Integrand size = 20, antiderivative size = 62 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {\log (x)}{a}-\frac {\log \left (a+b x+c x^2\right )}{2 a} \]
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=-\frac {\frac {2 b \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-2 \log (x)+\log (a+x (b+c x))}{2 a} \]
-1/2*((2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2* Log[x] + Log[a + x*(b + c*x)])/a
Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {9, 1144, 25, 1142, 1083, 219, 1103}
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.
\(\displaystyle \int \frac {x}{a x^2+b x^3+c x^4} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )}dx\) |
\(\Big \downarrow \) 1144 |
\(\displaystyle \frac {\int -\frac {b+c x}{c x^2+b x+a}dx}{a}+\frac {\log (x)}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\log (x)}{a}-\frac {\int \frac {b+c x}{c x^2+b x+a}dx}{a}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {\log (x)}{a}-\frac {\frac {1}{2} b \int \frac {1}{c x^2+b x+a}dx+\frac {1}{2} \int \frac {b+2 c x}{c x^2+b x+a}dx}{a}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {\log (x)}{a}-\frac {\frac {1}{2} \int \frac {b+2 c x}{c x^2+b x+a}dx-b \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\log (x)}{a}-\frac {\frac {1}{2} \int \frac {b+2 c x}{c x^2+b x+a}dx-\frac {b \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}}{a}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\log (x)}{a}-\frac {\frac {1}{2} \log \left (a+b x+c x^2\right )-\frac {b \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}}{a}\) |
Log[x]/a - (-((b*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c] ) + Log[a + b*x + c*x^2]/2)/a
3.1.15.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[e*(Log[RemoveContent[d + e*x, x]]/(c*d^2 - b*d*e + a*e^2)), x] + S imp[1/(c*d^2 - b*d*e + a*e^2) Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\ln \left (x \right )}{a}+\frac {-\frac {\ln \left (c \,x^{2}+b x +a \right )}{2}-\frac {b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{a}\) | \(61\) |
risch | \(-\frac {2 \ln \left (\left (8 a \,b^{2} c -2 b^{4}+6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) c}{4 a c -b^{2}}+\frac {\ln \left (\left (8 a \,b^{2} c -2 b^{4}+6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) b^{2}}{2 a \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (8 a \,b^{2} c -2 b^{4}+6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{2 a \left (4 a c -b^{2}\right )}-\frac {2 \ln \left (\left (8 a \,b^{2} c -2 b^{4}-6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) c}{4 a c -b^{2}}+\frac {\ln \left (\left (8 a \,b^{2} c -2 b^{4}-6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) b^{2}}{2 a \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (8 a \,b^{2} c -2 b^{4}-6 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a c +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b^{2}\right ) x +12 c b \,a^{2}-3 a \,b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{2 a \left (4 a c -b^{2}\right )}+\frac {\ln \left (x \right )}{a}\) | \(707\) |
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.40 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \]
[1/2*(sqrt(b^2 - 4*a*c)*b*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)) - (b^2 - 4*a*c)*log(c*x^2 + b*x + a) + 2*(b^2 - 4*a*c)*log(x))/(a*b^2 - 4*a^2*c), 1/2*(2*sqrt(-b^2 + 4*a* c)*b*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (b^2 - 4*a*c) *log(c*x^2 + b*x + a) + 2*(b^2 - 4*a*c)*log(x))/(a*b^2 - 4*a^2*c)]
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (54) = 108\).
Time = 4.49 (sec) , antiderivative size = 564, normalized size of antiderivative = 9.10 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) \log {\left (x + \frac {24 a^{4} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) + 2 a^{2} b^{4} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) \log {\left (x + \frac {24 a^{4} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) + 2 a^{2} b^{4} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \frac {\log {\left (x \right )}}{a} \]
(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))*log(x + (24*a**4*c **2*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 14*a**3*b **2*c*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 12*a**3 *c**2*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) + 2*a**2*b** 4*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 + 3*a**2*b**2 *c*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) - 12*a**2*c**2 + 11*a*b**2*c - 2*b**4)/(9*a*b*c**2 - 2*b**3*c)) + (b*sqrt(-4*a*c + b**2)/ (2*a*(4*a*c - b**2)) - 1/(2*a))*log(x + (24*a**4*c**2*(b*sqrt(-4*a*c + b** 2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 14*a**3*b**2*c*(b*sqrt(-4*a*c + b* *2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 12*a**3*c**2*(b*sqrt(-4*a*c + b** 2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) + 2*a**2*b**4*(b*sqrt(-4*a*c + b**2)/(2 *a*(4*a*c - b**2)) - 1/(2*a))**2 + 3*a**2*b**2*c*(b*sqrt(-4*a*c + b**2)/(2 *a*(4*a*c - b**2)) - 1/(2*a)) - 12*a**2*c**2 + 11*a*b**2*c - 2*b**4)/(9*a* b*c**2 - 2*b**3*c)) + log(x)/a
Exception generated. \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\text {Exception raised: ValueError} \]
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 = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=-\frac {b \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a} - \frac {\log \left (c x^{2} + b x + a\right )}{2 \, a} + \frac {\log \left ({\left | x \right |}\right )}{a} \]
-b*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a) - 1/2*log (c*x^2 + b*x + a)/a + log(abs(x))/a
Time = 8.72 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.44 \[ \int \frac {x}{a x^2+b x^3+c x^4} \, dx=\frac {\ln \left (x\right )}{a}-\ln \left (b\,c-\left (x\,\left (6\,a\,c^2-2\,b^2\,c\right )-a\,b\,c\right )\,\left (\frac {1}{2\,a}-\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )+3\,c^2\,x\right )\,\left (\frac {1}{2\,a}-\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )-\ln \left (\left (x\,\left (6\,a\,c^2-2\,b^2\,c\right )-a\,b\,c\right )\,\left (\frac {1}{2\,a}+\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )-b\,c-3\,c^2\,x\right )\,\left (\frac {1}{2\,a}+\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right ) \]
log(x)/a - log(b*c - (x*(6*a*c^2 - 2*b^2*c) - a*b*c)*(1/(2*a) - (b*(b^2 - 4*a*c)^(1/2))/(2*(a*b^2 - 4*a^2*c))) + 3*c^2*x)*(1/(2*a) - (b*(b^2 - 4*a*c )^(1/2))/(2*(a*b^2 - 4*a^2*c))) - log((x*(6*a*c^2 - 2*b^2*c) - a*b*c)*(1/( 2*a) + (b*(b^2 - 4*a*c)^(1/2))/(2*(a*b^2 - 4*a^2*c))) - b*c - 3*c^2*x)*(1/ (2*a) + (b*(b^2 - 4*a*c)^(1/2))/(2*(a*b^2 - 4*a^2*c)))