Optimal. Leaf size=208 \[ \frac{4 e^7 (a+b x)^2 (b d-a e)}{b^9}+\frac{28 e^6 x (b d-a e)^2}{b^8}-\frac{70 e^4 (b d-a e)^4}{b^9 (a+b x)}-\frac{28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac{28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}+\frac{56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}-\frac{2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac{(b d-a e)^8}{5 b^9 (a+b x)^5}+\frac{e^8 (a+b x)^3}{3 b^9} \]
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Rubi [A] time = 0.311493, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{4 e^7 (a+b x)^2 (b d-a e)}{b^9}+\frac{28 e^6 x (b d-a e)^2}{b^8}-\frac{70 e^4 (b d-a e)^4}{b^9 (a+b x)}-\frac{28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac{28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}+\frac{56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}-\frac{2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac{(b d-a e)^8}{5 b^9 (a+b x)^5}+\frac{e^8 (a+b x)^3}{3 b^9} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^8}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(d+e x)^8}{(a+b x)^6} \, dx\\ &=\int \left (\frac{28 e^6 (b d-a e)^2}{b^8}+\frac{(b d-a e)^8}{b^8 (a+b x)^6}+\frac{8 e (b d-a e)^7}{b^8 (a+b x)^5}+\frac{28 e^2 (b d-a e)^6}{b^8 (a+b x)^4}+\frac{56 e^3 (b d-a e)^5}{b^8 (a+b x)^3}+\frac{70 e^4 (b d-a e)^4}{b^8 (a+b x)^2}+\frac{56 e^5 (b d-a e)^3}{b^8 (a+b x)}+\frac{8 e^7 (b d-a e) (a+b x)}{b^8}+\frac{e^8 (a+b x)^2}{b^8}\right ) \, dx\\ &=\frac{28 e^6 (b d-a e)^2 x}{b^8}-\frac{(b d-a e)^8}{5 b^9 (a+b x)^5}-\frac{2 e (b d-a e)^7}{b^9 (a+b x)^4}-\frac{28 e^2 (b d-a e)^6}{3 b^9 (a+b x)^3}-\frac{28 e^3 (b d-a e)^5}{b^9 (a+b x)^2}-\frac{70 e^4 (b d-a e)^4}{b^9 (a+b x)}+\frac{4 e^7 (b d-a e) (a+b x)^2}{b^9}+\frac{e^8 (a+b x)^3}{3 b^9}+\frac{56 e^5 (b d-a e)^3 \log (a+b x)}{b^9}\\ \end{align*}
Mathematica [A] time = 0.170463, size = 195, normalized size = 0.94 \[ \frac{15 b e^6 x \left (21 a^2 e^2-48 a b d e+28 b^2 d^2\right )+15 b^2 e^7 x^2 (4 b d-3 a e)-\frac{1050 e^4 (b d-a e)^4}{a+b x}+\frac{420 e^3 (a e-b d)^5}{(a+b x)^2}-\frac{140 e^2 (b d-a e)^6}{(a+b x)^3}+840 e^5 (b d-a e)^3 \log (a+b x)+\frac{30 e (a e-b d)^7}{(a+b x)^4}-\frac{3 (b d-a e)^8}{(a+b x)^5}+5 b^3 e^8 x^3}{15 b^9} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 820, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.28623, size = 845, normalized size = 4.06 \begin{align*} -\frac{3 \, b^{8} d^{8} + 6 \, a b^{7} d^{7} e + 14 \, a^{2} b^{6} d^{6} e^{2} + 42 \, a^{3} b^{5} d^{5} e^{3} + 210 \, a^{4} b^{4} d^{4} e^{4} - 1918 \, a^{5} b^{3} d^{3} e^{5} + 3654 \, a^{6} b^{2} d^{2} e^{6} - 2754 \, a^{7} b d e^{7} + 743 \, a^{8} e^{8} + 1050 \,{\left (b^{8} d^{4} e^{4} - 4 \, a b^{7} d^{3} e^{5} + 6 \, a^{2} b^{6} d^{2} e^{6} - 4 \, a^{3} b^{5} d e^{7} + a^{4} b^{4} e^{8}\right )} x^{4} + 