Optimal. Leaf size=221 \[ \frac {3}{500} e x^4 \left (100 d^2-165 d e+27 e^2\right )+\frac {x^3 \left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right )}{1875}-\frac {x^2 \left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right )}{6250}+\frac {\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (5 x^2+2 x+3\right )}{156250}+\frac {x \left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right )}{15625}-\frac {\left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right ) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{78125 \sqrt {14}}+\frac {3}{125} e^2 x^5 (20 d-11 e)+\frac {2 e^3 x^6}{15} \]
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Rubi [A] time = 0.19, antiderivative size = 221, normalized size of antiderivative = 1.00, 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} \[ \frac {3}{500} e x^4 \left (100 d^2-165 d e+27 e^2\right )+\frac {x^3 \left (-2475 d^2 e+500 d^3+1215 d e^2+458 e^3\right )}{1875}-\frac {x^2 \left (-6075 d^2 e+4125 d^3-6870 d e^2+881 e^3\right )}{6250}+\frac {\left (-66075 d^2 e+57250 d^3-76620 d e^2+23431 e^3\right ) \log \left (5 x^2+2 x+3\right )}{156250}+\frac {x \left (34350 d^2 e+10125 d^3-13215 d e^2-5108 e^3\right )}{15625}-\frac {\left (449175 d^2 e+52875 d^3-274845 d e^2-53189 e^3\right ) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{78125 \sqrt {14}}+\frac {3}{125} e^2 x^5 (20 d-11 e)+\frac {2 e^3 x^6}{15} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1628
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{3+2 x+5 x^2} \, dx &=\int \left (\frac {10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3}{15625}-\frac {\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x}{3125}+\frac {1}{625} \left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^2+\frac {3}{125} e \left (100 d^2-165 d e+27 e^2\right ) x^3+\frac {3}{25} (20 d-11 e) e^2 x^4+\frac {4 e^3 x^5}{5}+\frac {875 d^3-103050 d^2 e+39645 d e^2+15324 e^3+\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) x}{15625 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac {\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac {\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac {\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac {3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac {3}{125} (20 d-11 e) e^2 x^5+\frac {2 e^3 x^6}{15}+\frac {\int \frac {875 d^3-103050 d^2 e+39645 d e^2+15324 e^3+\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) x}{3+2 x+5 x^2} \, dx}{15625}\\ &=\frac {\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac {\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac {\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac {3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac {3}{125} (20 d-11 e) e^2 x^5+\frac {2 e^3 x^6}{15}+\frac {\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \int \frac {2+10 x}{3+2 x+5 x^2} \, dx}{156250}+\frac {\left (-52875 d^3-449175 d^2 e+274845 d e^2+53189 e^3\right ) \int \frac {1}{3+2 x+5 x^2} \, dx}{78125}\\ &=\frac {\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac {\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac {\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac {3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac {3}{125} (20 d-11 e) e^2 x^5+\frac {2 e^3 x^6}{15}+\frac {\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (3+2 x+5 x^2\right )}{156250}+\frac {\left (2 \left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+10 x\right )}{78125}\\ &=\frac {\left (10125 d^3+34350 d^2 e-13215 d e^2-5108 e^3\right ) x}{15625}-\frac {\left (4125 d^3-6075 d^2 e-6870 d e^2+881 e^3\right ) x^2}{6250}+\frac {\left (500 d^3-2475 d^2 e+1215 d e^2+458 e^3\right ) x^3}{1875}+\frac {3}{500} e \left (100 d^2-165 d e+27 e^2\right ) x^4+\frac {3}{125} (20 d-11 e) e^2 