Optimal. Leaf size=38 \[ \frac{b (c+d x)^9}{9 d^2}-\frac{(c+d x)^8 (b c-a d)}{8 d^2} \]
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Rubi [A] time = 0.0164614, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{b (c+d x)^9}{9 d^2}-\frac{(c+d x)^8 (b c-a d)}{8 d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x) (c+d x)^7 \, dx &=\int \left (\frac{(-b c+a d) (c+d x)^7}{d}+\frac{b (c+d x)^8}{d}\right ) \, dx\\ &=-\frac{(b c-a d) (c+d x)^8}{8 d^2}+\frac{b (c+d x)^9}{9 d^2}\\ \end{align*}
Mathematica [B] time = 0.0170292, size = 151, normalized size = 3.97 \[ \frac{7}{6} c^2 d^4 x^6 (3 a d+5 b c)+7 c^3 d^3 x^5 (a d+b c)+\frac{7}{4} c^4 d^2 x^4 (5 a d+3 b c)+\frac{7}{3} c^5 d x^3 (3 a d+b c)+\frac{1}{2} c^6 x^2 (7 a d+b c)+\frac{1}{8} d^6 x^8 (a d+7 b c)+c d^5 x^7 (a d+3 b c)+a c^7 x+\frac{1}{9} b d^7 x^9 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 169, normalized size = 4.5 \begin{align*}{\frac{b{d}^{7}{x}^{9}}{9}}+{\frac{ \left ( a{d}^{7}+7\,bc{d}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,ac{d}^{6}+21\,b{c}^{2}{d}^{5} \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,a{c}^{2}{d}^{5}+35\,b{c}^{3}{d}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,a{c}^{3}{d}^{4}+35\,b{c}^{4}{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,a{c}^{4}{d}^{3}+21\,b{c}^{5}{d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,a{c}^{5}{d}^{2}+7\,b{c}^{6}d \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,a{c}^{6}d+b{c}^{7} \right ){x}^{2}}{2}}+a{c}^{7}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.945336, size = 220, normalized size = 5.79 \begin{align*} \frac{1}{9} \, b d^{7} x^{9} + a c^{7} x + \frac{1}{8} \,{\left (7 \, b c d^{6} + a d^{7}\right )} x^{8} +{\left (3 \, b c^{2} d^{5} + a c d^{6}\right )} x^{7} + \frac{7}{6} \,{\left (5 \, b c^{3} d^{4} + 3 \, a c^{2} d^{5}\right )} x^{6} + 7 \,{\left (b c^{4} d^{3} + a c^{3} d^{4}\right )} x^{5} + \frac{7}{4} \,{\left (3 \, b c^{5} d^{2} + 5 \, a c^{4} d^{3}\right )} x^{4} + \frac{7}{3} \,{\left (b c^{6} d + 3 \, a c^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{7} + 7 \, a c^{6} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85742, size = 378, normalized size = 9.95 \begin{align*} \frac{1}{9} x^{9} d^{7} b + \frac{7}{8} x^{8} d^{6} c b + \frac{1}{8} x^{8} d^{7} a + 3 x^{7} d^{5} c^{2} b + x^{7} d^{6} c a + \frac{35}{6} x^{6} d^{4} c^{3} b + \frac{7}{2} x^{6} d^{5} c^{2} a + 7 x^{5} d^{3} c^{4} b + 7 x^{5} d^{4} c^{3} a + \frac{21}{4} x^{4} d^{2} c^{5} b + \frac{35}{4} x^{4} d^{3} c^{4} a + \frac{7}{3} x^{3} d c^{6} b + 7 x^{3} d^{2} c^{5} a + \frac{1}{2} x^{2} c^{7} b + \frac{7}{2} x^{2} d c^{6} a + x c^{7} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.089849, size = 178, normalized size = 4.68 \begin{align*} a c^{7} x + \frac{b d^{7} x^{9}}{9} + x^{8} \left (\frac{a d^{7}}{8} + \frac{7 b c d^{6}}{8}\right ) + x^{7} \left (a c d^{6} + 3 b c^{2} d^{5}\right ) + x^{6} \left (\frac{7 a c^{2} d^{5}}{2} + \frac{35 b c^{3} d^{4}}{6}\right ) + x^{5} \left (7 a c^{3} d^{4} + 7 b c^{4} d^{3}\right ) + x^{4} \left (\frac{35 a c^{4} d^{3}}{4} + \frac{21 b c^{5} d^{2}}{4}\right ) + x^{3} \left (7 a c^{5} d^{2} + \frac{7 b c^{6} d}{3}\right ) + x^{2} \left (\frac{7 a c^{6} d}{2} + \frac{b c^{7}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07522, size = 228, normalized size = 6. \begin{align*} \frac{1}{9} \, b d^{7} x^{9} + \frac{7}{8} \, b c d^{6} x^{8} + \frac{1}{8} \, a d^{7} x^{8} + 3 \, b c^{2} d^{5} x^{7} + a c d^{6} x^{7} + \frac{35}{6} \, b c^{3} d^{4} x^{6} + \frac{7}{2} \, a c^{2} d^{5} x^{6} + 7 \, b c^{4} d^{3} x^{5} + 7 \, a c^{3} d^{4} x^{5} + \frac{21}{4} \, b c^{5} d^{2} x^{4} + \frac{35}{4} \, a c^{4} d^{3} x^{4} + \frac{7}{3} \, b c^{6} d x^{3} + 7 \, a c^{5} d^{2} x^{3} + \frac{1}{2} \, b c^{7} x^{2} + \frac{7}{2} \, a c^{6} d x^{2} + a c^{7} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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