Integrand size = 38, antiderivative size = 62 \[ \int \frac {(f+g x)^2}{\left (-2 d e f g+2 d^2 g^2-2 e g (e f-d g) x\right )^2} \, dx=\frac {x}{4 e^2 (e f-d g)^2}-\frac {1}{4 e^3 g^2 (d+e x)}+\frac {\log (d+e x)}{2 e^3 g (e f-d g)} \] Output:
1/4*x/e^2/(-d*g+e*f)^2-1/4/e^3/g^2/(e*x+d)+1/2*ln(e*x+d)/e^3/g/(-d*g+e*f)
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02 \[ \int \frac {(f+g x)^2}{\left (-2 d e f g+2 d^2 g^2-2 e g (e f-d g) x\right )^2} \, dx=\frac {e g^2 x-\frac {(e f-d g)^2}{d+e x}+2 g (e f-d g) \log (d+e x)}{4 e^3 g^2 (e f-d g)^2} \] Input:
Integrate[(f + g*x)^2/(-2*d*e*f*g + 2*d^2*g^2 - 2*e*g*(e*f - d*g)*x)^2,x]
Output:
(e*g^2*x - (e*f - d*g)^2/(d + e*x) + 2*g*(e*f - d*g)*Log[d + e*x])/(4*e^3* g^2*(e*f - d*g)^2)
Time = 0.36 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {49, 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.
\(\displaystyle \int \frac {(f+g x)^2}{\left (2 d^2 g^2-2 e g x (e f-d g)-2 d e f g\right )^2} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {1}{2 e^2 g (d+e x) (e f-d g)}+\frac {1}{4 e^2 (e f-d g)^2}+\frac {1}{4 e^2 g^2 (d+e x)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\log (d+e x)}{2 e^3 g (e f-d g)}-\frac {1}{4 e^3 g^2 (d+e x)}+\frac {x}{4 e^2 (e f-d g)^2}\) |
Input:
Int[(f + g*x)^2/(-2*d*e*f*g + 2*d^2*g^2 - 2*e*g*(e*f - d*g)*x)^2,x]
Output:
x/(4*e^2*(e*f - d*g)^2) - 1/(4*e^3*g^2*(d + e*x)) + Log[d + e*x]/(2*e^3*g* (e*f - d*g))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Time = 0.97 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {\frac {g^{2} x}{e^{2}}-\frac {2 g \left (d g -e f \right ) \ln \left (e x +d \right )}{e^{3}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{e^{3} \left (e x +d \right )}}{4 g^{2} \left (d g -e f \right )^{2}}\) | \(78\) |
parallelrisch | \(\frac {-2 \ln \left (e x +d \right ) x d e \,g^{2}+2 \ln \left (e x +d \right ) x \,e^{2} f g +e^{2} g^{2} x^{2}-2 \ln \left (e x +d \right ) d^{2} g^{2}+2 \ln \left (e x +d \right ) d e f g -2 d^{2} g^{2}+2 d e f g -e^{2} f^{2}}{4 e^{3} g^{2} \left (d g -e f \right )^{2} \left (e x +d \right )}\) | \(113\) |
norman | \(\frac {\frac {g \,x^{2}}{4 \left (d g -e f \right ) e}+\frac {-2 d^{2} g^{2}+2 d e f g -e^{2} f^{2}}{4 e^{3} g \left (d g -e f \right )}}{g \left (d e g x -e^{2} f x +d^{2} g -d e f \right )}-\frac {\ln \left (d e g x -e^{2} f x +d^{2} g -d e f \right )}{2 e^{3} g \left (d g -e f \right )}\) | \(134\) |
risch | \(\frac {x}{4 d^{2} e^{2} g^{2}-8 d \,e^{3} f g +4 e^{4} f^{2}}-\frac {g^{2} d^{4}}{e \left (4 d^{2} e^{2} g^{2}-8 d \,e^{3} f g +4 e^{4} f^{2}\right ) \left (d e g x -e^{2} f x +d^{2} g -d e f \right ) \left (d g -e f \right )}+\frac {4 g \,d^{3} f}{\left (4 d^{2} e^{2} g^{2}-8 d \,e^{3} f g +4 e^{4} f^{2}\right ) \left (d e g x -e^{2} f x +d^{2} g -d e f \right ) \left (d g -e f \right )}-\frac {6 e \,d^{2} f^{2}}{\left (4 d^{2} e^{2} g^{2}-8 d \,e^{3} f g +4 e^{4} f^{2}\right ) \left (d e g x -e^{2} f x +d^{2} g -d e f \right ) \left (d g -e f \right )}+\frac {4 e^{2} d \,f^{3}}{\left (4 d^{2} e^{2} g^{2}-8 d \,e^{3} f g +4 e^{4} f^{2}\right ) g \left (d e g x -e^{2} f x +d^{2} g -d e f \right ) \left (d g -e f \right )}-\frac {e^{3} f^{4}}{\left (4 d^{2} e^{2} g^{2}-8 d \,e^{3} f g +4 e^{4} f^{2}\right ) g^{2} \left (d e g x -e^{2} f x +d^{2} g -d e f \right ) \left (d g -e f \right )}-\frac {2 \ln \left (e x +d \right ) d}{\left (4 d^{2} e^{2} g^{2}-8 d \,e^{3} f g +4 e^{4} f^{2}\right ) e}+\frac {2 \ln \left (e x +d \right ) f}{\left (4 d^{2} e^{2} g^{2}-8 d \,e^{3} f g +4 e^{4} f^{2}\right ) g}\) | \(493\) |
Input:
int((g*x+f)^2/(-2*d*e*f*g+2*d^2*g^2-2*e*g*(-d*g+e*f)*x)^2,x,method=_RETURN VERBOSE)
Output:
1/4/g^2/(d*g-e*f)^2*(g^2*x/e^2-2/e^3*g*(d*g-e*f)*ln(e*x+d)-(d^2*g^2-2*d*e* f*g+e^2*f^2)/e^3/(e*x+d))
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (56) = 112\).
