Optimal. Leaf size=92 \[ \frac{3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac{b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac{(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac{b^3}{4 d^4 (c+d x)^4} \]
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Rubi [A] time = 0.057465, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac{b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac{(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac{b^3}{4 d^4 (c+d x)^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{(c+d x)^8} \, dx &=\int \left (\frac{(-b c+a d)^3}{d^3 (c+d x)^8}+\frac{3 b (b c-a d)^2}{d^3 (c+d x)^7}-\frac{3 b^2 (b c-a d)}{d^3 (c+d x)^6}+\frac{b^3}{d^3 (c+d x)^5}\right ) \, dx\\ &=\frac{(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac{b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac{3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac{b^3}{4 d^4 (c+d x)^4}\\ \end{align*}
Mathematica [A] time = 0.0292335, size = 94, normalized size = 1.02 \[ -\frac{10 a^2 b d^2 (c+7 d x)+20 a^3 d^3+4 a b^2 d \left (c^2+7 c d x+21 d^2 x^2\right )+b^3 \left (7 c^2 d x+c^3+21 c d^2 x^2+35 d^3 x^3\right )}{140 d^4 (c+d x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 122, normalized size = 1.3 \begin{align*} -{\frac{{a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3}}{7\,{d}^{4} \left ( dx+c \right ) ^{7}}}-{\frac{b \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{2\,{d}^{4} \left ( dx+c \right ) ^{6}}}-{\frac{3\,{b}^{2} \left ( ad-bc \right ) }{5\,{d}^{4} \left ( dx+c \right ) ^{5}}}-{\frac{{b}^{3}}{4\,{d}^{4} \left ( dx+c \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00277, size = 246, normalized size = 2.67 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \,{\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \,{\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76952, size = 382, normalized size = 4.15 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \,{\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \,{\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.48897, size = 194, normalized size = 2.11 \begin{align*} - \frac{20 a^{3} d^{3} + 10 a^{2} b c d^{2} + 4 a b^{2} c^{2} d + b^{3} c^{3} + 35 b^{3} d^{3} x^{3} + x^{2} \left (84 a b^{2} d^{3} + 21 b^{3} c d^{2}\right ) + x \left (70 a^{2} b d^{3} + 28 a b^{2} c d^{2} + 7 b^{3} c^{2} d\right )}{140 c^{7} d^{4} + 980 c^{6} d^{5} x + 2940 c^{5} d^{6} x^{2} + 4900 c^{4} d^{7} x^{3} + 4900 c^{3} d^{8} x^{4} + 2940 c^{2} d^{9} x^{5} + 980 c d^{10} x^{6} + 140 d^{11} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07896, size = 154, normalized size = 1.67 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + 21 \, b^{3} c d^{2} x^{2} + 84 \, a b^{2} d^{3} x^{2} + 7 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 70 \, a^{2} b d^{3} x + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3}}{140 \,{\left (d x + c\right )}^{7} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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