Integrand size = 21, antiderivative size = 114 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=-\frac {c \arctan \left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \arctan \left (1+\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}-\frac {c \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \]
1/12*c*arctan(-1+6^(1/4)*x)*6^(1/4)+1/12*c*arctan(1+6^(1/4)*x)*6^(1/4)+1/1 2*d*ln(3*x^4+2)+1/24*c*ln(-6^(3/4)*x+3*x^2+6^(1/2))*6^(1/4)-1/24*c*ln(6^(3 /4)*x+3*x^2+6^(1/2))*6^(1/4)
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.95 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \left (-2 \sqrt [4]{6} c \arctan \left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} c \arctan \left (1+\sqrt [4]{6} x\right )+\sqrt [4]{6} c \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )-\sqrt [4]{6} c \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+2 d \log \left (2+3 x^4\right )\right ) \]
(-2*6^(1/4)*c*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*c*ArcTan[1 + 6^(1/4)*x] + 6^(1/4)*c*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - 6^(1/4)*c*Log[2 + 2*6^(1/4) *x + Sqrt[6]*x^2] + 2*d*Log[2 + 3*x^4])/24
Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2027, 2370, 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.
\(\displaystyle \int \frac {c x^2+d x^3}{3 x^4+2} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x^2 (c+d x)}{3 x^4+2}dx\) |
\(\Big \downarrow \) 2370 |
\(\displaystyle \int \left (\frac {c x^2}{3 x^4+2}+\frac {d x^3}{3 x^4+2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {c \arctan \left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac {c \arctan \left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac {c \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{4\ 6^{3/4}}-\frac {c \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{4\ 6^{3/4}}+\frac {1}{12} d \log \left (3 x^4+2\right )\) |
-1/2*(c*ArcTan[1 - 6^(1/4)*x])/6^(3/4) + (c*ArcTan[1 + 6^(1/4)*x])/(2*6^(3 /4)) + (c*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(4*6^(3/4)) - (c*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(4*6^(3/4)) + (d*Log[2 + 3*x^4])/12
3.2.63.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[ {v = Sum[(c*x)^(m + ii)*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2) )/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{ a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.49 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.31
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{3} d +\textit {\_R}^{2} c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\) | \(35\) |
default | \(\frac {c \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{144}+\frac {d \ln \left (3 x^{4}+2\right )}{12}\) | \(106\) |
meijerg | \(\frac {d \ln \left (\frac {3 x^{4}}{2}+1\right )}{12}+\frac {54^{\frac {3}{4}} c \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{216}\) | \(183\) |
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (83) = 166\).
Time = 0.28 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.10 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \, {\left (\sqrt {2} \sqrt {\sqrt {6} \sqrt {-c^{4}}} + 2 \, d\right )} \log \left (6 \, c^{3} x + \sqrt {6} \sqrt {2} \sqrt {-c^{4}} \sqrt {\sqrt {6} \sqrt {-c^{4}}}\right ) - \frac {1}{24} \, {\left (\sqrt {2} \sqrt {\sqrt {6} \sqrt {-c^{4}}} - 2 \, d\right )} \log \left (6 \, c^{3} x - \sqrt {6} \sqrt {2} \sqrt {-c^{4}} \sqrt {\sqrt {6} \sqrt {-c^{4}}}\right ) - \frac {1}{24} \, {\left (\sqrt {2} \sqrt {-\sqrt {6} \sqrt {-c^{4}}} - 2 \, d\right )} \log \left (6 \, c^{3} x + \sqrt {6} \sqrt {2} \sqrt {-c^{4}} \sqrt {-\sqrt {6} \sqrt {-c^{4}}}\right ) + \frac {1}{24} \, {\left (\sqrt {2} \sqrt {-\sqrt {6} \sqrt {-c^{4}}} + 2 \, d\right )} \log \left (6 \, c^{3} x - \sqrt {6} \sqrt {2} \sqrt {-c^{4}} \sqrt {-\sqrt {6} \sqrt {-c^{4}}}\right ) \]
