Optimal. Leaf size=171 \[ \frac{3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{3 \left (-855175 d^2 e+353125 d^3+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{4900000 \sqrt{14}}+\frac{e^2 x (83065 d-126009 e)}{980000}+\frac{(d+e x)^2 (x (11015 d+49177 e)+3 (11449 d-2105 e))}{196000 \left (5 x^2+2 x+3\right )}-\frac{(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{2 e^3 x^2}{125} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.336016, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (5 x^2+2 x+3\right )}{6250}+\frac{3 \left (-855175 d^2 e+353125 d^3+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{4900000 \sqrt{14}}+\frac{e^2 x (83065 d-126009 e)}{980000}+\frac{(d+e x)^2 (x (11015 d+49177 e)+3 (11449 d-2105 e))}{196000 \left (5 x^2+2 x+3\right )}-\frac{(423 x+1367) (d+e x)^3}{7000 \left (5 x^2+2 x+3\right )^2}+\frac{2 e^3 x^2}{125} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1644
Rule 1628
Rule 634
Rule 618
Rule 204
Rule 628
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 )^3} \, dx &=-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{1}{112} \int \frac{(d+e x)^2 \left (\frac{6}{125} (1089 d+1367 e)-\frac{336}{125} (55 d-27 e) x+\frac{112}{25} (20 d-33 e) x^2+\frac{448 e x^3}{5}\right )}{\left (3+2 x+5 x^2\right )^2} \, dx\\ &=-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{(d+e x) \left (\frac{12}{25} \left (2825 d^2-5587 d e+842 e^2\right )+\frac{4}{25} (10341 d-22693 e) e x+\frac{25088 e^2 x^2}{25}\right )}{3+2 x+5 x^2} \, dx}{6272}\\ &=-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\int \left (\frac{4}{625} (83065 d-126009 e) e^2+\frac{25088 e^3 x}{125}+\frac{12 \left (70625 d^3-139675 d^2 e-62015 d e^2+126009 e^3+1568 e \left (100 d^2-245 d e+47 e^2\right ) x\right )}{625 \left (3+2 x+5 x^2\right )}\right ) \, dx}{6272}\\ &=\frac{(83065 d-126009 e) e^2 x}{980000}+\frac{2 e^3 x^2}{125}-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{3 \int \frac{70625 d^3-139675 d^2 e-62015 d e^2+126009 e^3+1568 e \left (100 d^2-245 d e+47 e^2\right ) x}{3+2 x+5 x^2} \, dx}{980000}\\ &=\frac{(83065 d-126009 e) e^2 x}{980000}+\frac{2 e^3 x^2}{125}-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{\left (3 e \left (100 d^2-245 d e+47 e^2\right )\right ) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{6250}+\frac{\left (3 \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right )\right ) \int \frac{1}{3+2 x+5 x^2} \, dx}{4900000}\\ &=\frac{(83065 d-126009 e) e^2 x}{980000}+\frac{2 e^3 x^2}{125}-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (3+2 x+5 x^2\right )}{6250}-\frac{\left (3 \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{2450000}\\ &=\frac{(83065 d-126009 e) e^2 x}{980000}+\frac{2 e^3 x^2}{125}-\frac{(1367+423 x) (d+e x)^3}{7000 \left (3+2 x+5 x^2\right )^2}+\frac{(d+e x)^2 (3 (11449 d-2105 e)+(11015 d+49177 e) x)}{196000 \left (3+2 x+5 x^2\right )}+\frac{3 \left (353125 d^3-855175 d^2 e+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{4900000 \sqrt{14}}+\frac{3 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (3+2 x+5 x^2\right )}{6250}\\ \end{align*}
