Integrand size = 35, antiderivative size = 269 \[ \int \frac {x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\frac {1}{14} \left (7-5 i \sqrt {7}\right ) x+\frac {1}{14} \left (7+5 i \sqrt {7}\right ) x+\frac {1}{28} \left (7-5 i \sqrt {7}\right ) x^2+\frac {1}{28} \left (7+5 i \sqrt {7}\right ) x^2-\frac {\left (53 i+\sqrt {7}\right ) \arctan \left (\frac {1-i \sqrt {7}+8 x}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35+i \sqrt {7}\right )}}+\frac {\left (53 i-\sqrt {7}\right ) \arctan \left (\frac {1+i \sqrt {7}+8 x}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{2 \sqrt {14 \left (35-i \sqrt {7}\right )}}-\frac {1}{56} \left (35+9 i \sqrt {7}\right ) \log \left (4+\left (1-i \sqrt {7}\right ) x+4 x^2\right )-\frac {1}{56} \left (35-9 i \sqrt {7}\right ) \log \left (4+\left (1+i \sqrt {7}\right ) x+4 x^2\right ) \]
1/14*x*(7-5*I*7^(1/2))+1/28*x^2*(7-5*I*7^(1/2))+1/14*x*(7+5*I*7^(1/2))+1/2 8*x^2*(7+5*I*7^(1/2))-1/56*ln(4+4*x^2+x*(1+I*7^(1/2)))*(35-9*I*7^(1/2))-1/ 56*ln(4+4*x^2+x*(1-I*7^(1/2)))*(35+9*I*7^(1/2))+1/2*arctan((1+8*x+I*7^(1/2 ))/(70-2*I*7^(1/2))^(1/2))*(53*I-7^(1/2))/(490-14*I*7^(1/2))^(1/2)-1/2*arc tan((1+8*x-I*7^(1/2))/(70+2*I*7^(1/2))^(1/2))*(53*I+7^(1/2))/(490+14*I*7^( 1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.38 \[ \int \frac {x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=x+\frac {x^2}{2}-\text {RootSum}\left [2+\text {$\#$1}+5 \text {$\#$1}^2+\text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {2 \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2+5 \log (x-\text {$\#$1}) \text {$\#$1}^3}{1+10 \text {$\#$1}+3 \text {$\#$1}^2+8 \text {$\#$1}^3}\&\right ] \]
x + x^2/2 - RootSum[2 + #1 + 5*#1^2 + #1^3 + 2*#1^4 & , (2*Log[x - #1] + 3 *Log[x - #1]*#1 + Log[x - #1]*#1^2 + 5*Log[x - #1]*#1^3)/(1 + 10*#1 + 3*#1 ^2 + 8*#1^3) & ]
Time = 0.58 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2492, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2 \left (2 x^3+3 x^2+x+5\right )}{2 x^4+x^3+5 x^2+x+2} \, dx\) |
\(\Big \downarrow \) 2492 |
\(\displaystyle \frac {1}{2} \int \left (2 x-\frac {2 \left (\left (9+5 i \sqrt {7}\right ) x+2 \left (5+i \sqrt {7}\right )\right )}{\sqrt {7} \left (4 i x^2+\left (i-\sqrt {7}\right ) x+4 i\right )}-\frac {2 \left (\left (35 i-9 \sqrt {7}\right ) x+2 \left (7 i-5 \sqrt {7}\right )\right )}{7 \left (4 i x^2+\left (i+\sqrt {7}\right ) x+4 i\right )}+2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (53+i \sqrt {7}\right ) \text {arctanh}\left (\frac {8 i x-\sqrt {7}+i}{\sqrt {2 \left (35-i \sqrt {7}\right )}}\right )}{\sqrt {14 \left (35-i \sqrt {7}\right )}}-\frac {\left (53-i \sqrt {7}\right ) \text {arctanh}\left (\frac {8 i x+\sqrt {7}+i}{\sqrt {2 \left (35+i \sqrt {7}\right )}}\right )}{\sqrt {14 \left (35+i \sqrt {7}\right )}}+x^2-\frac {1}{28} \left (35-9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (-\sqrt {7}+i\right ) x+4 i\right )-\frac {1}{28} \left (35+9 i \sqrt {7}\right ) \log \left (4 i x^2+\left (\sqrt {7}+i\right ) x+4 i\right )+2 x\right )\) |
