Optimal. Leaf size=38 \[ \frac{(a+b x)^7 (b d-a e)}{7 b^2}+\frac{e (a+b x)^8}{8 b^2} \]
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Rubi [A] time = 0.0163301, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ \frac{(a+b x)^7 (b d-a e)}{7 b^2}+\frac{e (a+b x)^8}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x) \, dx\\ &=\int \left (\frac{(b d-a e) (a+b x)^6}{b}+\frac{e (a+b x)^7}{b}\right ) \, dx\\ &=\frac{(b d-a e) (a+b x)^7}{7 b^2}+\frac{e (a+b x)^8}{8 b^2}\\ \end{align*}
Mathematica [B] time = 0.0357605, size = 122, normalized size = 3.21 \[ \frac{1}{56} x \left (70 a^4 b^2 x^2 (4 d+3 e x)+56 a^3 b^3 x^3 (5 d+4 e x)+28 a^2 b^4 x^4 (6 d+5 e x)+56 a^5 b x (3 d+2 e x)+28 a^6 (2 d+e x)+8 a b^5 x^5 (7 d+6 e x)+b^6 x^6 (8 d+7 e x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 145, normalized size = 3.8 \begin{align*}{\frac{e{b}^{6}{x}^{8}}{8}}+{\frac{ \left ( 6\,ea{b}^{5}+d{b}^{6} \right ){x}^{7}}{7}}+{\frac{ \left ( 15\,e{a}^{2}{b}^{4}+6\,da{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 20\,e{a}^{3}{b}^{3}+15\,d{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 15\,e{a}^{4}{b}^{2}+20\,d{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,e{a}^{5}b+15\,d{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{6}+6\,d{a}^{5}b \right ){x}^{2}}{2}}+d{a}^{6}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14097, size = 192, normalized size = 5.05 \begin{align*} \frac{1}{8} \, b^{6} e x^{8} + a^{6} d x + \frac{1}{7} \,{\left (b^{6} d + 6 \, a b^{5} e\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d + 5 \, a^{2} b^{4} e\right )} x^{6} +{\left (3 \, a^{2} b^{4} d + 4 \, a^{3} b^{3} e\right )} x^{5} + \frac{5}{4} \,{\left (4 \, a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{4} +{\left (5 \, a^{4} b^{2} d + 2 \, a^{5} b e\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d + a^{6} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.48967, size = 316, normalized size = 8.32 \begin{align*} \frac{1}{8} x^{8} e b^{6} + \frac{1}{7} x^{7} d b^{6} + \frac{6}{7} x^{7} e b^{5} a + x^{6} d b^{5} a + \frac{5}{2} x^{6} e b^{4} a^{2} + 3 x^{5} d b^{4} a^{2} + 4 x^{5} e b^{3} a^{3} + 5 x^{4} d b^{3} a^{3} + \frac{15}{4} x^{4} e b^{2} a^{4} + 5 x^{3} d b^{2} a^{4} + 2 x^{3} e b a^{5} + 3 x^{2} d b a^{5} + \frac{1}{2} x^{2} e a^{6} + x d a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.096173, size = 148, normalized size = 3.89 \begin{align*} a^{6} d x + \frac{b^{6} e x^{8}}{8} + x^{7} \left (\frac{6 a b^{5} e}{7} + \frac{b^{6} d}{7}\right ) + x^{6} \left (\frac{5 a^{2} b^{4} e}{2} + a b^{5} d\right ) + x^{5} \left (4 a^{3} b^{3} e + 3 a^{2} b^{4} d\right ) + x^{4} \left (\frac{15 a^{4} b^{2} e}{4} + 5 a^{3} b^{3} d\right ) + x^{3} \left (2 a^{5} b e + 5 a^{4} b^{2} d\right ) + x^{2} \left (\frac{a^{6} e}{2} + 3 a^{5} b d\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14366, size = 205, normalized size = 5.39 \begin{align*} \frac{1}{8} \, b^{6} x^{8} e + \frac{1}{7} \, b^{6} d x^{7} + \frac{6}{7} \, a b^{5} x^{7} e + a b^{5} d x^{6} + \frac{5}{2} \, a^{2} b^{4} x^{6} e + 3 \, a^{2} b^{4} d x^{5} + 4 \, a^{3} b^{3} x^{5} e + 5 \, a^{3} b^{3} d x^{4} + \frac{15}{4} \, a^{4} b^{2} x^{4} e + 5 \, a^{4} b^{2} d x^{3} + 2 \, a^{5} b x^{3} e + 3 \, a^{5} b d x^{2} + \frac{1}{2} \, a^{6} x^{2} e + a^{6} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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