Optimal. Leaf size=38 \[ \frac{(a+b x)^8 (b d-a e)}{8 b^2}+\frac{e (a+b x)^9}{9 b^2} \]
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Rubi [A] time = 0.0175583, antiderivative size = 38, 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} \[ \frac{(a+b x)^8 (b d-a e)}{8 b^2}+\frac{e (a+b x)^9}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x) \, dx\\ &=\int \left (\frac{(b d-a e) (a+b x)^7}{b}+\frac{e (a+b x)^8}{b}\right ) \, dx\\ &=\frac{(b d-a e) (a+b x)^8}{8 b^2}+\frac{e (a+b x)^9}{9 b^2}\\ \end{align*}
Mathematica [B] time = 0.0221783, size = 151, normalized size = 3.97 \[ \frac{7}{6} a^2 b^4 x^6 (5 a e+3 b d)+7 a^3 b^3 x^5 (a e+b d)+\frac{7}{4} a^4 b^2 x^4 (3 a e+5 b d)+\frac{7}{3} a^5 b x^3 (a e+3 b d)+\frac{1}{2} a^6 x^2 (a e+7 b d)+a^7 d x+\frac{1}{8} b^6 x^8 (7 a e+b d)+a b^5 x^7 (3 a e+b d)+\frac{1}{9} b^7 e x^9 \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 250, normalized size = 6.6 \begin{align*}{\frac{{b}^{7}e{x}^{9}}{9}}+{\frac{ \left ( \left ( ae+bd \right ){b}^{6}+6\,{b}^{6}ea \right ){x}^{8}}{8}}+{\frac{ \left ( ad{b}^{6}+6\, \left ( ae+bd \right ) a{b}^{5}+15\,{b}^{5}e{a}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{a}^{2}d{b}^{5}+15\, \left ( ae+bd \right ){a}^{2}{b}^{4}+20\,{b}^{4}e{a}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{a}^{3}d{b}^{4}+20\, \left ( ae+bd \right ){a}^{3}{b}^{3}+15\,{b}^{3}e{a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{a}^{4}d{b}^{3}+15\, \left ( ae+bd \right ){a}^{4}{b}^{2}+6\,{b}^{2}e{a}^{5} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{a}^{5}d{b}^{2}+6\, \left ( ae+bd \right ){a}^{5}b+be{a}^{6} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{a}^{6}db+ \left ( ae+bd \right ){a}^{6} \right ){x}^{2}}{2}}+{a}^{7}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.973726, size = 220, normalized size = 5.79 \begin{align*} \frac{1}{9} \, b^{7} e x^{9} + a^{7} d x + \frac{1}{8} \,{\left (b^{7} d + 7 \, a b^{6} e\right )} x^{8} +{\left (a b^{6} d + 3 \, a^{2} b^{5} e\right )} x^{7} + \frac{7}{6} \,{\left (3 \, a^{2} b^{5} d + 5 \, a^{3} b^{4} e\right )} x^{6} + 7 \,{\left (a^{3} b^{4} d + a^{4} b^{3} e\right )} x^{5} + \frac{7}{4} \,{\left (5 \, a^{4} b^{3} d + 3 \, a^{5} b^{2} e\right )} x^{4} + \frac{7}{3} \,{\left (3 \, a^{5} b^{2} d + a^{6} b e\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d + a^{7} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31108, size = 378, normalized size = 9.95 \begin{align*} \frac{1}{9} x^{9} e b^{7} + \frac{1}{8} x^{8} d b^{7} + \frac{7}{8} x^{8} e b^{6} a + x^{7} d b^{6} a + 3 x^{7} e b^{5} a^{2} + \frac{7}{2} x^{6} d b^{5} a^{2} + \frac{35}{6} x^{6} e b^{4} a^{3} + 7 x^{5} d b^{4} a^{3} + 7 x^{5} e b^{3} a^{4} + \frac{35}{4} x^{4} d b^{3} a^{4} + \frac{21}{4} x^{4} e b^{2} a^{5} + 7 x^{3} d b^{2} a^{5} + \frac{7}{3} x^{3} e b a^{6} + \frac{7}{2} x^{2} d b a^{6} + \frac{1}{2} x^{2} e a^{7} + x d a^{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.095187, size = 178, normalized size = 4.68 \begin{align*} a^{7} d x + \frac{b^{7} e x^{9}}{9} + x^{8} \left (\frac{7 a b^{6} e}{8} + \frac{b^{7} d}{8}\right ) + x^{7} \left (3 a^{2} b^{5} e + a b^{6} d\right ) + x^{6} \left (\frac{35 a^{3} b^{4} e}{6} + \frac{7 a^{2} b^{5} d}{2}\right ) + x^{5} \left (7 a^{4} b^{3} e + 7 a^{3} b^{4} d\right ) + x^{4} \left (\frac{21 a^{5} b^{2} e}{4} + \frac{35 a^{4} b^{3} d}{4}\right ) + x^{3} \left (\frac{7 a^{6} b e}{3} + 7 a^{5} b^{2} d\right ) + x^{2} \left (\frac{a^{7} e}{2} + \frac{7 a^{6} b d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10442, size = 239, normalized size = 6.29 \begin{align*} \frac{1}{9} \, b^{7} x^{9} e + \frac{1}{8} \, b^{7} d x^{8} + \frac{7}{8} \, a b^{6} x^{8} e + a b^{6} d x^{7} + 3 \, a^{2} b^{5} x^{7} e + \frac{7}{2} \, a^{2} b^{5} d x^{6} + \frac{35}{6} \, a^{3} b^{4} x^{6} e + 7 \, a^{3} b^{4} d x^{5} + 7 \, a^{4} b^{3} x^{5} e + \frac{35}{4} \, a^{4} b^{3} d x^{4} + \frac{21}{4} \, a^{5} b^{2} x^{4} e + 7 \, a^{5} b^{2} d x^{3} + \frac{7}{3} \, a^{6} b x^{3} e + \frac{7}{2} \, a^{6} b d x^{2} + \frac{1}{2} \, a^{7} x^{2} e + a^{7} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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