(2*x + x^2 + ((53 + I*Sqrt[7])*ArcTanh[(I - Sqrt[7] + (8*I)*x)/Sqrt[2*(35 - I*Sqrt[7])]])/Sqrt[14*(35 - I*Sqrt[7])] - ((53 - I*Sqrt[7])*ArcTanh[(I + Sqrt[7] + (8*I)*x)/Sqrt[2*(35 + I*Sqrt[7])]])/Sqrt[14*(35 + I*Sqrt[7])] - ((35 - (9*I)*Sqrt[7])*Log[4*I + (I - Sqrt[7])*x + (4*I)*x^2])/28 - ((35 + (9*I)*Sqrt[7])*Log[4*I + (I + Sqrt[7])*x + (4*I)*x^2])/28)/2
3.3.51.3.1 Defintions of rubi rules used
Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) ^(p_), x_Symbol] :> Simp[e^p Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( (b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.25
method | result | size |
default | \(\frac {x^{2}}{2}+x +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )}{\sum }\frac {\left (-5 \textit {\_R}^{3}-\textit {\_R}^{2}-3 \textit {\_R} -2\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}+3 \textit {\_R}^{2}+10 \textit {\_R} +1}\right )\) | \(67\) |
risch | \(\frac {x^{2}}{2}+x +\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}+\textit {\_Z}^{3}+5 \textit {\_Z}^{2}+\textit {\_Z} +2\right )}{\sum }\frac {\left (-5 \textit {\_R}^{3}-\textit {\_R}^{2}-3 \textit {\_R} -2\right ) \ln \left (x -\textit {\_R} \right )}{8 \textit {\_R}^{3}+3 \textit {\_R}^{2}+10 \textit {\_R} +1}\right )\) | \(67\) |
1/2*x^2+x+sum((-5*_R^3-_R^2-3*_R-2)/(8*_R^3+3*_R^2+10*_R+1)*ln(x-_R),_R=Ro otOf(2*_Z^4+_Z^3+5*_Z^2+_Z+2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1145 vs. \(2 (171) = 342\).
Time = 1.02 (sec) , antiderivative size = 1145, normalized size of antiderivative = 4.26 \[ \int \frac {x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\text {Too large to display} \]
1/2*x^2 - 1/56*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35)*log (49/4*(135*I*sqrt(7) + 420*sqrt(-37/392*I*sqrt(7) + 79/56) - 1459)*(9/56*I *sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 10290*(-9/56*I*sq rt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^3 - 25725*(-9/56*I*sqrt (7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 + 3/64*(3920*(-9/56*I*s qrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1575*I*sqrt(7) - 4 900*sqrt(-37/392*I*sqrt(7) + 79/56) + 5587)*(-9*I*sqrt(7) + 28*sqrt(37/392 *I*sqrt(7) + 79/56) + 35) + 8384*x + 6615/2*I*sqrt(7) + 10290*sqrt(-37/392 *I*sqrt(7) + 79/56) + 13373/2) + 1/8*(2*sqrt(-12*(9/56*I*sqrt(7) - 1/2*sqr t(37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 12*(-9/56*I*sqrt(7) - 1/2*sqrt(-37/ 392*I*sqrt(7) + 79/56) - 5/8)^2 - 1/392*(9*I*sqrt(7) + 28*sqrt(-37/392*I*s qrt(7) + 79/56) - 105)*(-9*I*sqrt(7) + 28*sqrt(37/392*I*sqrt(7) + 79/56) + 35) + 45/14*I*sqrt(7) + 10*sqrt(-37/392*I*sqrt(7) + 79/56) + 11/2) + 2*sq rt(37/392*I*sqrt(7) + 79/56) + 2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5)*log( -49/4*(135*I*sqrt(7) + 420*sqrt(-37/392*I*sqrt(7) + 79/56) - 1459)*(9/56*I *sqrt(7) - 1/2*sqrt(37/392*I*sqrt(7) + 79/56) - 5/8)^2 + 24304*(-9/56*I*sq rt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 3/64*(3920*(-9/56*I *sqrt(7) - 1/2*sqrt(-37/392*I*sqrt(7) + 79/56) - 5/8)^2 - 1575*I*sqrt(7) - 4900*sqrt(-37/392*I*sqrt(7) + 79/56) + 5587)*(-9*I*sqrt(7) + 28*sqrt(37/3 92*I*sqrt(7) + 79/56) + 35) + 7/64*sqrt(-12*(9/56*I*sqrt(7) - 1/2*sqrt(...
