Integrand size = 20, antiderivative size = 81 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\frac {x^2}{2 c}-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^2+c x^4\right )}{4 c^2} \]
1/2*x^2/c-1/4*b*ln(c*x^4+b*x^2+a)/c^2-1/2*(-2*a*c+b^2)*arctanh((2*c*x^2+b) /(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2)^(1/2)
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\frac {2 c x^2+\frac {2 \left (b^2-2 a c\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-b \log \left (a+b x^2+c x^4\right )}{4 c^2} \]
(2*c*x^2 + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt [-b^2 + 4*a*c] - b*Log[a + b*x^2 + c*x^4])/(4*c^2)
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {9, 1434, 1143, 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.
| \(\displaystyle \int \frac {x^6}{a x+b x^3+c x^5} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {x^5}{a+b x^2+c x^4}dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{c x^4+b x^2+a}dx^2\) |
| \(\Big \downarrow \) 1143 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {1}{c}-\frac {b x^2+a}{c \left (c x^4+b x^2+a\right )}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^2+c x^4\right )}{2 c^2}+\frac {x^2}{c}\right )\) |
(x^2/c - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqr t[b^2 - 4*a*c]) - (b*Log[a + b*x^2 + c*x^4])/(2*c^2))/2
3.1.80.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02
| method | result | size |
| default | \(\frac {x^{2}}{2 c}+\frac {-\frac {b \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (-a +\frac {b^{2}}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c}\) | \(83\) |
| risch | \(\frac {x^{2}}{2 c}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{4 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{4 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{4 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{4 c^{2} \left (4 a c -b^{2}\right )}\) | \(681\) |
1/2*x^2/c+1/2/c*(-1/2*b/c*ln(c*x^4+b*x^2+a)+2*(-a+1/2/c*b^2)/(4*a*c-b^2)^( 1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.14 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - {\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - 2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \]
[1/4*(2*(b^2*c - 4*a*c^2)*x^2 - (b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*log((2*c^2 *x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) - (b^3 - 4*a*b*c)*log(c*x^4 + b*x^2 + a))/(b^2*c^2 - 4*a*c^3) , 1/4*(2*(b^2*c - 4*a*c^2)*x^2 - 2*(b^2 - 2*a*c)*sqrt(-b^2 + 4*a*c)*arctan (-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (b^3 - 4*a*b*c)*log(c* x^4 + b*x^2 + a))/(b^2*c^2 - 4*a*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (71) = 142\).
Time = 1.11 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.90 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\left (- \frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- a b - 8 a c^{2} \left (- \frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- a b - 8 a c^{2} \left (- \frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac {x^{2}}{2 c} \]
(-b/(4*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2))) *log(x**2 + (-a*b - 8*a*c**2*(-b/(4*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b **2)/(4*c**2*(4*a*c - b**2))) + 2*b**2*c*(-b/(4*c**2) - sqrt(-4*a*c + b**2 )*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + (-b/(4*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2)))*log(x**2 + ( -a*b - 8*a*c**2*(-b/(4*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2* (4*a*c - b**2))) + 2*b**2*c*(-b/(4*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b* *2)/(4*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + x**2/(2*c)
\[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\int { \frac {x^{6}}{c x^{5} + b x^{3} + a x} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\frac {x^{2}}{2 \, c} - \frac {b \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \]
1/2*x^2/c - 1/4*b*log(c*x^4 + b*x^2 + a)/c^2 + 1/2*(b^2 - 2*a*c)*arctan((2 *c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2)
Time = 8.75 (sec) , antiderivative size = 655, normalized size of antiderivative = 8.09 \[ \int \frac {x^6}{a x+b x^3+c x^5} \, dx=\frac {x^2}{2\,c}+\frac {\ln \left (c\,x^4+b\,x^2+a\right )\,\left (2\,b^3-8\,a\,b\,c\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}+\frac {\mathrm {atan}\left (\frac {2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (\frac {\frac {\left (8\,a\,b+\frac {8\,a\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )\,\left (2\,a\,c-b^2\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}+\frac {a\,\left (2\,b^3-8\,a\,b\,c\right )\,\left (2\,a\,c-b^2\right )}{\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}-x^2\,\left (\frac {\frac {\left (2\,a\,c-b^2\right )\,\left (\frac {4\,a\,c^3-6\,b^2\,c^2}{c^2}-\frac {4\,b\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}-\frac {b\,\left (2\,b^3-8\,a\,b\,c\right )\,\left (2\,a\,c-b^2\right )}{2\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}+\frac {b\,\left (\frac {\left (2\,b^3-8\,a\,b\,c\right )\,\left (\frac {4\,a\,c^3-6\,b^2\,c^2}{c^2}-\frac {4\,b\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {b^3-a\,b\,c}{c^2}+\frac {b\,{\left (2\,a\,c-b^2\right )}^2}{2\,c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,\left (\frac {a\,b^2}{c^2}+\frac {\left (2\,b^3-8\,a\,b\,c\right )\,\left (8\,a\,b+\frac {8\,a\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {a\,{\left (2\,a\,c-b^2\right )}^2}{c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )}{4\,a^2\,c^2-4\,a\,b^2\,c+b^4}\right )\,\left (2\,a\,c-b^2\right )}{2\,c^2\,\sqrt {4\,a\,c-b^2}} \]
x^2/(2*c) + (log(a + b*x^2 + c*x^4)*(2*b^3 - 8*a*b*c))/(2*(16*a*c^3 - 4*b^ 2*c^2)) + (atan((2*c^2*(4*a*c - b^2)*((((8*a*b + (8*a*c^2*(2*b^3 - 8*a*b*c ))/(16*a*c^3 - 4*b^2*c^2))*(2*a*c - b^2))/(8*c^2*(4*a*c - b^2)^(1/2)) + (a *(2*b^3 - 8*a*b*c)*(2*a*c - b^2))/((4*a*c - b^2)^(1/2)*(16*a*c^3 - 4*b^2*c ^2)))/a - x^2*((((2*a*c - b^2)*((4*a*c^3 - 6*b^2*c^2)/c^2 - (4*b*c^2*(2*b^ 3 - 8*a*b*c))/(16*a*c^3 - 4*b^2*c^2)))/(8*c^2*(4*a*c - b^2)^(1/2)) - (b*(2 *b^3 - 8*a*b*c)*(2*a*c - b^2))/(2*(4*a*c - b^2)^(1/2)*(16*a*c^3 - 4*b^2*c^ 2)))/a + (b*(((2*b^3 - 8*a*b*c)*((4*a*c^3 - 6*b^2*c^2)/c^2 - (4*b*c^2*(2*b ^3 - 8*a*b*c))/(16*a*c^3 - 4*b^2*c^2)))/(2*(16*a*c^3 - 4*b^2*c^2)) - (b^3 - a*b*c)/c^2 + (b*(2*a*c - b^2)^2)/(2*c^2*(4*a*c - b^2))))/(2*a*(4*a*c - b ^2)^(1/2))) + (b*((a*b^2)/c^2 + ((2*b^3 - 8*a*b*c)*(8*a*b + (8*a*c^2*(2*b^ 3 - 8*a*b*c))/(16*a*c^3 - 4*b^2*c^2)))/(2*(16*a*c^3 - 4*b^2*c^2)) - (a*(2* a*c - b^2)^2)/(c^2*(4*a*c - b^2))))/(2*a*(4*a*c - b^2)^(1/2))))/(b^4 + 4*a ^2*c^2 - 4*a*b^2*c))*(2*a*c - b^2))/(2*c^2*(4*a*c - b^2)^(1/2))