Integrand size = 22, antiderivative size = 89 \[ \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx=-\frac {b x}{c^2}+\frac {x^2}{2 c}+\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3} \]
-b*x/c^2+1/2*x^2/c+1/2*(-a*c+b^2)*ln(c*x^2+b*x+a)/c^3+b*(-3*a*c+b^2)*arcta nh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(1/2)
Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94 \[ \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx=\frac {c x (-2 b+c x)-\frac {2 b \left (b^2-3 a c\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\left (b^2-a c\right ) \log (a+x (b+c x))}{2 c^3} \]
(c*x*(-2*b + c*x) - (2*b*(b^2 - 3*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a* c]])/Sqrt[-b^2 + 4*a*c] + (b^2 - a*c)*Log[a + x*(b + c*x)])/(2*c^3)
Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {9, 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.
\(\displaystyle \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {x^3}{a+b x+c x^2}dx\) |
\(\Big \downarrow \) 1143 |
\(\displaystyle \int \left (\frac {x \left (b^2-a c\right )+a b}{c^2 \left (a+b x+c x^2\right )}-\frac {b}{c^2}+\frac {x}{c}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {b x}{c^2}+\frac {x^2}{2 c}\) |
-((b*x)/c^2) + x^2/(2*c) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) + ((b^2 - a*c)*Log[a + b*x + c*x^2])/(2* c^3)
3.1.11.3.1 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]
Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10
method | result | size |
default | \(-\frac {-\frac {1}{2} c \,x^{2}+b x}{c^{2}}+\frac {\frac {\left (-a c +b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (a b -\frac {\left (-a c +b^{2}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{2}}\) | \(98\) |
risch | \(\frac {x^{2}}{2 c}-\frac {b x}{c^{2}}-\frac {2 \ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{c \left (4 a c -b^{2}\right )}+\frac {5 \ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{2 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 c^{3} \left (4 a c -b^{2}\right )}+\frac {\ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x -\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}}{2 c^{3} \left (4 a c -b^{2}\right )}-\frac {2 \ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) a^{2}}{c \left (4 a c -b^{2}\right )}+\frac {5 \ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) a \,b^{2}}{2 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) b^{4}}{2 c^{3} \left (4 a c -b^{2}\right )}-\frac {\ln \left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, c x +\sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right ) \left (3 a c -b^{2}\right )^{2}}}{2 c^{3} \left (4 a c -b^{2}\right )}\) | \(915\) |
-1/c^2*(-1/2*c*x^2+b*x)+1/c^2*(1/2*(-a*c+b^2)/c*ln(c*x^2+b*x+a)+2*(a*b-1/2 *(-a*c+b^2)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.34 \[ \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx=\left [\frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - {\left (b^{3} - 3 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 2 \, {\left (b^{3} - 3 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \]
[1/2*((b^2*c^2 - 4*a*c^3)*x^2 - (b^3 - 3*a*b*c)*sqrt(b^2 - 4*a*c)*log((2*c ^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b *x + a)) - 2*(b^3*c - 4*a*b*c^2)*x + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*log(c*x ^2 + b*x + a))/(b^2*c^3 - 4*a*c^4), 1/2*((b^2*c^2 - 4*a*c^3)*x^2 + 2*(b^3 - 3*a*b*c)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 2*(b^3*c - 4*a*b*c^2)*x + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*log(c* x^2 + b*x + a))/(b^2*c^3 - 4*a*c^4)]
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (83) = 166\).
Time = 0.48 (sec) , antiderivative size = 381, normalized size of antiderivative = 4.28 \[ \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx=- \frac {b x}{c^{2}} + \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right ) \log {\left (x + \frac {2 a^{2} c - a b^{2} + 4 a c^{3} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right ) - b^{2} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right ) \log {\left (x + \frac {2 a^{2} c - a b^{2} + 4 a c^{3} \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right ) - b^{2} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}} \cdot \left (3 a c - b^{2}\right )}{2 c^{3} \cdot \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{2 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \frac {x^{2}}{2 c} \]
-b*x/c**2 + (-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*c**3*(4*a*c - b**2)) - (a*c - b**2)/(2*c**3))*log(x + (2*a**2*c - a*b**2 + 4*a*c**3*(-b*sqrt(- 4*a*c + b**2)*(3*a*c - b**2)/(2*c**3*(4*a*c - b**2)) - (a*c - b**2)/(2*c** 3)) - b**2*c**2*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*c**3*(4*a*c - b* *2)) - (a*c - b**2)/(2*c**3)))/(3*a*b*c - b**3)) + (b*sqrt(-4*a*c + b**2)* (3*a*c - b**2)/(2*c**3*(4*a*c - b**2)) - (a*c - b**2)/(2*c**3))*log(x + (2 *a**2*c - a*b**2 + 4*a*c**3*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*c**3* (4*a*c - b**2)) - (a*c - b**2)/(2*c**3)) - b**2*c**2*(b*sqrt(-4*a*c + b**2 )*(3*a*c - b**2)/(2*c**3*(4*a*c - b**2)) - (a*c - b**2)/(2*c**3)))/(3*a*b* c - b**3)) + x**2/(2*c)
Exception generated. \[ \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.97 \[ \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx=\frac {c x^{2} - 2 \, b x}{2 \, c^{2}} + \frac {{\left (b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac {{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{3}} \]
1/2*(c*x^2 - 2*b*x)/c^2 + 1/2*(b^2 - a*c)*log(c*x^2 + b*x + a)/c^3 - (b^3 - 3*a*b*c)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^3)
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.26 \[ \int \frac {x^5}{a x^2+b x^3+c x^4} \, dx=\frac {x^2}{2\,c}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,a^2\,c^2-5\,a\,b^2\,c+b^4\right )}{2\,\left (4\,a\,c^4-b^2\,c^3\right )}-\frac {b\,x}{c^2}+\frac {b\,\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (3\,a\,c-b^2\right )}{c^3\,\sqrt {4\,a\,c-b^2}} \]