Leaf count of result is larger than twice the leaf count of optimal. 3662 vs. \(2 (219) = 438\).
Time = 1.59 (sec) , antiderivative size = 3662, normalized size of antiderivative = 13.61 \[ \int \frac {x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\text {Too large to display} \]
x**2/2 + x + (-5/8 + sqrt(79/448 + sqrt(77)/49))*log(x**2 + x*(-1459*sqrt( 14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/536576 - 15 *sqrt(77)*sqrt(553 + 64*sqrt(77))/2096 - 10391*sqrt(553 + 64*sqrt(77))/268 288 + 1459*sqrt(77)/8384 + 522933/268288 + 45*sqrt(14)*sqrt(553 + 64*sqrt( 77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/536576) - 510895297*sqrt(14)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(7 7))/71978450944 - 6009493*sqrt(22)*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 219 75 + 7648*sqrt(77))/1124663296 - 38714551*sqrt(77)*sqrt(553 + 64*sqrt(77)) /2249326592 - 4417610843*sqrt(553 + 64*sqrt(77))/35989225472 + 153195*sqrt (22)*sqrt(553 + 64*sqrt(77))*sqrt(-333*sqrt(553 + 64*sqrt(77)) + 21975 + 7 648*sqrt(77))/2249326592 + 8313499*sqrt(14)*sqrt(553 + 64*sqrt(77))*sqrt(- 333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/71978450944 + 2908324 44193/35989225472 + 2303470247*sqrt(77)/2249326592) + (-5/8 - sqrt(79/448 + sqrt(77)/49))*log(x**2 + x*(-45*sqrt(14)*sqrt(553 + 64*sqrt(77))*sqrt(33 3*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/536576 - 1459*sqrt(14)* sqrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/536576 + 10391*s qrt(553 + 64*sqrt(77))/268288 + 1459*sqrt(77)/8384 + 522933/268288 + 15*sq rt(77)*sqrt(553 + 64*sqrt(77))/2096) - 510895297*sqrt(14)*sqrt(333*sqrt(55 3 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/71978450944 - 6009493*sqrt(22)*s qrt(333*sqrt(553 + 64*sqrt(77)) + 21975 + 7648*sqrt(77))/1124663296 - 8...
\[ \int \frac {x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\int { \frac {{\left (2 \, x^{3} + 3 \, x^{2} + x + 5\right )} x^{2}}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2} \,d x } \]
\[ \int \frac {x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=\int { \frac {{\left (2 \, x^{3} + 3 \, x^{2} + x + 5\right )} x^{2}}{2 \, x^{4} + x^{3} + 5 \, x^{2} + x + 2} \,d x } \]
Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (5+x+3 x^2+2 x^3\right )}{2+x+5 x^2+x^3+2 x^4} \, dx=x+\frac {x^2}{2}+\left (\sum _{k=1}^4\ln \left (-\frac {179\,\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}{8}-7\,x-\frac {\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )\,x\,459}{8}-\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^2\,x\,665}{8}-\frac {{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^3\,x\,147}{4}-\frac {35\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^2}{32}+\frac {49\,{\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )}^3}{16}-15\right )\,\mathrm {root}\left (z^4+\frac {5\,z^3}{2}+2\,z^2+\frac {32\,z}{49}+\frac {128}{343},z,k\right )\right ) \]
x + x^2/2 + symsum(log((49*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/ 343, z, k)^3)/16 - 7*x - (459*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 1 28/343, z, k)*x)/8 - (665*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/3 43, z, k)^2*x)/8 - (147*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343 , z, k)^3*x)/4 - (35*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z , k)^2)/32 - (179*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k ))/8 - 15)*root(z^4 + (5*z^3)/2 + 2*z^2 + (32*z)/49 + 128/343, z, k), k, 1 , 4)