Integrand size = 22, antiderivative size = 149 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=-\frac {\left (b^2 d-2 a c d-b c e\right ) \text {arctanh}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac {(b d-c e) \log \left (c+b x+a x^2\right )}{2 a \left (a d^2-e (b d-c e)\right )} \]
d^2*ln(e*x+d)/e/(a*d^2-b*d*e+c*e^2)-1/2*(b*d-c*e)*ln(a*x^2+b*x+c)/a/(a*d^2 -e*(b*d-c*e))-(-2*a*c*d+b^2*d-b*c*e)*arctanh((2*a*x+b)/(-4*a*c+b^2)^(1/2)) /a/(a*d^2-e*(b*d-c*e))/(-4*a*c+b^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=-\frac {2 e \left (-b^2 d+2 a c d+b c e\right ) \arctan \left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )+\sqrt {-b^2+4 a c} \left (-2 a d^2 \log (d+e x)+e (b d-c e) \log (c+x (b+a x))\right )}{2 a \sqrt {-b^2+4 a c} e \left (a d^2+e (-b d+c e)\right )} \]
-1/2*(2*e*(-(b^2*d) + 2*a*c*d + b*c*e)*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a* c]] + Sqrt[-b^2 + 4*a*c]*(-2*a*d^2*Log[d + e*x] + e*(b*d - c*e)*Log[c + x* (b + a*x)]))/(a*Sqrt[-b^2 + 4*a*c]*e*(a*d^2 + e*(-(b*d) + c*e)))
Time = 0.37 (sec) , antiderivative size = 149, 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 = {1775, 1200, 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.
\(\displaystyle \int \frac {1}{(d+e x) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )} \, dx\) |
\(\Big \downarrow \) 1775 |
\(\displaystyle \int \frac {x^2}{(d+e x) \left (a x^2+b x+c\right )}dx\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \int \left (\frac {-x (b d-c e)-c d}{\left (a x^2+b x+c\right ) \left (a d^2-e (b d-c e)\right )}+\frac {d^2}{(d+e x) \left (a d^2-e (b d-c e)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right ) \left (-2 a c d+b^2 d-b c e\right )}{a \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {d^2 \log (d+e x)}{e \left (a d^2-b d e+c e^2\right )}-\frac {(b d-c e) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )}\) |
-(((b^2*d - 2*a*c*d - b*c*e)*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(a*Sq rt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e)))) + (d^2*Log[d + e*x])/(e*(a*d^2 - b*d*e + c*e^2)) - ((b*d - c*e)*Log[c + b*x + a*x^2])/(2*a*(a*d^2 - e*(b*d - c*e)))
3.1.64.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x _)^(n_.))^(q_.), x_Symbol] :> Int[((d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p) /x^(2*n*p), x] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[mn, -n] && EqQ[mn2 , 2*mn] && IntegerQ[p]
Time = 0.74 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\frac {\left (-b d +e c \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (-c d -\frac {\left (-b d +e c \right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{a \,d^{2}-b d e +c \,e^{2}}+\frac {d^{2} \ln \left (e x +d \right )}{e \left (a \,d^{2}-b d e +c \,e^{2}\right )}\) | \(130\) |
risch | \(\text {Expression too large to display}\) | \(7752\) |
1/(a*d^2-b*d*e+c*e^2)*(1/2*(-b*d+c*e)/a*ln(a*x^2+b*x+c)+2*(-c*d-1/2*(-b*d+ c*e)*b/a)/(4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2)))+d^2*ln(e* x+d)/e/(a*d^2-b*d*e+c*e^2)
