Optimal. Leaf size=99 \[ \frac{1}{75} x^3 (20 d-33 e)-\frac{3}{250} x^2 (55 d-27 e)+\frac{(2290 d-881 e) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{1}{625} x (405 d+458 e)-\frac{(2115 d+5989 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{3125 \sqrt{14}}+\frac{e x^4}{5} \]
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Rubi [A] time = 0.108266, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {1628, 634, 618, 204, 628} \[ \frac{1}{75} x^3 (20 d-33 e)-\frac{3}{250} x^2 (55 d-27 e)+\frac{(2290 d-881 e) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{1}{625} x (405 d+458 e)-\frac{(2115 d+5989 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{3125 \sqrt{14}}+\frac{e x^4}{5} \]
Antiderivative was successfully verified.
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Rule 1628
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx &=\int \left (\frac{1}{625} (405 d+458 e)-\frac{3}{125} (55 d-27 e) x+\frac{1}{25} (20 d-33 e) x^2+\frac{4 e x^3}{5}+\frac{35 d-1374 e+(2290 d-881 e) x}{625 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac{1}{625} (405 d+458 e) x-\frac{3}{250} (55 d-27 e) x^2+\frac{1}{75} (20 d-33 e) x^3+\frac{e x^4}{5}+\frac{1}{625} \int \frac{35 d-1374 e+(2290 d-881 e) x}{3+2 x+5 x^2} \, dx\\ &=\frac{1}{625} (405 d+458 e) x-\frac{3}{250} (55 d-27 e) x^2+\frac{1}{75} (20 d-33 e) x^3+\frac{e x^4}{5}+\frac{(-2115 d-5989 e) \int \frac{1}{3+2 x+5 x^2} \, dx}{3125}+\frac{(2290 d-881 e) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{6250}\\ &=\frac{1}{625} (405 d+458 e) x-\frac{3}{250} (55 d-27 e) x^2+\frac{1}{75} (20 d-33 e) x^3+\frac{e x^4}{5}+\frac{(2290 d-881 e) \log \left (3+2 x+5 x^2\right )}{6250}+\frac{(2 (2115 d+5989 e)) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{3125}\\ &=\frac{1}{625} (405 d+458 e) x-\frac{3}{250} (55 d-27 e) x^2+\frac{1}{75} (20 d-33 e) x^3+\frac{e x^4}{5}-\frac{(2115 d+5989 e) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{3125 \sqrt{14}}+\frac{(2290 d-881 e) \log \left (3+2 x+5 x^2\right )}{6250}\\ \end{align*}
Mathematica [A] time = 0.0503886, size = 86, normalized size = 0.87 \[ \frac{35 x \left (5 d \left (200 x^2-495 x+486\right )+3 e \left (250 x^3-550 x^2+405 x+916\right )\right )+21 (2290 d-881 e) \log \left (5 x^2+2 x+3\right )-3 \sqrt{14} (2115 d+5989 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{131250} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 102, normalized size = 1. \begin{align*}{\frac{e{x}^{4}}{5}}+{\frac{4\,d{x}^{3}}{15}}-{\frac{11\,{x}^{3}e}{25}}-{\frac{33\,d{x}^{2}}{50}}+{\frac{81\,e{x}^{2}}{250}}+{\frac{81\,dx}{125}}+{\frac{458\,ex}{625}}+{\frac{229\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) d}{625}}-{\frac{881\,e\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) }{6250}}-{\frac{423\,\sqrt{14}d}{8750}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{5989\,\sqrt{14}e}{43750}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56544, size = 113, normalized size = 1.14 \begin{align*} \frac{1}{5} \, e x^{4} + \frac{1}{75} \,{\left (20 \, d - 33 \, e\right )} x^{3} - \frac{3}{250} \,{\left (55 \, d - 27 \, e\right )} x^{2} - \frac{1}{43750} \, \sqrt{14}{\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{625} \,{\left (405 \, d + 458 \, e\right )} x + \frac{1}{6250} \,{\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31163, size = 275, normalized size = 2.78 \begin{align*} \frac{1}{5} \, e x^{4} + \frac{1}{75} \,{\left (20 \, d - 33 \, e\right )} x^{3} - \frac{3}{250} \,{\left (55 \, d - 27 \, e\right )} x^{2} - \frac{1}{43750} \, \sqrt{14}{\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{625} \,{\left (405 \, d + 458 \, e\right )} x + \frac{1}{6250} \,{\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.686389, size = 163, normalized size = 1.65 \begin{align*} \frac{e x^{4}}{5} + x^{3} \left (\frac{4 d}{15} - \frac{11 e}{25}\right ) + x^{2} \left (- \frac{33 d}{50} + \frac{81 e}{250}\right ) + x \left (\frac{81 d}{125} + \frac{458 e}{625}\right ) + \left (\frac{229 d}{625} - \frac{881 e}{6250} - \frac{\sqrt{14} i \left (2115 d + 5989 e\right )}{87500}\right ) \log{\left (x + \frac{423 d + \frac{5989 e}{5} + \frac{\sqrt{14} i \left (2115 d + 5989 e\right )}{5}}{2115 d + 5989 e} \right )} + \left (\frac{229 d}{625} - \frac{881 e}{6250} + \frac{\sqrt{14} i \left (2115 d + 5989 e\right )}{87500}\right ) \log{\left (x + \frac{423 d + \frac{5989 e}{5} - \frac{\sqrt{14} i \left (2115 d + 5989 e\right )}{5}}{2115 d + 5989 e} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1224, size = 119, normalized size = 1.2 \begin{align*} \frac{1}{5} \, x^{4} e + \frac{4}{15} \, d x^{3} - \frac{11}{25} \, x^{3} e - \frac{33}{50} \, d x^{2} + \frac{81}{250} \, x^{2} e - \frac{1}{43750} \, \sqrt{14}{\left (2115 \, d + 5989 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{81}{125} \, d x + \frac{458}{625} \, x e + \frac{1}{6250} \,{\left (2290 \, d - 881 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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