420 \,{\left (b^{8} d^{5} e^{3} + 5 \, a b^{7} d^{4} e^{4} - 30 \, a^{2} b^{6} d^{3} e^{5} + 50 \, a^{3} b^{5} d^{2} e^{6} - 35 \, a^{4} b^{4} d e^{7} + 9 \, a^{5} b^{3} e^{8}\right )} x^{3} + 140 \,{\left (b^{8} d^{6} e^{2} + 3 \, a b^{7} d^{5} e^{3} + 15 \, a^{2} b^{6} d^{4} e^{4} - 110 \, a^{3} b^{5} d^{3} e^{5} + 195 \, a^{4} b^{4} d^{2} e^{6} - 141 \, a^{5} b^{3} d e^{7} + 37 \, a^{6} b^{2} e^{8}\right )} x^{2} + 10 \,{\left (3 \, b^{8} d^{7} e + 7 \, a b^{7} d^{6} e^{2} + 21 \, a^{2} b^{6} d^{5} e^{3} + 105 \, a^{3} b^{5} d^{4} e^{4} - 875 \, a^{4} b^{4} d^{3} e^{5} + 1617 \, a^{5} b^{3} d^{2} e^{6} - 1197 \, a^{6} b^{2} d e^{7} + 319 \, a^{7} b e^{8}\right )} x}{15 \,{\left (b^{14} x^{5} + 5 \, a b^{13} x^{4} + 10 \, a^{2} b^{12} x^{3} + 10 \, a^{3} b^{11} x^{2} + 5 \, a^{4} b^{10} x + a^{5} b^{9}\right )}} + \frac{b^{2} e^{8} x^{3} + 3 \,{\left (4 \, b^{2} d e^{7} - 3 \, a b e^{8}\right )} x^{2} + 3 \,{\left (28 \, b^{2} d^{2} e^{6} - 48 \, a b d e^{7} + 21 \, a^{2} e^{8}\right )} x}{3 \, b^{8}} + \frac{56 \,{\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \log \left (b x + a\right )}{b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86638, size = 1952, normalized size = 9.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17656, size = 737, normalized size = 3.54 \begin{align*} \frac{56 \,{\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{9}} - \frac{3 \, b^{8} d^{8} + 6 \, a b^{7} d^{7} e + 14 \, a^{2} b^{6} d^{6} e^{2} + 42 \, a^{3} b^{5} d^{5} e^{3} + 210 \, a^{4} b^{4} d^{4} e^{4} - 1918 \, a^{5} b^{3} d^{3} e^{5} + 3654 \, a^{6} b^{2} d^{2} e^{6} - 2754 \, a^{7} b d e^{7} + 743 \, a^{8} e^{8} + 1050 \,{\left (b^{8} d^{4} e^{4} - 4 \, a b^{7} d^{3} e^{5} + 6 \, a^{2} b^{6} d^{2} e^{6} - 4 \, a^{3} b^{5} d e^{7} + a^{4} b^{4} e^{8}\right )} x^{4} + 420 \,{\left (b^{8} d^{5} e^{3} + 5 \, a b^{7} d^{4} e^{4} - 30 \, a^{2} b^{6} d^{3} e^{5} + 50 \, a^{3} b^{5} d^{2} e^{6} - 35 \, a^{4} b^{4} d e^{7} + 9 \, a^{5} b^{3} e^{8}\right )} x^{3} + 140 \,{\left (b^{8} d^{6} e^{2} + 3 \, a b^{7} d^{5} e^{3} + 15 \, a^{2} b^{6} d^{4} e^{4} - 110 \, a^{3} b^{5} d^{3} e^{5} + 195 \, a^{4} b^{4} d^{2} e^{6} - 141 \, a^{5} b^{3} d e^{7} + 37 \, a^{6} b^{2} e^{8}\right )} x^{2} + 10 \,{\left (3 \, b^{8} d^{7} e + 7 \, a b^{7} d^{6} e^{2} + 21 \, a^{2} b^{6} d^{5} e^{3} + 105 \, a^{3} b^{5} d^{4} e^{4} - 875 \, a^{4} b^{4} d^{3} e^{5} + 1617 \, a^{5} b^{3} d^{2} e^{6} - 1197 \, a^{6} b^{2} d e^{7} + 319 \, a^{7} b e^{8}\right )} x}{15 \,{\left (b x + a\right )}^{5} b^{9}} + \frac{b^{12} x^{3} e^{8} + 12 \, b^{12} d x^{2} e^{7} + 84 \, b^{12} d^{2} x e^{6} - 9 \, a b^{11} x^{2} e^{8} - 144 \, a b^{11} d x e^{7} + 63 \, a^{2} b^{10} x e^{8}}{3 \, b^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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