left ( \left ( a{e}^{3}+3\,bd{e}^{2} \right ){b}^{2}+2\,a{b}^{2}{e}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){b}^{2}+2\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ) ab+b{e}^{3}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){b}^{2}+2\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ) ab+ \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( a{d}^{3}{b}^{2}+2\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ) ab+ \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{d}^{3}b+ \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965722, size = 225, normalized size = 2.45 \begin{align*} \frac{1}{7} \, b^{3} e^{3} x^{7} + a^{3} d^{3} x + \frac{1}{2} \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (b^{3} d^{2} e + 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{3} + 9 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} x^{4} +{\left (a b^{2} d^{3} + 3 \, a^{2} b d^{2} e + a^{3} d e^{2}\right )} x^{3} + \frac{3}{2} \,{\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31409, size = 409, normalized size = 4.45 \begin{align*} \frac{1}{7} x^{7} e^{3} b^{3} + \frac{1}{2} x^{6} e^{2} d b^{3} + \frac{1}{2} x^{6} e^{3} b^{2} a + \frac{3}{5} x^{5} e d^{2} b^{3} + \frac{9}{5} x^{5} e^{2} d b^{2} a + \frac{3}{5} x^{5} e^{3} b a^{2} + \frac{1}{4} x^{4} d^{3} b^{3} + \frac{9}{4} x^{4} e d^{2} b^{2} a + \frac{9}{4} x^{4} e^{2} d b a^{2} + \frac{1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} b^{2} a + 3 x^{3} e d^{2} b a^{2} + x^{3} e^{2} d a^{3} + \frac{3}{2} x^{2} d^{3} b a^{2} + \frac{3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.087387, size = 190, normalized size = 2.07 \begin{align*} a^{3} d^{3} x + \frac{b^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac{a b^{2} e^{3}}{2} + \frac{b^{3} d e^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} b e^{3}}{5} + \frac{9 a b^{2} d e^{2}}{5} + \frac{3 b^{3} d^{2} e}{5}\right ) + x^{4} \left (\frac{a^{3} e^{3}}{4} + \frac{9 a^{2} b d e^{2}}{4} + \frac{9 a b^{2} d^{2} e}{4} + \frac{b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{3} d^{2} e}{2} + \frac{3 a^{2} b d^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27304, size = 248, normalized size = 2.7 \begin{align*} \frac{1}{7} \, b^{3} x^{7} e^{3} + \frac{1}{2} \, b^{3} d x^{6} e^{2} + \frac{3}{5} \, b^{3} d^{2} x^{5} e + \frac{1}{4} \, b^{3} d^{3} x^{4} + \frac{1}{2} \, a b^{2} x^{6} e^{3} + \frac{9}{5} \, a b^{2} d x^{5} e^{2} + \frac{9}{4} \, a b^{2} d^{2} x^{4} e + a b^{2} d^{3} x^{3} + \frac{3}{5} \, a^{2} b x^{5} e^{3} + \frac{9}{4} \, a^{2} b d x^{4} e^{2} + 3 \, a^{2} b d^{2} x^{3} e + \frac{3}{2} \, a^{2} b d^{3} x^{2} + \frac{1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac{3}{2} \, a^{3} d^{2} x^{2} e + a^{3} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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