Optimal. Leaf size=189 \[ \frac {e x^2 \left (840 d^2-1722 d e+373 e^2\right )}{3500}-\frac {\left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac {x \left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right )}{17500}+\frac {\left (32825 d^3+317565 d^2 e-221643 d e^2-67499 e^3\right ) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{87500 \sqrt {14}}+\frac {1}{375} e^2 x^3 (60 d-41 e)-\frac {(423 x+1367) (d+e x)^3}{3500 \left (5 x^2+2 x+3\right )}+\frac {e^3 x^4}{25} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1644, 1628, 634, 618, 204, 628} \[ \frac {e x^2 \left (840 d^2-1722 d e+373 e^2\right )}{3500}-\frac {\left (-1545 d^2 e+1025 d^3-2601 d e^2+832 e^3\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac {x \left (-17220 d^2 e+2800 d^3+9921 d e^2+6053 e^3\right )}{17500}+\frac {\left (317565 d^2 e+32825 d^3-221643 d e^2-67499 e^3\right ) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{87500 \sqrt {14}}+\frac {1}{375} e^2 x^3 (60 d-41 e)-\frac {(423 x+1367) (d+e x)^3}{3500 \left (5 x^2+2 x+3\right )}+\frac {e^3 x^4}{25} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1628
Rule 1644
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (2+x+3 x^2-5 x^3+4 x^4\right )}{\left (3+2 x+5 x^2\right )^2} \, dx &=-\frac {(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac {1}{56} \int \frac {(d+e x)^2 \left (\frac {6}{125} (615 d+1367 e)-\frac {12}{125} (770 d-519 e) x+\frac {56}{25} (20 d-33 e) x^2+\frac {224 e x^3}{5}\right )}{3+2 x+5 x^2} \, dx\\ &=-\frac {(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac {1}{56} \int \left (\frac {2}{625} \left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right )+\frac {4}{125} e \left (840 d^2-1722 d e+373 e^2\right ) x+\frac {56}{125} (60 d-41 e) e^2 x^2+\frac {224 e^3 x^3}{25}+\frac {2 \left (3 \left (275 d^3+24055 d^2 e-9921 d e^2-6053 e^3\right )-28 \left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) x\right )}{625 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac {\left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right ) x}{17500}+\frac {e \left (840 d^2-1722 d e+373 e^2\right ) x^2}{3500}+\frac {1}{375} (60 d-41 e) e^2 x^3+\frac {e^3 x^4}{25}-\frac {(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac {\int \frac {3 \left (275 d^3+24055 d^2 e-9921 d e^2-6053 e^3\right )-28 \left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) x}{3+2 x+5 x^2} \, dx}{17500}\\ &=\frac {\left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right ) x}{17500}+\frac {e \left (840 d^2-1722 d e+373 e^2\right ) x^2}{3500}+\frac {1}{375} (60 d-41 e) e^2 x^3+\frac {e^3 x^4}{25}-\frac {(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac {\left (32825 d^3+317565 d^2 e-221643 d e^2-67499 e^3\right ) \int \frac {1}{3+2 x+5 x^2} \, dx}{87500}+\frac {\left (-1025 d^3+1545 d^2 e+2601 d e^2-832 e^3\right ) \int \frac {2+10 x}{3+2 x+5 x^2} \, dx}{6250}\\ &=\frac {\left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right ) x}{17500}+\frac {e \left (840 d^2-1722 d e+373 e^2\right ) x^2}{3500}+\frac {1}{375} (60 d-41 e) e^2 x^3+\frac {e^3 x^4}{25}-\frac {(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}-\frac {\left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) \log \left (3+2 x+5 x^2\right )}{6250}+\frac {\left (-32825 d^3-317565 d^2 e+221643 d e^2+67499 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{-56-x^2} \, dx,x,2+10 x\right )}{43750}\\ &=\frac {\left (2800 d^3-17220 d^2 e+9921 d e^2+6053 e^3\right ) x}{17500}+\frac {e \left (840 d^2-1722 d e+373 e^2\right ) x^2}{3500}+\frac {1}{375} (60 d-41 e) e^2 x^3+\frac {e^3 x^4}{25}-\frac {(1367+423 x) (d+e x)^3}{3500 \left (3+2 x+5 x^2\right )}+\frac {\left (32825 d^3+317565 d^2 e-221643 d e^2-67499 e^3\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{87500 \sqrt {14}}-\frac {\left (1025 d^3-1545 d^2 e-2601 d e^2+832 e^3\right ) \log \left (3+2 x+5 x^2\right )}{6250}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 