Optimal. Leaf size=38 \[ \frac{(a+b x)^6 (b d-a e)}{6 b^2}+\frac{e (a+b x)^7}{7 b^2} \]
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Rubi [A] time = 0.0167707, 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} \[ \frac{(a+b x)^6 (b d-a e)}{6 b^2}+\frac{e (a+b x)^7}{7 b^2} \]
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 )^2 \, dx &=\int (a+b x)^5 (d+e x) \, dx\\ &=\int \left (\frac{(b d-a e) (a+b x)^5}{b}+\frac{e (a+b x)^6}{b}\right ) \, dx\\ &=\frac{(b d-a e) (a+b x)^6}{6 b^2}+\frac{e (a+b x)^7}{7 b^2}\\ \end{align*}
Mathematica [B] time = 0.016037, size = 109, normalized size = 2.87 \[ \frac{5}{2} a^2 b^2 x^4 (a e+b d)+\frac{5}{3} a^3 b x^3 (a e+2 b d)+\frac{1}{2} a^4 x^2 (a e+5 b d)+a^5 d x+\frac{1}{6} b^4 x^6 (5 a e+b d)+a b^3 x^5 (2 a e+b d)+\frac{1}{7} b^5 e x^7 \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 172, normalized size = 4.5 \begin{align*}{\frac{{b}^{5}e{x}^{7}}{7}}+{\frac{ \left ( \left ( ae+bd \right ){b}^{4}+4\,{b}^{4}ea \right ){x}^{6}}{6}}+{\frac{ \left ( ad{b}^{4}+4\, \left ( ae+bd \right ) a{b}^{3}+6\,{b}^{3}e{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{2}d{b}^{3}+6\, \left ( ae+bd \right ){a}^{2}{b}^{2}+4\,{b}^{2}e{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{3}d{b}^{2}+4\, \left ( ae+bd \right ){a}^{3}b+be{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{4}db+ \left ( ae+bd \right ){a}^{4} \right ){x}^{2}}{2}}+{a}^{5}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02191, size = 155, normalized size = 4.08 \begin{align*} \frac{1}{7} \, b^{5} e x^{7} + a^{5} d x + \frac{1}{6} \,{\left (b^{5} d + 5 \, a b^{4} e\right )} x^{6} +{\left (a b^{4} d + 2 \, a^{2} b^{3} e\right )} x^{5} + \frac{5}{2} \,{\left (a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d + a^{4} b e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d + a^{5} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31888, size = 275, normalized size = 7.24 \begin{align*} \frac{1}{7} x^{7} e b^{5} + \frac{1}{6} x^{6} d b^{5} + \frac{5}{6} x^{6} e b^{4} a + x^{5} d b^{4} a + 2 x^{5} e b^{3} a^{2} + \frac{5}{2} x^{4} d b^{3} a^{2} + \frac{5}{2} x^{4} e b^{2} a^{3} + \frac{10}{3} x^{3} d b^{2} a^{3} + \frac{5}{3} x^{3} e b a^{4} + \frac{5}{2} x^{2} d b a^{4} + \frac{1}{2} x^{2} e a^{5} + x d a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.082625, size = 129, normalized size = 3.39 \begin{align*} a^{5} d x + \frac{b^{5} e x^{7}}{7} + x^{6} \left (\frac{5 a b^{4} e}{6} + \frac{b^{5} d}{6}\right ) + x^{5} \left (2 a^{2} b^{3} e + a b^{4} d\right ) + x^{4} \left (\frac{5 a^{3} b^{2} e}{2} + \frac{5 a^{2} b^{3} d}{2}\right ) + x^{3} \left (\frac{5 a^{4} b e}{3} + \frac{10 a^{3} b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{5} e}{2} + \frac{5 a^{4} b d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12944, size = 171, normalized size = 4.5 \begin{align*} \frac{1}{7} \, b^{5} x^{7} e + \frac{1}{6} \, b^{5} d x^{6} + \frac{5}{6} \, a b^{4} x^{6} e + a b^{4} d x^{5} + 2 \, a^{2} b^{3} x^{5} e + \frac{5}{2} \, a^{2} b^{3} d x^{4} + \frac{5}{2} \, a^{3} b^{2} x^{4} e + \frac{10}{3} \, a^{3} b^{2} d x^{3} + \frac{5}{3} \, a^{4} b x^{3} e + \frac{5}{2} \, a^{4} b d x^{2} + \frac{1}{2} \, a^{5} x^{2} e + a^{5} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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