x^5+\frac {2 e^3 x^6}{15}-\frac {\left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{78125 \sqrt {14}}+\frac {\left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (3+2 x+5 x^2\right )}{156250}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 178, normalized size = 0.81 \[ \frac {42 \left (57250 d^3-66075 d^2 e-76620 d e^2+23431 e^3\right ) \log \left (5 x^2+2 x+3\right )+35 x \left (250 d^3 \left (200 x^2-495 x+486\right )+450 d^2 e \left (250 x^3-550 x^2+405 x+916\right )+45 d e^2 \left (2000 x^4-4125 x^3+2700 x^2+4580 x-3524\right )+e^3 \left (25000 x^5-49500 x^4+30375 x^3+45800 x^2-26430 x-61296\right )\right )-6 \sqrt {14} \left (52875 d^3+449175 d^2 e-274845 d e^2-53189 e^3\right ) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{6562500} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 206, normalized size = 0.93 \[ \frac {2}{15} \, e^{3} x^{6} + \frac {3}{125} \, {\left (20 \, d e^{2} - 11 \, e^{3}\right )} x^{5} + \frac {3}{500} \, {\left (100 \, d^{2} e - 165 \, d e^{2} + 27 \, e^{3}\right )} x^{4} + \frac {1}{1875} \, {\left (500 \, d^{3} - 2475 \, d^{2} e + 1215 \, d e^{2} + 458 \, e^{3}\right )} x^{3} - \frac {1}{6250} \, {\left (4125 \, d^{3} - 6075 \, d^{2} e - 6870 \, d e^{2} + 881 \, e^{3}\right )} x^{2} - \frac {1}{1093750} \, \sqrt {14} {\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{15625} \, {\left (10125 \, d^{3} + 34350 \, d^{2} e - 13215 \, d e^{2} - 5108 \, e^{3}\right )} x + \frac {1}{156250} \, {\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 212, normalized size = 0.96 \[ \frac {2}{15} \, x^{6} e^{3} + \frac {12}{25} \, d x^{5} e^{2} + \frac {3}{5} \, d^{2} x^{4} e + \frac {4}{15} \, d^{3} x^{3} - \frac {33}{125} \, x^{5} e^{3} - \frac {99}{100} \, d x^{4} e^{2} - \frac {33}{25} \, d^{2} x^{3} e - \frac {33}{50} \, d^{3} x^{2} + \frac {81}{500} \, x^{4} e^{3} + \frac {81}{125} \, d x^{3} e^{2} + \frac {243}{250} \, d^{2} x^{2} e + \frac {81}{125} \, d^{3} x + \frac {458}{1875} \, x^{3} e^{3} + \frac {687}{625} \, d x^{2} e^{2} + \frac {1374}{625} \, d^{2} x e - \frac {881}{6250} \, x^{2} e^{3} - \frac {2643}{3125} \, d x e^{2} - \frac {1}{1093750} \, \sqrt {14} {\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {5108}{15625} \, x e^{3} + \frac {1}{156250} \, {\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 291, normalized size = 1.32 \[ \frac {2 e^{3} x^{6}}{15}+\frac {12 d \,e^{2} x^{5}}{25}-\frac {33 e^{3} x^{5}}{125}+\frac {3 d^{2} e \,x^{4}}{5}-\frac {99 d \,e^{2} x^{4}}{100}+\frac {81 e^{3} x^{4}}{500}+\frac {4 d^{3} x^{3}}{15}-\frac {33 d^{2} e \,x^{3}}{25}+\frac {81 d \,e^{2} x^{3}}{125}+\frac {458 e^{3} x^{3}}{1875}-\frac {33 d^{3} x^{2}}{50}+\frac {243 d^{2} e \,x^{2}}{250}+\frac {687 d \,e^{2} x^{2}}{625}-\frac {881 e^{3} x^{2}}{6250}+\frac {81 d^{3} x}{125}-\frac {423 \sqrt {14}\, d^{3} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{8750}+\frac {229 d^{3} \ln \left (5 x^{2}+2 x +3\right )}{625}+\frac {1374 d^{2} e x}{625}-\frac {17967 \sqrt {14}\, d^{2} e \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{43750}-\frac {2643 d^{2} e \ln \left (5 x^{2}+2 x +3\right )}{6250}-\frac {2643 d \,e^{2} x}{3125}+\frac {54969 \sqrt {14}\, d \,e^{2} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{218750}-\frac {7662 d \,e^{2} \ln \left (5 x^{2}+2 x +3\right )}{15625}-\frac {5108 e^{3} x}{15625}+\frac {53189 \sqrt {14}\, e^{3} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{1093750}+\frac {23431 e^{3} \ln \left (5 x^{2}+2 x +3\right )}{156250} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 206, normalized size = 0.93 \[ \frac {2}{15} \, e^{3} x^{6} + \frac {3}{125} \, {\left (20 \, d e^{2} - 11 \, e^{3}\right )} x^{5} + \frac {3}{500} \, {\left (100 \, d^{2} e - 165 \, d e^{2} + 27 \, e^{3}\right )} x^{4} + \frac {1}{1875} \, {\left (500 \, d^{3} - 2475 \, d^{2} e + 1215 \, d e^{2} + 458 \, e^{3}\right )} x^{3} - \frac {1}{6250} \, {\left (4125 \, d^{3} - 6075 \, d^{2} e - 6870 \, d e^{2} + 881 \, e^{3}\right )} x^{2} - \frac {1}{1093750} \, \sqrt {14} {\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{15625} \, {\left (10125 \, d^{3} + 34350 \, d^{2} e - 13215 \, d e^{2} - 5108 \, e^{3}\right )} x + \frac {1}{156250} \, {\left (57250 \, d^{3} - 66075 \, d^{2} e - 76620 \, d e^{2} + 23431 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.18, size = 397, normalized size = 1.80 \[ x^2\,\left (\frac {26\,e^2\,\left (12\,d-5\,e\right )}{625}-\frac {33\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{250}-\frac {3\,d\,e^2}{50}+\frac {3\,d^2\,e}{2}-\frac {33\,d^3}{50}+\frac {622\,e^3}{3125}\right )-x^3\,\left (\frac {11\,e^2\,\left (12\,d-5\,e\right )}{375}+\frac {2\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{25}-\frac {3\,d\,e^2}{5}+d^2\,e-\frac {4\,d^3}{15}-\frac {111\,e^3}{625}\right )+x^5\,\left (\frac {e^2\,\left (12\,d-5\,e\right )}{25}-\frac {8\,e^3}{125}\right )-\ln \left (5\,x^2+2\,x+3\right )\,\left (-\frac {229\,d^3}{625}+\frac {2643\,d^2\,e}{6250}+\frac {7662\,d\,e^2}{15625}-\frac {23431\,e^3}{156250}\right )-x^4\,\left (\frac {e^2\,\left (12\,d-5\,e\right )}{50}-\frac {3\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{20}+\frac {11\,e^3}{125}\right )+\frac {2\,e^3\,x^6}{15}+x\,\left (\frac {61\,e^2\,\left (12\,d-5\,e\right )}{3125}+\frac {3\,d\,\left (d^2+d\,e+2\,e^2\right )}{5}+\frac {156\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{625}-\frac {129\,d\,e^2}{125}+\frac {3\,d^2\,e}{5}+\frac {6\,d^3}{125}-\frac {7483\,e^3}{15625}\right )+\frac {\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {\sqrt {14}\,\left (-52875\,d^3-449175\,d^2\,e+274845\,d\,e^2+53189\,e^3\right )}{1093750}+\frac {\sqrt {14}\,x\,\left (-52875\,d^3-449175\,d^2\,e+274845\,d\,e^2+53189\,e^3\right )}{218750}}{-\frac {423\,d^3}{625}-\frac {17967\,d^2\,e}{3125}+\frac {54969\,d\,e^2}{15625}+\frac {53189\,e^3}{78125}}\right )\,\left (-52875\,d^3-449175\,d^2\,e+274845\,d\,e^2+53189\,e^3\right )}{1093750} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.58, size = 450, normalized size = 2.04 \[ \frac {2 e^{3} x^{6}}{15} + x^{5} \left (\frac {12 d e^{2}}{25} - \frac {33 e^{3}}{125}\right ) + x^{4} \left (\frac {3 d^{2} e}{5} - \frac {99 d e^{2}}{100} + \frac {81 e^{3}}{500}\right ) + x^{3} \left (\frac {4 d^{3}}{15} - \frac {33 d^{2} e}{25} + \frac {81 d e^{2}}{125} + \frac {458 e^{3}}{1875}\right ) + x^{2} \left (- \frac {33 d^{3}}{50} + \frac {243 d^{2} e}{250} + \frac {687 d e^{2}}{625} - \frac {881 e^{3}}{6250}\right ) + x \left (\frac {81 d^{3}}{125} + \frac {1374 d^{2} e}{625} - \frac {2643 d e^{2}}{3125} - \frac {5108 e^{3}}{15625}\right ) + \left (\frac {229 d^{3}}{625} - \frac {2643 d^{2} e}{6250} - \frac {7662 d e^{2}}{15625} + \frac {23431 e^{3}}{156250} - \frac {\sqrt {14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{2187500}\right ) \log {\left (x + \frac {10575 d^{3} + 89835 d^{2} e - 54969 d e^{2} - \frac {53189 e^{3}}{5} + \frac {\sqrt {14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{5}}{52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}} \right )} + \left (\frac {229 d^{3}}{625} - \frac {2643 d^{2} e}{6250} - \frac {7662 d e^{2}}{15625} + \frac {23431 e^{3}}{156250} + \frac {\sqrt {14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{2187500}\right ) \log {\left (x + \frac {10575 d^{3} + 89835 d^{2} e - 54969 d e^{2} - \frac {53189 e^{3}}{5} - \frac {\sqrt {14} i \left (52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}\right )}{5}}{52875 d^{3} + 449175 d^{2} e - 274845 d e^{2} - 53189 e^{3}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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