209, normalized size = 1.11 \[ \frac {14700 e x^2 \left (300 d^2-615 d e+103 e^2\right )-\frac {42 \left (125 d^3 (423 x+1367)+75 d^2 e (5989 x-1269)-15 d e^2 (18323 x+17967)+e^3 (54969-53189 x)\right )}{5 x^2+2 x+3}+2940 \left (-1025 d^3+1545 d^2 e+2601 d e^2-832 e^3\right ) \log \left (5 x^2+2 x+3\right )+5880 x \left (500 d^3-3075 d^2 e+1545 d e^2+867 e^3\right )+15 \sqrt {14} \left (32825 d^3+317565 d^2 e-221643 d e^2-67499 e^3\right ) \tan ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )+49000 e^2 x^3 (60 d-41 e)+735000 e^3 x^4}{18375000} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.87, size = 350, normalized size = 1.85 \[ \frac {3675000 \, e^{3} x^{6} + 1225000 \, {\left (12 \, d e^{2} - 7 \, e^{3}\right )} x^{5} + 122500 \, {\left (180 \, d^{2} e - 321 \, d e^{2} + 47 \, e^{3}\right )} x^{4} + 147000 \, {\left (100 \, d^{3} - 555 \, d^{2} e + 246 \, d e^{2} + 153 \, e^{3}\right )} x^{3} - 7176750 \, d^{3} + 3997350 \, d^{2} e + 11319210 \, d e^{2} - 2308698 \, e^{3} + 2940 \, {\left (2000 \, d^{3} - 7800 \, d^{2} e - 3045 \, d e^{2} + 5013 \, e^{3}\right )} x^{2} + 15 \, \sqrt {14} {\left (98475 \, d^{3} + 952695 \, d^{2} e - 664929 \, d e^{2} - 202497 \, e^{3} + 5 \, {\left (32825 \, d^{3} + 317565 \, d^{2} e - 221643 \, d e^{2} - 67499 \, e^{3}\right )} x^{2} + 2 \, {\left (32825 \, d^{3} + 317565 \, d^{2} e - 221643 \, d e^{2} - 67499 \, e^{3}\right )} x\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + 42 \, {\left (157125 \, d^{3} - 1740675 \, d^{2} e + 923745 \, d e^{2} + 417329 \, e^{3}\right )} x - 2940 \, {\left (3075 \, d^{3} - 4635 \, d^{2} e - 7803 \, d e^{2} + 2496 \, e^{3} + 5 \, {\left (1025 \, d^{3} - 1545 \, d^{2} e - 2601 \, d e^{2} + 832 \, e^{3}\right )} x^{2} + 2 \, {\left (1025 \, d^{3} - 1545 \, d^{2} e - 2601 \, d e^{2} + 832 \, e^{3}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{18375000 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 206, normalized size = 1.09 \[ \frac {1}{25} \, x^{4} e^{3} + \frac {4}{25} \, d x^{3} e^{2} + \frac {6}{25} \, d^{2} x^{2} e + \frac {4}{25} \, d^{3} x - \frac {41}{375} \, x^{3} e^{3} - \frac {123}{250} \, d x^{2} e^{2} - \frac {123}{125} \, d^{2} x e + \frac {103}{1250} \, x^{2} e^{3} + \frac {309}{625} \, d x e^{2} + \frac {1}{1225000} \, \sqrt {14} {\left (32825 \, d^{3} + 317565 \, d^{2} e - 221643 \, d e^{2} - 67499 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {867}{3125} \, x e^{3} - \frac {1}{6250} \, {\left (1025 \, d^{3} - 1545 \, d^{2} e - 2601 \, d e^{2} + 832 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - \frac {170875 \, d^{3} - 95175 \, d^{2} e + {\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} x - 269505 \, d e^{2} + 54969 \, e^{3}}{437500 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 283, normalized size = 1.50 \[ \frac {e^{3} x^{4}}{25}+\frac {4 d \,e^{2} x^{3}}{25}-\frac {41 e^{3} x^{3}}{375}+\frac {6 d^{2} e \,x^{2}}{25}-\frac {123 d \,e^{2} x^{2}}{250}+\frac {103 e^{3} x^{2}}{1250}+\frac {4 d^{3} x}{25}+\frac {1313 \sqrt {14}\, d^{3} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{49000}-\frac {41 d^{3} \ln \left (5 x^{2}+2 x +3\right )}{250}-\frac {123 d^{2} e x}{125}+\frac {63513 \sqrt {14}\, d^{2} e \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{245000}+\frac {309 d^{2} e \ln \left (5 x^{2}+2 x +3\right )}{1250}+\frac {309 d \,e^{2} x}{625}-\frac {221643 \sqrt {14}\, d \,e^{2} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{1225000}+\frac {2601 d \,e^{2} \ln \left (5 x^{2}+2 x +3\right )}{6250}+\frac {867 e^{3} x}{3125}-\frac {67499 \sqrt {14}\, e^{3} \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{1225000}-\frac {416 e^{3} \ln \left (5 x^{2}+2 x +3\right )}{3125}-\frac {\frac {6835 d^{3}}{28}-\frac {3807 d^{2} e}{28}-\frac {53901 d \,e^{2}}{140}+\frac {54969 e^{3}}{700}+\left (\frac {2115}{28} d^{3}+\frac {17967}{28} d^{2} e -\frac {54969}{140} d \,e^{2}-\frac {53189}{700} e^{3}\right ) x}{3125 \left (x^{2}+\frac {2}{5} x +\frac {3}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 212, normalized size = 1.12 \[ \frac {1}{25} \, e^{3} x^{4} + \frac {1}{375} \, {\left (60 \, d e^{2} - 41 \, e^{3}\right )} x^{3} + \frac {1}{1250} \, {\left (300 \, d^{2} e - 615 \, d e^{2} + 103 \, e^{3}\right )} x^{2} + \frac {1}{1225000} \, \sqrt {14} {\left (32825 \, d^{3} + 317565 \, d^{2} e - 221643 \, d e^{2} - 67499 \, e^{3}\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {1}{3125} \, {\left (500 \, d^{3} - 3075 \, d^{2} e + 1545 \, d e^{2} + 867 \, e^{3}\right )} x - \frac {1}{6250} \, {\left (1025 \, d^{3} - 1545 \, d^{2} e - 2601 \, d e^{2} + 832 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - \frac {170875 \, d^{3} - 95175 \, d^{2} e - 269505 \, d e^{2} + 54969 \, e^{3} + {\left (52875 \, d^{3} + 449175 \, d^{2} e - 274845 \, d e^{2} - 53189 \, e^{3}\right )} x}{437500 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 333, normalized size = 1.76 \[ \frac {\frac {53901\,d\,e^2}{28}+\frac {19035\,d^2\,e}{28}+x\,\left (-\frac {10575\,d^3}{28}-\frac {89835\,d^2\,e}{28}+\frac {54969\,d\,e^2}{28}+\frac {53189\,e^3}{140}\right )-\frac {34175\,d^3}{28}-\frac {54969\,e^3}{140}}{15625\,x^2+6250\,x+9375}+x^3\,\left (\frac {e^2\,\left (12\,d-5\,e\right )}{75}-\frac {16\,e^3}{375}\right )-x\,\left (\frac {18\,e^2\,\left (12\,d-5\,e\right )}{625}+\frac {12\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{125}-\frac {9\,d\,e^2}{25}+\frac {3\,d^2\,e}{5}-\frac {4\,d^3}{25}-\frac {717\,e^3}{3125}\right )+\ln \left (5\,x^2+2\,x+3\right )\,\left (-\frac {41\,d^3}{250}+\frac {309\,d^2\,e}{1250}+\frac {2601\,d\,e^2}{6250}-\frac {416\,e^3}{3125}\right )-x^2\,\left (\frac {2\,e^2\,\left (12\,d-5\,e\right )}{125}-\frac {3\,e\,\left (4\,d^2-5\,d\,e+e^2\right )}{50}+\frac {36\,e^3}{625}\right )+\frac {e^3\,x^4}{25}-\frac {\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {\sqrt {14}\,\left (-32825\,d^3-317565\,d^2\,e+221643\,d\,e^2+67499\,e^3\right )}{1225000}+\frac {\sqrt {14}\,x\,\left (-32825\,d^3-317565\,d^2\,e+221643\,d\,e^2+67499\,e^3\right )}{245000}}{-\frac {1313\,d^3}{3500}-\frac {63513\,d^2\,e}{17500}+\frac {221643\,d\,e^2}{87500}+\frac {67499\,e^3}{87500}}\right )\,\left (-32825\,d^3-317565\,d^2\,e+221643\,d\,e^2+67499\,e^3\right )}{1225000} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 2.77, size = 444, normalized size = 2.35 \[ \frac {e^{3} x^{4}}{25} + x^{3} \left (\frac {4 d e^{2}}{25} - \frac {41 e^{3}}{375}\right ) + x^{2} \left (\frac {6 d^{2} e}{25} - \frac {123 d e^{2}}{250} + \frac {103 e^{3}}{1250}\right ) + x \left (\frac {4 d^{3}}{25} - \frac {123 d^{2} e}{125} + \frac {309 d e^{2}}{625} + \frac {867 e^{3}}{3125}\right ) + \left (- \frac {41 d^{3}}{250} + \frac {309 d^{2} e}{1250} + \frac {2601 d e^{2}}{6250} - \frac {416 e^{3}}{3125} - \frac {\sqrt {14} i \left (32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}\right )}{2450000}\right ) \log {\left (x + \frac {6565 d^{3} + 63513 d^{2} e - \frac {221643 d e^{2}}{5} - \frac {67499 e^{3}}{5} - \frac {\sqrt {14} i \left (32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}\right )}{5}}{32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}} \right )} + \left (- \frac {41 d^{3}}{250} + \frac {309 d^{2} e}{1250} + \frac {2601 d e^{2}}{6250} - \frac {416 e^{3}}{3125} + \frac {\sqrt {14} i \left (32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}\right )}{2450000}\right ) \log {\left (x + \frac {6565 d^{3} + 63513 d^{2} e - \frac {221643 d e^{2}}{5} - \frac {67499 e^{3}}{5} + \frac {\sqrt {14} i \left (32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}\right )}{5}}{32825 d^{3} + 317565 d^{2} e - 221643 d e^{2} - 67499 e^{3}} \right )} + \frac {- 170875 d^{3} + 95175 d^{2} e + 269505 d e^{2} - 54969 e^{3} + x \left (- 52875 d^{3} - 449175 d^{2} e + 274845 d e^{2} + 53189 e^{3}\right )}{2187500 x^{2} + 875000 x + 1312500} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________