Optimal. Leaf size=317 \[ \frac{\left (-21 d^2 e+458 d^3-816 d e^2+113 e^3\right ) \log \left (5 x^2+2 x+3\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{28 d^3 e^2+44 d^2 e^3+d^4 e+40 d^5-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (228 d^4 e^2-242 d^3 e^3+141 d^2 e^4-120 d^5 e+100 d^6+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (-4101 d^2 e+423 d^3+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{5 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3} \]
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Rubi [A] time = 0.290028, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {1628, 634, 618, 204, 628} \[ \frac{\left (-21 d^2 e+458 d^3-816 d e^2+113 e^3\right ) \log \left (5 x^2+2 x+3\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{28 d^3 e^2+44 d^2 e^3+d^4 e+40 d^5-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (228 d^4 e^2-242 d^3 e^3+141 d^2 e^4-120 d^5 e+100 d^6+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (-4101 d^2 e+423 d^3+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{5 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 1628
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{2+x+3 x^2-5 x^3+4 x^4}{(d+e x)^3 \left (3+2 x+5 x^2\right )} \, dx &=\int \left (\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{e^2 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^3}+\frac{-40 d^5-d^4 e-28 d^3 e^2-44 d^2 e^3+2 d e^4-e^5}{e^2 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)^2}+\frac{100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6}{e^2 \left (5 d^2-2 d e+3 e^2\right )^3 (d+e x)}+\frac{7 d^3+816 d^2 e-339 d e^2-118 e^3+\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) x}{\left (5 d^2-2 d e+3 e^2\right )^3 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\int \frac{7 d^3+816 d^2 e-339 d e^2-118 e^3+\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) x}{3+2 x+5 x^2} \, dx}{\left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{10 \left (5 d^2-2 d e+3 e^2\right )^3}-\frac{\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{5 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}+\frac{\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (2 \left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{5 \left (5 d^2-2 d e+3 e^2\right )^3}\\ &=-\frac{4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4}{2 e^3 \left (5 d^2-2 d e+3 e^2\right ) (d+e x)^2}+\frac{40 d^5+d^4 e+28 d^3 e^2+44 d^2 e^3-2 d e^4+e^5}{e^3 \left (5 d^2-2 d e+3 e^2\right )^2 (d+e x)}-\frac{\left (423 d^3-4101 d^2 e+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{5 \sqrt{14} \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (100 d^6-120 d^5 e+228 d^4 e^2-242 d^3 e^3+141 d^2 e^4+120 d e^5-e^6\right ) \log (d+e x)}{e^3 \left (5 d^2-2 d e+3 e^2\right )^3}+\frac{\left (458 d^3-21 d^2 e-816 d e^2+113 e^3\right ) \log \left (3+2 x+5 x^2\right )}{10 \left (5 d^2-2 d e+3 e^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.432427, size = 278, normalized size = 0.88 \[ -\frac{-7 \left (-21 d^2 e+458 d^3-816 d e^2+113 e^3\right ) \log \left (5 x^2+2 x+3\right )+\frac{35 \left (3 d^2 e^2+5 d^3 e+4 d^4-d e^3+2 e^4\right ) \left (5 d^2-2 d e+3 e^2\right )^2}{e^3 (d+e x)^2}-\frac{70 \left (28 d^3 e^2+44 d^2 e^3+d^4 e+40 d^5-2 d e^4+e^5\right ) \left (5 d^2-2 d e+3 e^2\right )}{e^3 (d+e x)}+\frac{70 \left (-228 d^4 e^2+242 d^3 e^3-141 d^2 e^4+120 d^5 e-100 d^6-120 d e^5+e^6\right ) \log (d+e x)}{e^3}+\sqrt{14} \left (-4101 d^2 e+423 d^3+879 d e^2+703 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{70 \left (5 d^2-2 d e+3 e^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 819, normalized size = 2.6 \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 [A] time = 1.50774, size = 672, normalized size = 2.12 \begin{align*} -\frac{\sqrt{14}{\left (423 \, d^{3} - 4101 \, d^{2} e + 879 \, d e^{2} + 703 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{70 \,{\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac{{\left (100 \, d^{6} - 120 \, d^{5} e + 228 \, d^{4} e^{2} - 242 \, d^{3} e^{3} + 141 \, d^{2} e^{4} + 120 \, d e^{5} - e^{6}\right )} \log \left (e x + d\right )}{125 \, d^{6} e^{3} - 150 \, d^{5} e^{4} + 285 \, d^{4} e^{5} - 188 \, d^{3} e^{6} + 171 \, d^{2} e^{7} - 54 \, d e^{8} + 27 \, e^{9}} + \frac{{\left (458 \, d^{3} - 21 \, d^{2} e - 816 \, d e^{2} + 113 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{10 \,{\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac{60 \, d^{6} - 15 \, d^{5} e + 39 \, d^{4} e^{2} + 84 \, d^{3} e^{3} - 25 \, d^{2} e^{4} + 9 \, d e^{5} - 6 \, e^{6} + 2 \,{\left (40 \, d^{5} e + d^{4} e^{2} + 28 \, d^{3} e^{3} + 44 \, d^{2} e^{4} - 2 \, d e^{5} + e^{6}\right )} x}{2 \,{\left (25 \, d^{6} e^{3} - 20 \, d^{5} e^{4} + 34 \, d^{4} e^{5} - 12 \, d^{3} e^{6} + 9 \, d^{2} e^{7} +{\left (25 \, d^{4} e^{5} - 20 \, d^{3} e^{6} + 34 \, d^{2} e^{7} - 12 \, d e^{8} + 9 \, e^{9}\right )} x^{2} + 2 \,{\left (25 \, d^{5} e^{4} - 20 \, d^{4} e^{5} + 34 \, d^{3} e^{6} - 12 \, d^{2} e^{7} + 9 \, d e^{8}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30544, size = 1674, normalized size = 5.28 \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 [A] time = 1.16568, size = 548, normalized size = 1.73 \begin{align*} -\frac{\sqrt{14}{\left (423 \, d^{3} - 4101 \, d^{2} e + 879 \, d e^{2} + 703 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right )}{70 \,{\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac{{\left (458 \, d^{3} - 21 \, d^{2} e - 816 \, d e^{2} + 113 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{10 \,{\left (125 \, d^{6} - 150 \, d^{5} e + 285 \, d^{4} e^{2} - 188 \, d^{3} e^{3} + 171 \, d^{2} e^{4} - 54 \, d e^{5} + 27 \, e^{6}\right )}} + \frac{{\left (100 \, d^{6} - 120 \, d^{5} e + 228 \, d^{4} e^{2} - 242 \, d^{3} e^{3} + 141 \, d^{2} e^{4} + 120 \, d e^{5} - e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{125 \, d^{6} e^{3} - 150 \, d^{5} e^{4} + 285 \, d^{4} e^{5} - 188 \, d^{3} e^{6} + 171 \, d^{2} e^{7} - 54 \, d e^{8} + 27 \, e^{9}} + \frac{{\left (2 \,{\left (200 \, d^{7} - 75 \, d^{6} e + 258 \, d^{5} e^{2} + 167 \, d^{4} e^{3} - 14 \, d^{3} e^{4} + 141 \, d^{2} e^{5} - 8 \, d e^{6} + 3 \, e^{7}\right )} x +{\left (300 \, d^{8} - 195 \, d^{7} e + 405 \, d^{6} e^{2} + 297 \, d^{5} e^{3} - 176 \, d^{4} e^{4} + 347 \, d^{3} e^{5} - 123 \, d^{2} e^{6} + 39 \, d e^{7} - 18 \, e^{8}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (5 \, d^{2} - 2 \, d e + 3 \, e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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