Integrand size = 19, antiderivative size = 110 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {1}{b^2 d x}-\frac {c^2}{b^2 (c d-b e) (b+c x)}-\frac {(2 c d+b e) \log (x)}{b^3 d^2}+\frac {c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}+\frac {e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
-1/b^2/d/x-c^2/b^2/(-b*e+c*d)/(c*x+b)-(b*e+2*c*d)*ln(x)/b^3/d^2+c^2*(-3*b* e+2*c*d)*ln(c*x+b)/b^3/(-b*e+c*d)^2+e^3*ln(e*x+d)/d^2/(-b*e+c*d)^2
Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {1}{b^2 d x}+\frac {c^2}{b^2 (-c d+b e) (b+c x)}+\frac {(-2 c d-b e) \log (x)}{b^3 d^2}+\frac {\left (2 c^3 d-3 b c^2 e\right ) \log (b+c x)}{b^3 (-c d+b e)^2}+\frac {e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
-(1/(b^2*d*x)) + c^2/(b^2*(-(c*d) + b*e)*(b + c*x)) + ((-2*c*d - b*e)*Log[ x])/(b^3*d^2) + ((2*c^3*d - 3*b*c^2*e)*Log[b + c*x])/(b^3*(-(c*d) + b*e)^2 ) + (e^3*Log[d + e*x])/(d^2*(c*d - b*e)^2)
Time = 0.34 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1141, 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.
\(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^2 (d+e x)} \, dx\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle c^2 \int \left (\frac {e^4}{c^2 d^2 (c d-b e)^2 (d+e x)}-\frac {2 c d+b e}{b^3 c^2 d^2 x}+\frac {c (2 c d-3 b e)}{b^3 (c d-b e)^2 (b+c x)}+\frac {1}{b^2 c^2 d x^2}+\frac {c}{b^2 (c d-b e) (b+c x)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle c^2 \left (-\frac {\log (x) (b e+2 c d)}{b^3 c^2 d^2}+\frac {(2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}-\frac {1}{b^2 c^2 d x}-\frac {1}{b^2 (b+c x) (c d-b e)}+\frac {e^3 \log (d+e x)}{c^2 d^2 (c d-b e)^2}\right )\) |
c^2*(-(1/(b^2*c^2*d*x)) - 1/(b^2*(c*d - b*e)*(b + c*x)) - ((2*c*d + b*e)*L og[x])/(b^3*c^2*d^2) + ((2*c*d - 3*b*e)*Log[b + c*x])/(b^3*(c*d - b*e)^2) + (e^3*Log[d + e*x])/(c^2*d^2*(c*d - b*e)^2))
3.3.73.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Time = 1.90 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {1}{b^{2} d x}+\frac {\left (-b e -2 c d \right ) \ln \left (x \right )}{b^{3} d^{2}}+\frac {c^{2}}{\left (b e -c d \right ) b^{2} \left (c x +b \right )}-\frac {c^{2} \left (3 b e -2 c d \right ) \ln \left (c x +b \right )}{\left (b e -c d \right )^{2} b^{3}}+\frac {e^{3} \ln \left (e x +d \right )}{d^{2} \left (b e -c d \right )^{2}}\) | \(111\) |
norman | \(\frac {\frac {\left (b c e -2 c^{2} d \right ) c \,x^{2}}{d \,b^{3} \left (b e -c d \right )}-\frac {1}{b d}}{x \left (c x +b \right )}+\frac {e^{3} \ln \left (e x +d \right )}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\left (b e +2 c d \right ) \ln \left (x \right )}{b^{3} d^{2}}-\frac {c^{2} \left (3 b e -2 c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}\) | \(154\) |
risch | \(\frac {-\frac {c \left (b e -2 c d \right ) x}{b^{2} d \left (b e -c d \right )}-\frac {1}{b d}}{x \left (c x +b \right )}+\frac {e^{3} \ln \left (-e x -d \right )}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\ln \left (-x \right ) e}{b^{2} d^{2}}-\frac {2 \ln \left (-x \right ) c}{b^{3} d}-\frac {3 c^{2} \ln \left (c x +b \right ) e}{b^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {2 c^{3} \ln \left (c x +b \right ) d}{b^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}\) | \(191\) |
parallelrisch | \(-\frac {\ln \left (x \right ) x^{2} b^{3} c^{2} e^{3}-3 \ln \left (x \right ) x^{2} b \,c^{4} d^{2} e +2 \ln \left (x \right ) x^{2} c^{5} d^{3}+3 \ln \left (c x +b \right ) x^{2} b \,c^{4} d^{2} e -2 \ln \left (c x +b \right ) x^{2} c^{5} d^{3}-\ln \left (e x +d \right ) x^{2} b^{3} c^{2} e^{3}+\ln \left (x \right ) x \,b^{4} c \,e^{3}-3 \ln \left (x \right ) x \,b^{2} c^{3} d^{2} e +2 \ln \left (x \right ) x b \,c^{4} d^{3}+3 \ln \left (c x +b \right ) x \,b^{2} c^{3} d^{2} e -2 \ln \left (c x +b \right ) x b \,c^{4} d^{3}-\ln \left (e x +d \right ) x \,b^{4} c \,e^{3}+x \,b^{3} c^{2} d \,e^{2}-3 x \,b^{2} c^{3} d^{2} e +2 x b \,c^{4} d^{3}+b^{4} c d \,e^{2}-2 b^{3} c^{2} d^{2} e +b^{2} c^{3} d^{3}}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \left (c x +b \right ) x \,b^{3} c \,d^{2}}\) | \(300\) |
-1/b^2/d/x+(-b*e-2*c*d)/b^3/d^2*ln(x)+c^2/(b*e-c*d)/b^2/(c*x+b)-c^2*(3*b*e -2*c*d)/(b*e-c*d)^2/b^3*ln(c*x+b)+e^3/d^2/(b*e-c*d)^2*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (110) = 220\).
