Optimal. Leaf size=156 \[ \frac {\left (2025 d^2+4580 d e-881 e^2\right ) x}{3125}-\frac {\left (825 d^2-810 d e-458 e^2\right ) x^2}{1250}+\frac {1}{375} \left (100 d^2-330 d e+81 e^2\right ) x^3+\frac {1}{100} (40 d-33 e) e x^4+\frac {4 e^2 x^5}{25}-\frac {\left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{15625 \sqrt {14}}+\frac {\left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (3+2 x+5 x^2\right )}{15625} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 156, 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 = {1642, 648, 632,
210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {5 x+1}{\sqrt {14}}\right ) \left (10575 d^2+59890 d e-18323 e^2\right )}{15625 \sqrt {14}}+\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 {1}{100} e x^4 (40 d-33 e)+\frac {4 e^2 x^5}{25} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1642
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 ) \text {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 = 0.06, size = 130, normalized size = 0.83 \begin {gather*} \frac {35 x \left (50 d^2 \left (486-495 x+200 x^2\right )+60 d e \left (916+405 x-550 x^2+250 x^3\right )+3 e^2 \left (-3524+4580 x+2700 x^2-4125 x^3+2000 x^4\right )\right )-6 \sqrt {14} \left (10575 d^2+59890 d e-18323 e^2\right ) \tan ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )+84 \left (5725 d^2-4405 d e-2554 e^2\right ) \log \left (3+2 x+5 x^2\right )}{1312500} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 147, normalized size = 0.94
method | result | size |
default | \(\frac {4 e^{2} x^{5}}{25}+\frac {2 x^{4} d e}{5}-\frac {33 x^{4} e^{2}}{100}+\frac {4 x^{3} d^{2}}{15}-\frac {22 x^{3} d e}{25}+\frac {27 e^{2} x^{3}}{125}-\frac {33 d^{2} x^{2}}{50}+\frac {81 d e \,x^{2}}{125}+\frac {229 e^{2} x^{2}}{625}+\frac {81 d^{2} x}{125}+\frac {916 d e x}{625}-\frac {881 e^{2} x}{3125}+\frac {\left (11450 d^{2}-8810 d e -5108 e^{2}\right ) \ln \left (5 x^{2}+2 x +3\right )}{31250}+\frac {\left (-2115 d^{2}-11978 d e +\frac {18323}{5} e^{2}\right ) \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{43750}\) | \(147\) |
risch | \(-\frac {33 x^{4} e^{2}}{100}+\frac {4 x^{3} d^{2}}{15}-\frac {33 d^{2} x^{2}}{50}-\frac {22 x^{3} d e}{25}+\frac {2 x^{4} d e}{5}+\frac {229 d^{2} \ln \left (350 x^{2}+140 x +210\right )}{625}-\frac {2554 e^{2} \ln \left (350 x^{2}+140 x +210\right )}{15625}+\frac {916 d e x}{625}-\frac {5989 \sqrt {14}\, d e \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{21875}+\frac {27 e^{2} x^{3}}{125}+\frac {81 d^{2} x}{125}+\frac {229 e^{2} x^{2}}{625}+\frac {81 d e \,x^{2}}{125}-\frac {881 d e \ln \left (350 x^{2}+140 x +210\right )}{3125}-\frac {423 \sqrt {14}\, d^{2} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{8750}+\frac {18323 \sqrt {14}\, e^{2} \arctan \left (\frac {5 \sqrt {14}\, x}{14}+\frac {\sqrt {14}}{14}\right )}{218750}-\frac {881 e^{2} x}{3125}+\frac {4 e^{2} x^{5}}{25}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.56, size = 140, normalized size = 0.90 \begin {gather*} \frac {4}{25} \, x^{5} e^{2} + \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 137, normalized size = 0.88 \begin {gather*} \frac {4}{15} \, d^{2} x^{3} - \frac {33}{50} \, d^{2} x^{2} + \frac {81}{125} \, d^{2} x - \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}{12500} \, {\left (2000 \, x^{5} - 4125 \, x^{4} + 2700 \, x^{3} + 4580 \, x^{2} - 3524 \, x\right )} e^{2} + \frac {1}{625} \, {\left (250 \, d x^{4} - 550 \, d x^{3} + 405 \, d x^{2} + 916 \, d x\right )} e + \frac {1}{15625} \, {\left (5725 \, d^{2} - 4405 \, d e - 2554 \, e^{2}\right )} \log \left (5 \, x^{2} + 2 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.52, size = 303, normalized size = 1.94 \begin {gather*} \frac {4 e^{2} x^{5}}{25} + x^{4} \cdot \left (\frac {2 d e}{5} - \frac {33 e^{2}}{100}\right ) + x^{3} \cdot \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.30, size = 145, normalized size = 0.93 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.10, size = 223, normalized size = 1.43 \begin {gather*} x\,\left (\frac {4\,d\,e}{5}+\frac {52\,e\,\left (8\,d-5\,e\right )}{625}+\frac {81\,d^2}{125}+\frac {419\,e^2}{3125}\right )-\ln \left (5\,x^2+2\,x+3\right )\,\left (-\frac {229\,d^2}{625}+\frac {881\,d\,e}{3125}+\frac {2554\,e^2}{15625}\right )+x^4\,\left (\frac {e\,\left (8\,d-5\,e\right )}{20}-\frac {2\,e^2}{25}\right )-x^3\,\left (\frac {2\,d\,e}{3}+\frac {2\,e\,\left (8\,d-5\,e\right )}{75}-\frac {4\,d^2}{15}-\frac {31\,e^2}{375}\right )+x^2\,\left (d\,e-\frac {11\,e\,\left (8\,d-5\,e\right )}{250}-\frac {33\,d^2}{50}+\frac {183\,e^2}{1250}\right )+\frac {4\,e^2\,x^5}{25}-\frac {\sqrt {14}\,\mathrm {atan}\left (\frac {\frac {\sqrt {14}\,\left (10575\,d^2+59890\,d\,e-18323\,e^2\right )}{218750}+\frac {\sqrt {14}\,x\,\left (10575\,d^2+59890\,d\,e-18323\,e^2\right )}{43750}}{\frac {423\,d^2}{625}+\frac {11978\,d\,e}{3125}-\frac {18323\,e^2}{15625}}\right )\,\left (10575\,d^2+59890\,d\,e-18323\,e^2\right )}{218750} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________