Mathematica [A] time = 0.20124, size = 209, normalized size = 1.22 \[ \frac{-\frac{392 \left (75 d^2 e (5989 x-1269)+125 d^3 (423 x+1367)-15 d e^2 (18323 x+17967)+e^3 (54969-53189 x)\right )}{\left (5 x^2+2 x+3\right )^2}+\frac{14 \left (75 d^2 e (181765 x-44399)+125 d^3 (11015 x+34347)-15 d e^2 (647195 x+809167)+e^3 (2639639-3109005 x)\right )}{5 x^2+2 x+3}+164640 e \left (100 d^2-245 d e+47 e^2\right ) \log \left (5 x^2+2 x+3\right )+15 \sqrt{14} \left (-855175 d^2 e+353125 d^3+74085 d e^2+556349 e^3\right ) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )+548800 e^2 x (60 d-49 e)+5488000 e^3 x^2}{343000000} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.059, size = 267, normalized size = 1.6 \begin{align*}{\frac{2\,{e}^{3}{x}^{2}}{125}}+{\frac{12\,xd{e}^{2}}{125}}-{\frac{49\,{e}^{3}x}{625}}+{\frac{1}{25\, \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{2}} \left ( \left ({\frac{11015\,{d}^{3}}{1568}}+{\frac{109059\,{d}^{2}e}{1568}}-{\frac{388317\,d{e}^{2}}{7840}}-{\frac{621801\,{e}^{3}}{39200}} \right ){x}^{3}+ \left ({\frac{38753\,{d}^{3}}{1568}}+{\frac{84921\,{d}^{2}e}{7840}}-{\frac{640827\,d{e}^{2}}{7840}}+{\frac{1396037\,{e}^{3}}{196000}} \right ){x}^{2}+ \left ({\frac{17979\,{d}^{3}}{1568}}+{\frac{173283\,{d}^{2}e}{7840}}-{\frac{73125\,d{e}^{2}}{1568}}-{\frac{511689\,{e}^{3}}{196000}} \right ) x+{\frac{12953\,{d}^{3}}{1568}}-{\frac{58599\,{d}^{2}e}{7840}}-{\frac{230931\,d{e}^{2}}{7840}}+{\frac{1275957\,{e}^{3}}{196000}} \right ) }+{\frac{6\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){d}^{2}e}{125}}-{\frac{147\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) d{e}^{2}}{1250}}+{\frac{141\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ){e}^{3}}{6250}}+{\frac{339\,\sqrt{14}{d}^{3}}{21952}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }-{\frac{102621\,\sqrt{14}{d}^{2}e}{2744000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{44451\,\sqrt{14}d{e}^{2}}{13720000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{1669047\,\sqrt{14}{e}^{3}}{68600000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.55006, size = 300, normalized size = 1.75 \begin{align*} \frac{2}{125} \, e^{3} x^{2} + \frac{3}{68600000} \, \sqrt{14}{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{625} \,{\left (60 \, d e^{2} - 49 \, e^{3}\right )} x + \frac{3}{6250} \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{5 \,{\left (275375 \, d^{3} + 2726475 \, d^{2} e - 1941585 \, d e^{2} - 621801 \, e^{3}\right )} x^{3} + 1619125 \, d^{3} - 1464975 \, d^{2} e - 5773275 \, d e^{2} + 1275957 \, e^{3} +{\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 16020675 \, d e^{2} + 1396037 \, e^{3}\right )} x^{2} + 3 \,{\left (749125 \, d^{3} + 1444025 \, d^{2} e - 3046875 \, d e^{2} - 170563 \, e^{3}\right )} x}{4900000 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.25751, size = 1339, normalized size = 7.83 \begin{align*} \frac{27440000 \, e^{3} x^{6} + 2744000 \,{\left (60 \, d e^{2} - 41 \, e^{3}\right )} x^{5} + 8780800 \,{\left (15 \, d e^{2} - 8 \, e^{3}\right )} x^{4} + 70 \,{\left (275375 \, d^{3} + 2726475 \, d^{2} e + 1257135 \, d e^{2} - 3045929 \, e^{3}\right )} x^{3} + 22667750 \, d^{3} - 20509650 \, d^{2} e - 80825850 \, d e^{2} + 17863398 \, e^{3} + 14 \,{\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 10375875 \, d e^{2} - 2508283 \, e^{3}\right )} x^{2} + 3 \, \sqrt{14}{\left (25 \,{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{4} + 20 \,{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{3} + 