Time = 2.38 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.62 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {b^{2} c^{2} d^{3} - 2 \, b^{3} c d^{2} e + b^{4} d e^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{3} c d e^{2}\right )} x - {\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e\right )} x^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e\right )} x\right )} \log \left (c x + b\right ) - {\left (b^{3} c e^{3} x^{2} + b^{4} e^{3} x\right )} \log \left (e x + d\right ) + {\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{3} c e^{3}\right )} x^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{4} e^{3}\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e + b^{5} c d^{2} e^{2}\right )} x^{2} + {\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} x} \]
-(b^2*c^2*d^3 - 2*b^3*c*d^2*e + b^4*d*e^2 + (2*b*c^3*d^3 - 3*b^2*c^2*d^2*e + b^3*c*d*e^2)*x - ((2*c^4*d^3 - 3*b*c^3*d^2*e)*x^2 + (2*b*c^3*d^3 - 3*b^ 2*c^2*d^2*e)*x)*log(c*x + b) - (b^3*c*e^3*x^2 + b^4*e^3*x)*log(e*x + d) + ((2*c^4*d^3 - 3*b*c^3*d^2*e + b^3*c*e^3)*x^2 + (2*b*c^3*d^3 - 3*b^2*c^2*d^ 2*e + b^4*e^3)*x)*log(x))/((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5*c*d^2*e^2) *x^2 + (b^4*c^2*d^4 - 2*b^5*c*d^3*e + b^6*d^2*e^2)*x)
Timed out. \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\frac {e^{3} \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}} + \frac {{\left (2 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}} - \frac {b c d - b^{2} e + {\left (2 \, c^{2} d - b c e\right )} x}{{\left (b^{2} c^{2} d^{2} - b^{3} c d e\right )} x^{2} + {\left (b^{3} c d^{2} - b^{4} d e\right )} x} - \frac {{\left (2 \, c d + b e\right )} \log \left (x\right )}{b^{3} d^{2}} \]
e^3*log(e*x + d)/(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2) + (2*c^3*d - 3*b*c^ 2*e)*log(c*x + b)/(b^3*c^2*d^2 - 2*b^4*c*d*e + b^5*e^2) - (b*c*d - b^2*e + (2*c^2*d - b*c*e)*x)/((b^2*c^2*d^2 - b^3*c*d*e)*x^2 + (b^3*c*d^2 - b^4*d* e)*x) - (2*c*d + b*e)*log(x)/(b^3*d^2)
Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}} + \frac {{\left (2 \, c^{4} d - 3 \, b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3} d^{2} - 2 \, b^{4} c^{2} d e + b^{5} c e^{2}} - \frac {{\left (2 \, c d + b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{2}} - \frac {b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2}\right )} x}{{\left (c d - b e\right )}^{2} {\left (c x + b\right )} b^{2} d^{2} x} \]
e^4*log(abs(e*x + d))/(c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3) + (2*c^4*d - 3*b*c^3*e)*log(abs(c*x + b))/(b^3*c^3*d^2 - 2*b^4*c^2*d*e + b^5*c*e^2) - (2*c*d + b*e)*log(abs(x))/(b^3*d^2) - (b*c^2*d^3 - 2*b^2*c*d^2*e + b^3*d *e^2 + (2*c^3*d^3 - 3*b*c^2*d^2*e + b^2*c*d*e^2)*x)/((c*d - b*e)^2*(c*x + b)*b^2*d^2*x)
Time = 9.85 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\frac {\ln \left (b+c\,x\right )\,\left (2\,c^3\,d-3\,b\,c^2\,e\right )}{b^5\,e^2-2\,b^4\,c\,d\,e+b^3\,c^2\,d^2}-\frac {\frac {1}{b\,d}-\frac {x\,\left (2\,c^2\,d-b\,c\,e\right )}{b^2\,d\,\left (b\,e-c\,d\right )}}{c\,x^2+b\,x}+\frac {e^3\,\ln \left (d+e\,x\right )}{d^2\,{\left (b\,e-c\,d\right )}^2}-\frac {\ln \left (x\right )\,\left (b\,e+2\,c\,d\right )}{b^3\,d^2} \]