1/24*(sqrt(2)*sqrt(sqrt(6)*sqrt(-c^4)) + 2*d)*log(6*c^3*x + sqrt(6)*sqrt(2 )*sqrt(-c^4)*sqrt(sqrt(6)*sqrt(-c^4))) - 1/24*(sqrt(2)*sqrt(sqrt(6)*sqrt(- c^4)) - 2*d)*log(6*c^3*x - sqrt(6)*sqrt(2)*sqrt(-c^4)*sqrt(sqrt(6)*sqrt(-c ^4))) - 1/24*(sqrt(2)*sqrt(-sqrt(6)*sqrt(-c^4)) - 2*d)*log(6*c^3*x + sqrt( 6)*sqrt(2)*sqrt(-c^4)*sqrt(-sqrt(6)*sqrt(-c^4))) + 1/24*(sqrt(2)*sqrt(-sqr t(6)*sqrt(-c^4)) + 2*d)*log(6*c^3*x - sqrt(6)*sqrt(2)*sqrt(-c^4)*sqrt(-sqr t(6)*sqrt(-c^4)))
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.61 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\operatorname {RootSum} {\left (41472 t^{4} - 13824 t^{3} d + 1728 t^{2} d^{2} - 96 t d^{3} + 3 c^{4} + 2 d^{4}, \left ( t \mapsto t \log {\left (x + \frac {3456 t^{3} - 864 t^{2} d + 72 t d^{2} - 2 d^{3}}{3 c^{3}} \right )} \right )\right )} \]
RootSum(41472*_t**4 - 13824*_t**3*d + 1728*_t**2*d**2 - 96*_t*d**3 + 3*c** 4 + 2*d**4, Lambda(_t, _t*log(x + (3456*_t**3 - 864*_t**2*d + 72*_t*d**2 - 2*d**3)/(3*c**3))))
Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.33 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{72} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} d - \sqrt {3} c\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{72} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (3^{\frac {1}{4}} 2^{\frac {3}{4}} d + \sqrt {3} c\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{12} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} c \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{12} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} c \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) \]
1/72*3^(3/4)*2^(1/4)*(3^(1/4)*2^(3/4)*d - sqrt(3)*c)*log(sqrt(3)*x^2 + 3^( 1/4)*2^(3/4)*x + sqrt(2)) + 1/72*3^(3/4)*2^(1/4)*(3^(1/4)*2^(3/4)*d + sqrt (3)*c)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/12*3^(1/4)*2^(1/ 4)*c*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/12*3^ (1/4)*2^(1/4)*c*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4)) )
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{12} \cdot 6^{\frac {1}{4}} c \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{12} \cdot 6^{\frac {1}{4}} c \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{24} \, {\left (6^{\frac {1}{4}} c - 2 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) + \frac {1}{24} \, {\left (6^{\frac {1}{4}} c + 2 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
1/12*6^(1/4)*c*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/12*6^(1/4)*c*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4 ))) - 1/24*(6^(1/4)*c - 2*d)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) + 1/24*(6^(1/4)*c + 2*d)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))
Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03 \[ \int \frac {c x^2+d x^3}{2+3 x^4} \, dx=\ln \left (x-\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}+\frac {6^{1/4}\,\sqrt {-\frac {1}{2}{}\mathrm {i}}\,c}{12}\right )+\ln \left (x+\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}-\frac {6^{1/4}\,\sqrt {-\frac {1}{2}{}\mathrm {i}}\,c}{12}\right )+\ln \left (x-\frac {{\left (-1\right )}^{3/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}-\frac {6^{1/4}\,\sqrt {\frac {1}{2}{}\mathrm {i}}\,c}{12}\right )+\ln \left (x+\frac {{\left (-1\right )}^{3/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}+\frac {6^{1/4}\,\sqrt {\frac {1}{2}{}\mathrm {i}}\,c}{12}\right ) \]
log(x - ((-1)^(1/4)*2^(1/4)*3^(3/4))/3)*(d/12 + (6^(1/4)*(-1i/2)^(1/2)*c)/ 12) + log(x + ((-1)^(1/4)*2^(1/4)*3^(3/4))/3)*(d/12 - (6^(1/4)*(-1i/2)^(1/ 2)*c)/12) + log(x - ((-1)^(3/4)*2^(1/4)*3^(3/4))/3)*(d/12 - (6^(1/4)*(1i/2 )^(1/2)*c)/12) + log(x + ((-1)^(3/4)*2^(1/4)*3^(3/4))/3)*(d/12 + (6^(1/4)* (1i/2)^(1/2)*c)/12)