Optimal. Leaf size=97 \[ -\frac{(423 x+1367) (d+e x)}{3500 \left (5 x^2+2 x+3\right )}-\frac{(205 d-103 e) \log \left (5 x^2+2 x+3\right )}{1250}+\frac{1}{125} x (20 d-41 e)+\frac{(6565 d+21171 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{17500 \sqrt{14}}+\frac{2 e x^2}{25} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.193461, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1644, 1657, 634, 618, 204, 628} \[ -\frac{(423 x+1367) (d+e x)}{3500 \left (5 x^2+2 x+3\right )}-\frac{(205 d-103 e) \log \left (5 x^2+2 x+3\right )}{1250}+\frac{1}{125} x (20 d-41 e)+\frac{(6565 d+21171 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{17500 \sqrt{14}}+\frac{2 e x^2}{25} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1644
Rule 1657
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 )}{\left (3+2 x+5 x^2\right )^2} \, dx &=-\frac{(1367+423 x) (d+e x)}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \frac{\frac{2}{125} (1845 d+1367 e)-\frac{168}{125} (55 d-27 e) x+\frac{56}{25} (20 d-33 e) x^2+\frac{224 e x^3}{5}}{3+2 x+5 x^2} \, dx\\ &=-\frac{(1367+423 x) (d+e x)}{3500 \left (3+2 x+5 x^2\right )}+\frac{1}{56} \int \left (\frac{56}{125} (20 d-41 e)+\frac{224 e x}{25}+\frac{2 (165 d+4811 e-28 (205 d-103 e) x)}{125 \left (3+2 x+5 x^2\right )}\right ) \, dx\\ &=\frac{1}{125} (20 d-41 e) x+\frac{2 e x^2}{25}-\frac{(1367+423 x) (d+e x)}{3500 \left (3+2 x+5 x^2\right )}+\frac{\int \frac{165 d+4811 e-28 (205 d-103 e) x}{3+2 x+5 x^2} \, dx}{3500}\\ &=\frac{1}{125} (20 d-41 e) x+\frac{2 e x^2}{25}-\frac{(1367+423 x) (d+e x)}{3500 \left (3+2 x+5 x^2\right )}+\frac{(-205 d+103 e) \int \frac{2+10 x}{3+2 x+5 x^2} \, dx}{1250}+\frac{(6565 d+21171 e) \int \frac{1}{3+2 x+5 x^2} \, dx}{17500}\\ &=\frac{1}{125} (20 d-41 e) x+\frac{2 e x^2}{25}-\frac{(1367+423 x) (d+e x)}{3500 \left (3+2 x+5 x^2\right )}-\frac{(205 d-103 e) \log \left (3+2 x+5 x^2\right )}{1250}+\frac{(-6565 d-21171 e) \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+10 x\right )}{8750}\\ &=\frac{1}{125} (20 d-41 e) x+\frac{2 e x^2}{25}-\frac{(1367+423 x) (d+e x)}{3500 \left (3+2 x+5 x^2\right )}+\frac{(6565 d+21171 e) \tan ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{17500 \sqrt{14}}-\frac{(205 d-103 e) \log \left (3+2 x+5 x^2\right )}{1250}\\ \end{align*}
Mathematica [A] time = 0.0662719, size = 96, normalized size = 0.99 \[ \frac{-\frac{14 (5 d (423 x+1367)+e (5989 x-1269))}{5 x^2+2 x+3}+196 (103 e-205 d) \log \left (5 x^2+2 x+3\right )+1960 x (20 d-41 e)+\sqrt{14} (6565 d+21171 e) \tan ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )+19600 e x^2}{245000} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 106, normalized size = 1.1 \begin{align*}{\frac{2\,e{x}^{2}}{25}}+{\frac{4\,dx}{25}}-{\frac{41\,ex}{125}}-{\frac{1}{125} \left ( \left ({\frac{423\,d}{140}}+{\frac{5989\,e}{700}} \right ) x+{\frac{1367\,d}{140}}-{\frac{1269\,e}{700}} \right ) \left ({x}^{2}+{\frac{2\,x}{5}}+{\frac{3}{5}} \right ) ^{-1}}-{\frac{41\,\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) d}{250}}+{\frac{103\,e\ln \left ( 5\,{x}^{2}+2\,x+3 \right ) }{1250}}+{\frac{1313\,\sqrt{14}d}{49000}\arctan \left ({\frac{ \left ( 10\,x+2 \right ) \sqrt{14}}{28}} \right ) }+{\frac{21171\,\sqrt{14}e}{245000}\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 = 1.49595, size = 122, normalized size = 1.26 \begin{align*} \frac{2}{25} \, e x^{2} + \frac{1}{245000} \, \sqrt{14}{\left (6565 \, d + 21171 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{1}{125} \,{\left (20 \, d - 41 \, e\right )} x - \frac{1}{1250} \,{\left (205 \, d - 103 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - \frac{{\left (2115 \, d + 5989 \, e\right )} x + 6835 \, d - 1269 \, e}{17500 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.20687, size = 462, normalized size = 4.76 \begin{align*} \frac{98000 \, e x^{4} + 9800 \,{\left (20 \, d - 37 \, e\right )} x^{3} + 7840 \,{\left (10 \, d - 13 \, e\right )} x^{2} + \sqrt{14}{\left (5 \,{\left (6565 \, d + 21171 \, e\right )} x^{2} + 2 \,{\left (6565 \, d + 21171 \, e\right )} x + 19695 \, d + 63513 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + 14 \,{\left (6285 \, d - 23209 \, e\right )} x - 196 \,{\left (5 \,{\left (205 \, d - 103 \, e\right )} x^{2} + 2 \,{\left (205 \, d - 103 \, e\right )} x + 615 \, d - 309 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - 95690 \, d + 17766 \, e}{245000 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.11261, size = 163, normalized size = 1.68 \begin{align*} \frac{2 e x^{2}}{25} + x \left (\frac{4 d}{25} - \frac{41 e}{125}\right ) + \left (- \frac{41 d}{250} + \frac{103 e}{1250} - \frac{\sqrt{14} i \left (6565 d + 21171 e\right )}{490000}\right ) \log{\left (x + \frac{1313 d + \frac{21171 e}{5} - \frac{\sqrt{14} i \left (6565 d + 21171 e\right )}{5}}{6565 d + 21171 e} \right )} + \left (- \frac{41 d}{250} + \frac{103 e}{1250} + \frac{\sqrt{14} i \left (6565 d + 21171 e\right )}{490000}\right ) \log{\left (x + \frac{1313 d + \frac{21171 e}{5} + \frac{\sqrt{14} i \left (6565 d + 21171 e\right )}{5}}{6565 d + 21171 e} \right )} - \frac{6835 d - 1269 e + x \left (2115 d + 5989 e\right )}{87500 x^{2} + 35000 x + 52500} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14407, size = 127, normalized size = 1.31 \begin{align*} \frac{2}{25} \, x^{2} e + \frac{1}{245000} \, \sqrt{14}{\left (6565 \, d + 21171 \, e\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{4}{25} \, d x - \frac{41}{125} \, x e - \frac{1}{1250} \,{\left (205 \, d - 103 \, e\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) - \frac{{\left (2115 \, d + 5989 \, e\right )} x + 6835 \, d - 1269 \, e}{17500 \,{\left (5 \, x^{2} + 2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]