Optimal. Leaf size=38 \[ \frac{b c-a d}{7 d^2 (c+d x)^7}-\frac{b}{6 d^2 (c+d x)^6} \]
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Rubi [A] time = 0.0220319, 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} \[ \frac{b c-a d}{7 d^2 (c+d x)^7}-\frac{b}{6 d^2 (c+d x)^6} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{a+b x}{(c+d x)^8} \, dx &=\int \left (\frac{-b c+a d}{d (c+d x)^8}+\frac{b}{d (c+d x)^7}\right ) \, dx\\ &=\frac{b c-a d}{7 d^2 (c+d x)^7}-\frac{b}{6 d^2 (c+d x)^6}\\ \end{align*}
Mathematica [A] time = 0.009272, size = 27, normalized size = 0.71 \[ -\frac{6 a d+b (c+7 d x)}{42 d^2 (c+d x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 0.9 \begin{align*} -{\frac{ad-bc}{7\,{d}^{2} \left ( dx+c \right ) ^{7}}}-{\frac{b}{6\,{d}^{2} \left ( dx+c \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.975201, size = 127, normalized size = 3.34 \begin{align*} -\frac{7 \, b d x + b c + 6 \, a d}{42 \,{\left (d^{9} x^{7} + 7 \, c d^{8} x^{6} + 21 \, c^{2} d^{7} x^{5} + 35 \, c^{3} d^{6} x^{4} + 35 \, c^{4} d^{5} x^{3} + 21 \, c^{5} d^{4} x^{2} + 7 \, c^{6} d^{3} x + c^{7} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78749, size = 198, normalized size = 5.21 \begin{align*} -\frac{7 \, b d x + b c + 6 \, a d}{42 \,{\left (d^{9} x^{7} + 7 \, c d^{8} x^{6} + 21 \, c^{2} d^{7} x^{5} + 35 \, c^{3} d^{6} x^{4} + 35 \, c^{4} d^{5} x^{3} + 21 \, c^{5} d^{4} x^{2} + 7 \, c^{6} d^{3} x + c^{7} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.880056, size = 100, normalized size = 2.63 \begin{align*} - \frac{6 a d + b c + 7 b d x}{42 c^{7} d^{2} + 294 c^{6} d^{3} x + 882 c^{5} d^{4} x^{2} + 1470 c^{4} d^{5} x^{3} + 1470 c^{3} d^{6} x^{4} + 882 c^{2} d^{7} x^{5} + 294 c d^{8} x^{6} + 42 d^{9} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06551, size = 34, normalized size = 0.89 \begin{align*} -\frac{7 \, b d x + b c + 6 \, a d}{42 \,{\left (d x + c\right )}^{7} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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