Optimal. Leaf size=156 \[ \frac{1}{375} x^3 \left (100 d^2-330 d e+81 e^2\right )-\frac{x^2 \left (825 d^2-810 d e-458 e^2\right )}{1250}+\frac{\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (5 x^2+2 x+3\right )}{15625}+\frac{x \left (2025 d^2+4580 d e-881 e^2\right )}{3125}-\frac{\left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{15625 \sqrt{14}}+\frac{1}{100} e x^4 (40 d-33 e)+\frac{4 e^2 x^5}{25} \]
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Rubi [A] time = 0.162131, antiderivative size = 156, 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} \[ \frac{1}{375} x^3 \left (100 d^2-330 d e+81 e^2\right )-\frac{x^2 \left (825 d^2-810 d e-458 e^2\right )}{1250}+\frac{\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (5 x^2+2 x+3\right )}{15625}+\frac{x \left (2025 d^2+4580 d e-881 e^2\right )}{3125}-\frac{\left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{15625 \sqrt{14}}+\frac{1}{100} e x^4 (40 d-33 e)+\frac{4 e^2 x^5}{25} \]
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)^2 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx &=\int \left (\frac{2025 d^2+4580 d e-881 e^2}{3125}-\frac{1}{625} \left (825 d^2-810 d e-458 e^2\right ) x+\frac{1}{125} \left (100 d^2-330 d e+81 e^2\right ) x^2+\frac{1}{25} (40 d-33 e) e x^3+\frac{4 e^2 x^4}{5}+\frac{175 d^2-13740 d e+2643 e^2+2 \left (5725 d^2-4405 d e-2554 e^2\right ) x}{3125 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac{\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac{\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac{1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac{1}{100} (40 d-33 e) e x^4+\frac{4 e^2 x^5}{25}+\frac{\int \frac{175 d^2-13740 d e+2643 e^2+2 \left (5725 d^2-4405 d e-2554 e^2\right ) x}{3+2 x+5 x^2} \, dx}{3125}\\ &=\frac{\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac{\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac{1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac{1}{100} (40 d-33 e) e x^4+\frac{4 e^2 x^5}{25}+\frac{\left (5725 d^2-4405 d e-2554 e^2\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{15625}+\frac{\left (-10575 d^2-59890 d e+18323 e^2\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{15625}\\ &=\frac{\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac{\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac{1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac{1}{100} (40 d-33 e) e x^4+\frac{4 e^2 x^5}{25}+\frac{\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (3+2 x+5 x^2\right )}{15625}+\frac{\left (2 \left (10575 d^2+59890 d e-18323 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{15625}\\ &=\frac{\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac{\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac{1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac{1}{100} (40 d-33 e) e x^4+\frac{4 e^2 x^5}{25}-\frac{\left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{15625 \sqrt{14}}+\frac{\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (3+2 x+5 x^2\right )}{15625}\\ \end{align*}
Mathematica [A] time = 0.0839269, size = 130, normalized size = 0.83 \[ \frac{35 x \left (50 d^2 \left (200 x^2-495 x+486\right )+60 d e \left (250 x^3-550 x^2+405 x+916\right )+3 e^2 \left (2000 x^4-4125 x^3+2700 x^2+4580 x-3524\right )\right )+84 \left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (5 x^2+2 x+3\right )-6 \sqrt{14} \left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1312500} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 191, normalized size = 1.2 \begin{align*}{\frac{4\,{e}^{2}{x}^{5}}{25}}+{\frac{2\,{x}^{4}de}{5}}-{\frac{33\,{x}^{4}{e}^{2}}{100}}+{\frac{4\,{x}^{3}{d}^{2}}{15}}-{\frac{22\,{x}^{3}de}{25}}+{\frac{27\,{e}^{2}{x}^{3}}{125}}-{\frac{33\,{x}^{2}{d}^{2}}{50}}+{\frac{81\,{x}^{2}de}{125}}+{\frac{229\,{x}^{2}{e}^{2}}{625}}+{\frac{81\,{d}^{2}x}{125}}+{\frac{916\,xde}{625}}-{\frac{881\,{e}^{2}x}{3125}}+{\frac{229\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{2}}{625}}-{\frac{881\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) de}{3125}}-{\frac{2554\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{2}}{15625}}-{\frac{423\,\sqrt{14}{d}^{2}}{8750}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{5989\,\sqrt{14}de}{21875}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{18323\,\sqrt{14}{e}^{2}}{218750}\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 = 1.53978, size = 190, normalized size = 1.22 \begin{align*} \frac{4}{25} \, e^{2} x^{5} + \frac{1}{100} \,{\left (40 \, d e - 33 \, e^{2}\right )} x^{4} + \frac{1}{375} \,{\left (100 \, d^{2} - 330 \, d e + 81 \, e^{2}\right )} x^{3} - \frac{1}{1250} \,{\left (825 \, d^{2} - 810 \, d e - 458 \, e^{2}\right )} x^{2} - \frac{1}{218750} \, \sqrt{14}{\left (10575 \, d^{2} + 59890 \, d e - 18323 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{3125} \,{\left (2025 \, d^{2} + 4580 \, d e - 881 \, e^{2}\right )} x + \frac{1}{15625} \,{\left (5725 \, d^{2} - 4405 \, d e - 2554 \, e^{2}\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 = 1.00717, size = 436, normalized size = 2.79 \begin{align*} \frac{4}{25} \, e^{2} x^{5} + \frac{1}{100} \,{\left (40 \, d e - 33 \, e^{2}\right )} x^{4} + \frac{1}{375} \,{\left (100 \, d^{2} - 330 \, d e + 81 \, e^{2}\right )} x^{3} - \frac{1}{1250} \,{\left (825 \, d^{2} - 810 \, d e - 458 \, e^{2}\right )} x^{2} - \frac{1}{218750} \, \sqrt{14}{\left (10575 \, d^{2} + 59890 \, d e - 18323 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{3125} \,{\left (2025 \, d^{2} + 4580 \, d e - 881 \, e^{2}\right )} x + \frac{1}{15625} \,{\left (5725 \, d^{2} - 4405 \, d e - 2554 \, e^{2}\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 = 0.998055, size = 303, normalized size = 1.94 \begin{align*} \frac{4 e^{2} x^{5}}{25} + x^{4} \left (\frac{2 d e}{5} - \frac{33 e^{2}}{100}\right ) + x^{3} \left (\frac{4 d^{2}}{15} - \frac{22 d e}{25} + \frac{27 e^{2}}{125}\right ) + x^{2} \left (- \frac{33 d^{2}}{50} + \frac{81 d e}{125} + \frac{229 e^{2}}{625}\right ) + x \left (\frac{81 d^{2}}{125} + \frac{916 d e}{625} - \frac{881 e^{2}}{3125}\right ) + \left (\frac{229 d^{2}}{625} - \frac{881 d e}{3125} - \frac{2554 e^{2}}{15625} - \frac{\sqrt{14} i \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{437500}\right ) \log{\left (x + \frac{2115 d^{2} + 11978 d e - \frac{18323 e^{2}}{5} + \frac{\sqrt{14} i \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{5}}{10575 d^{2} + 59890 d e - 18323 e^{2}} \right )} + \left (\frac{229 d^{2}}{625} - \frac{881 d e}{3125} - \frac{2554 e^{2}}{15625} + \frac{\sqrt{14} i \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{437500}\right ) \log{\left (x + \frac{2115 d^{2} + 11978 d e - \frac{18323 e^{2}}{5} - \frac{\sqrt{14} i \left (10575 d^{2} + 59890 d e - 18323 e^{2}\right )}{5}}{10575 d^{2} + 59890 d e - 18323 e^{2}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13646, size = 196, normalized size = 1.26 \begin{align*} \frac{4}{25} \, x^{5} e^{2} + \frac{2}{5} \, d x^{4} e + \frac{4}{15} \, d^{2} x^{3} - \frac{33}{100} \, x^{4} e^{2} - \frac{22}{25} \, d x^{3} e - \frac{33}{50} \, d^{2} x^{2} + \frac{27}{125} \, x^{3} e^{2} + \frac{81}{125} \, d x^{2} e + \frac{81}{125} \, d^{2} x + \frac{229}{625} \, x^{2} e^{2} + \frac{916}{625} \, d x e - \frac{1}{218750} \, \sqrt{14}{\left (10575 \, d^{2} + 59890 \, d e - 18323 \, e^{2}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{881}{3125} \, x e^{2} + \frac{1}{15625} \,{\left (5725 \, d^{2} - 4405 \, d e - 2554 \, e^{2}\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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