Time = 1.07 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.72 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\left [\frac {2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} \log \left (e x + d\right ) + {\left (b c e^{2} - {\left (b^{2} - 2 \, a c\right )} d e\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, a x + b\right )}}{a x^{2} + b x + c}\right ) - {\left ({\left (b^{3} - 4 \, a b c\right )} d e - {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} e - {\left (a b^{3} - 4 \, a^{2} b c\right )} d e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{3}\right )}}, \frac {2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} \log \left (e x + d\right ) + 2 \, {\left (b c e^{2} - {\left (b^{2} - 2 \, a c\right )} d e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{3} - 4 \, a b c\right )} d e - {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} e - {\left (a b^{3} - 4 \, a^{2} b c\right )} d e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} e^{3}\right )}}\right ] \]
[1/2*(2*(a*b^2 - 4*a^2*c)*d^2*log(e*x + d) + (b*c*e^2 - (b^2 - 2*a*c)*d*e) *sqrt(b^2 - 4*a*c)*log((2*a^2*x^2 + 2*a*b*x + b^2 - 2*a*c + sqrt(b^2 - 4*a *c)*(2*a*x + b))/(a*x^2 + b*x + c)) - ((b^3 - 4*a*b*c)*d*e - (b^2*c - 4*a* c^2)*e^2)*log(a*x^2 + b*x + c))/((a^2*b^2 - 4*a^3*c)*d^2*e - (a*b^3 - 4*a^ 2*b*c)*d*e^2 + (a*b^2*c - 4*a^2*c^2)*e^3), 1/2*(2*(a*b^2 - 4*a^2*c)*d^2*lo g(e*x + d) + 2*(b*c*e^2 - (b^2 - 2*a*c)*d*e)*sqrt(-b^2 + 4*a*c)*arctan(-sq rt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c)) - ((b^3 - 4*a*b*c)*d*e - (b^2* c - 4*a*c^2)*e^2)*log(a*x^2 + b*x + c))/((a^2*b^2 - 4*a^3*c)*d^2*e - (a*b^ 3 - 4*a^2*b*c)*d*e^2 + (a*b^2*c - 4*a^2*c^2)*e^3)]
Timed out. \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\text {Exception raised: ValueError} \]
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 = 0.34 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\frac {d^{2} \log \left ({\left | e x + d \right |}\right )}{a d^{2} e - b d e^{2} + c e^{3}} - \frac {{\left (b d - c e\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a^{2} d^{2} - a b d e + a c e^{2}\right )}} + \frac {{\left (b^{2} d - 2 \, a c d - b c e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{2} d^{2} - a b d e + a c e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} \]
d^2*log(abs(e*x + d))/(a*d^2*e - b*d*e^2 + c*e^3) - 1/2*(b*d - c*e)*log(a* x^2 + b*x + c)/(a^2*d^2 - a*b*d*e + a*c*e^2) + (b^2*d - 2*a*c*d - b*c*e)*a rctan((2*a*x + b)/sqrt(-b^2 + 4*a*c))/((a^2*d^2 - a*b*d*e + a*c*e^2)*sqrt( -b^2 + 4*a*c))
Time = 9.72 (sec) , antiderivative size = 966, normalized size of antiderivative = 6.48 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)} \, dx=\frac {d^2\,\ln \left (d+e\,x\right )}{a\,d^2\,e-b\,d\,e^2+c\,e^3}-\frac {\ln \left (a\,b^2\,d^4-2\,c^3\,e^4-4\,a^2\,c\,d^4+b^3\,d^3\,e+c^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}+10\,a\,c^2\,d^2\,e^2-4\,b^2\,c\,d^2\,e^2-b^3\,d^2\,e^2\,x+a\,b\,d^4\,\sqrt {b^2-4\,a\,c}+3\,b\,c^2\,d\,e^3-b\,c^2\,e^4\,x+b^2\,d^3\,e\,\sqrt {b^2-4\,a\,c}+3\,c^2\,d\,e^3\,\sqrt {b^2-4\,a\,c}+2\,a^2\,d^4\,x\,\sqrt {b^2-4\,a\,c}+3\,a\,b^2\,d^3\,e\,x+6\,a\,c^2\,d\,e^3\,x-10\,a^2\,c\,d^3\,e\,x-2\,b\,c\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}-3\,a\,b\,c\,d^3\,e+b^2\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}-5\,a\,c\,d^3\,e\,\sqrt {b^2-4\,a\,c}-a\,b\,d^3\,e\,x\,\sqrt {b^2-4\,a\,c}+a\,b\,c\,d^2\,e^2\,x-5\,a\,c\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (e\,\left (\frac {b^2\,c}{2}-2\,a\,c^2+\frac {b\,c\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^3\,d}{2}-\frac {b^2\,d\,\sqrt {b^2-4\,a\,c}}{2}+a\,c\,d\,\sqrt {b^2-4\,a\,c}+2\,a\,b\,c\,d\right )}{4\,a^3\,c\,d^2-a^2\,b^2\,d^2-4\,a^2\,b\,c\,d\,e+4\,a^2\,c^2\,e^2+a\,b^3\,d\,e-a\,b^2\,c\,e^2}+\frac {\ln \left (2\,c^3\,e^4-a\,b^2\,d^4+4\,a^2\,c\,d^4-b^3\,d^3\,e+c^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}-10\,a\,c^2\,d^2\,e^2+4\,b^2\,c\,d^2\,e^2+b^3\,d^2\,e^2\,x+a\,b\,d^4\,\sqrt {b^2-4\,a\,c}-3\,b\,c^2\,d\,e^3+b\,c^2\,e^4\,x+b^2\,d^3\,e\,\sqrt {b^2-4\,a\,c}+3\,c^2\,d\,e^3\,\sqrt {b^2-4\,a\,c}+2\,a^2\,d^4\,x\,\sqrt {b^2-4\,a\,c}-3\,a\,b^2\,d^3\,e\,x-6\,a\,c^2\,d\,e^3\,x+10\,a^2\,c\,d^3\,e\,x-2\,b\,c\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}+3\,a\,b\,c\,d^3\,e+b^2\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}-5\,a\,c\,d^3\,e\,\sqrt {b^2-4\,a\,c}-a\,b\,d^3\,e\,x\,\sqrt {b^2-4\,a\,c}-a\,b\,c\,d^2\,e^2\,x-5\,a\,c\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^3\,d}{2}+e\,\left (2\,a\,c^2-\frac {b^2\,c}{2}+\frac {b\,c\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^2\,d\,\sqrt {b^2-4\,a\,c}}{2}+a\,c\,d\,\sqrt {b^2-4\,a\,c}-2\,a\,b\,c\,d\right )}{4\,a^3\,c\,d^2-a^2\,b^2\,d^2-4\,a^2\,b\,c\,d\,e+4\,a^2\,c^2\,e^2+a\,b^3\,d\,e-a\,b^2\,c\,e^2} \]
(d^2*log(d + e*x))/(c*e^3 + a*d^2*e - b*d*e^2) - (log(a*b^2*d^4 - 2*c^3*e^ 4 - 4*a^2*c*d^4 + b^3*d^3*e + c^2*e^4*x*(b^2 - 4*a*c)^(1/2) + 10*a*c^2*d^2 *e^2 - 4*b^2*c*d^2*e^2 - b^3*d^2*e^2*x + a*b*d^4*(b^2 - 4*a*c)^(1/2) + 3*b *c^2*d*e^3 - b*c^2*e^4*x + b^2*d^3*e*(b^2 - 4*a*c)^(1/2) + 3*c^2*d*e^3*(b^ 2 - 4*a*c)^(1/2) + 2*a^2*d^4*x*(b^2 - 4*a*c)^(1/2) + 3*a*b^2*d^3*e*x + 6*a *c^2*d*e^3*x - 10*a^2*c*d^3*e*x - 2*b*c*d^2*e^2*(b^2 - 4*a*c)^(1/2) - 3*a* b*c*d^3*e + b^2*d^2*e^2*x*(b^2 - 4*a*c)^(1/2) - 5*a*c*d^3*e*(b^2 - 4*a*c)^ (1/2) - a*b*d^3*e*x*(b^2 - 4*a*c)^(1/2) + a*b*c*d^2*e^2*x - 5*a*c*d^2*e^2* x*(b^2 - 4*a*c)^(1/2))*(e*((b^2*c)/2 - 2*a*c^2 + (b*c*(b^2 - 4*a*c)^(1/2)) /2) - (b^3*d)/2 - (b^2*d*(b^2 - 4*a*c)^(1/2))/2 + a*c*d*(b^2 - 4*a*c)^(1/2 ) + 2*a*b*c*d))/(4*a^3*c*d^2 - a^2*b^2*d^2 + 4*a^2*c^2*e^2 + a*b^3*d*e - a *b^2*c*e^2 - 4*a^2*b*c*d*e) + (log(2*c^3*e^4 - a*b^2*d^4 + 4*a^2*c*d^4 - b ^3*d^3*e + c^2*e^4*x*(b^2 - 4*a*c)^(1/2) - 10*a*c^2*d^2*e^2 + 4*b^2*c*d^2* e^2 + b^3*d^2*e^2*x + a*b*d^4*(b^2 - 4*a*c)^(1/2) - 3*b*c^2*d*e^3 + b*c^2* e^4*x + b^2*d^3*e*(b^2 - 4*a*c)^(1/2) + 3*c^2*d*e^3*(b^2 - 4*a*c)^(1/2) + 2*a^2*d^4*x*(b^2 - 4*a*c)^(1/2) - 3*a*b^2*d^3*e*x - 6*a*c^2*d*e^3*x + 10*a ^2*c*d^3*e*x - 2*b*c*d^2*e^2*(b^2 - 4*a*c)^(1/2) + 3*a*b*c*d^3*e + b^2*d^2 *e^2*x*(b^2 - 4*a*c)^(1/2) - 5*a*c*d^3*e*(b^2 - 4*a*c)^(1/2) - a*b*d^3*e*x *(b^2 - 4*a*c)^(1/2) - a*b*c*d^2*e^2*x - 5*a*c*d^2*e^2*x*(b^2 - 4*a*c)^(1/ 2))*((b^3*d)/2 + e*(2*a*c^2 - (b^2*c)/2 + (b*c*(b^2 - 4*a*c)^(1/2))/2) ...