3178125 \, d^{3} - 7696575 \, d^{2} e + 666765 \, d e^{2} + 5007141 \, e^{3} + 34 \,{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x^{2} + 12 \,{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} x\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 42 \,{\left (749125 \, d^{3} + 1444025 \, d^{2} e - 1635675 \, d e^{2} - 1323043 \, e^{3}\right )} x + 32928 \,{\left (25 \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{4} + 20 \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{3} + 900 \, d^{2} e - 2205 \, d e^{2} + 423 \, e^{3} + 34 \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x^{2} + 12 \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} x\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right )}{68600000 \,{\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 5.28771, size = 469, normalized size = 2.74 \begin{align*} \frac{2 e^{3} x^{2}}{125} + x \left (\frac{12 d e^{2}}{125} - \frac{49 e^{3}}{625}\right ) + \left (\frac{3 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{6250} - \frac{3 \sqrt{14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{137200000}\right ) \log{\left (x + \frac{211875 d^{3} - 1830225 d^{2} e + 3271395 d e^{2} - 285237 e^{3} + \frac{65856 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{5} - \frac{3 \sqrt{14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{5}}{1059375 d^{3} - 2565525 d^{2} e + 222255 d e^{2} + 1669047 e^{3}} \right )} + \left (\frac{3 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{6250} + \frac{3 \sqrt{14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{137200000}\right ) \log{\left (x + \frac{211875 d^{3} - 1830225 d^{2} e + 3271395 d e^{2} - 285237 e^{3} + \frac{65856 e \left (100 d^{2} - 245 d e + 47 e^{2}\right )}{5} + \frac{3 \sqrt{14} i \left (353125 d^{3} - 855175 d^{2} e + 74085 d e^{2} + 556349 e^{3}\right )}{5}}{1059375 d^{3} - 2565525 d^{2} e + 222255 d e^{2} + 1669047 e^{3}} \right )} + \frac{1619125 d^{3} - 1464975 d^{2} e - 5773275 d e^{2} + 1275957 e^{3} + x^{3} \left (1376875 d^{3} + 13632375 d^{2} e - 9707925 d e^{2} - 3109005 e^{3}\right ) + x^{2} \left (4844125 d^{3} + 2123025 d^{2} e - 16020675 d e^{2} + 1396037 e^{3}\right ) + x \left (2247375 d^{3} + 4332075 d^{2} e - 9140625 d e^{2} - 511689 e^{3}\right )}{122500000 x^{4} + 98000000 x^{3} + 166600000 x^{2} + 58800000 x + 44100000} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14852, size = 271, normalized size = 1.58 \begin{align*} \frac{2}{125} \, x^{2} e^{3} + \frac{12}{125} \, d x e^{2} + \frac{3}{68600000} \, \sqrt{14}{\left (353125 \, d^{3} - 855175 \, d^{2} e + 74085 \, d e^{2} + 556349 \, e^{3}\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{49}{625} \, x e^{3} + \frac{3}{6250} \,{\left (100 \, d^{2} e - 245 \, d e^{2} + 47 \, e^{3}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) + \frac{5 \,{\left (275375 \, d^{3} + 2726475 \, d^{2} e - 1941585 \, d e^{2} - 621801 \, e^{3}\right )} x^{3} + 1619125 \, d^{3} +{\left (4844125 \, d^{3} + 2123025 \, d^{2} e - 16020675 \, d e^{2} + 1396037 \, e^{3}\right )} x^{2} - 1464975 \, d^{2} e + 3 \,{\left (749125 \, d^{3} + 1444025 \, d^{2} e - 3046875 \, d e^{2} - 170563 \, e^{3}\right )} x - 5773275 \, d e^{2} + 1275957 \, e^{3}}{4900000 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]