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.40 \[ \int \frac {(f+g x)^2}{\left (-2 d e f g+2 d^2 g^2-2 e g (e f-d g) x\right )^2} \, dx=\frac {e^{2} g^{2} x^{2} + d e g^{2} x - e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2} + 2 \, {\left (d e f g - d^{2} g^{2} + {\left (e^{2} f g - d e g^{2}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (d e^{5} f^{2} g^{2} - 2 \, d^{2} e^{4} f g^{3} + d^{3} e^{3} g^{4} + {\left (e^{6} f^{2} g^{2} - 2 \, d e^{5} f g^{3} + d^{2} e^{4} g^{4}\right )} x\right )}} \] Input:
integrate((g*x+f)^2/(-2*d*e*f*g+2*d^2*g^2-2*e*g*(-d*g+e*f)*x)^2,x, algorit hm="fricas")
Output:
1/4*(e^2*g^2*x^2 + d*e*g^2*x - e^2*f^2 + 2*d*e*f*g - d^2*g^2 + 2*(d*e*f*g - d^2*g^2 + (e^2*f*g - d*e*g^2)*x)*log(e*x + d))/(d*e^5*f^2*g^2 - 2*d^2*e^ 4*f*g^3 + d^3*e^3*g^4 + (e^6*f^2*g^2 - 2*d*e^5*f*g^3 + d^2*e^4*g^4)*x)
Time = 0.42 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {(f+g x)^2}{\left (-2 d e f g+2 d^2 g^2-2 e g (e f-d g) x\right )^2} \, dx=\frac {x}{4 d^{2} e^{2} g^{2} - 8 d e^{3} f g + 4 e^{4} f^{2}} - \frac {1}{4 d e^{3} g^{2} + 4 e^{4} g^{2} x} - \frac {\log {\left (d + e x \right )}}{2 e^{3} g \left (d g - e f\right )} \] Input:
integrate((g*x+f)**2/(-2*d*e*f*g+2*d**2*g**2-2*e*g*(-d*g+e*f)*x)**2,x)
Output:
x/(4*d**2*e**2*g**2 - 8*d*e**3*f*g + 4*e**4*f**2) - 1/(4*d*e**3*g**2 + 4*e **4*g**2*x) - log(d + e*x)/(2*e**3*g*(d*g - e*f))
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.27 \[ \int \frac {(f+g x)^2}{\left (-2 d e f g+2 d^2 g^2-2 e g (e f-d g) x\right )^2} \, dx=\frac {x}{4 \, {\left (e^{4} f^{2} - 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )}} + \frac {\log \left (e x + d\right )}{2 \, {\left (e^{4} f g - d e^{3} g^{2}\right )}} - \frac {1}{4 \, {\left (e^{4} g^{2} x + d e^{3} g^{2}\right )}} \] Input:
integrate((g*x+f)^2/(-2*d*e*f*g+2*d^2*g^2-2*e*g*(-d*g+e*f)*x)^2,x, algorit hm="maxima")
Output:
1/4*x/(e^4*f^2 - 2*d*e^3*f*g + d^2*e^2*g^2) + 1/2*log(e*x + d)/(e^4*f*g - d*e^3*g^2) - 1/4/(e^4*g^2*x + d*e^3*g^2)
Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (56) = 112\).
Time = 0.17 (sec) , antiderivative size = 553, normalized size of antiderivative = 8.92 \[ \int \frac {(f+g x)^2}{\left (-2 d e f g+2 d^2 g^2-2 e g (e f-d g) x\right )^2} \, dx=-\frac {\frac {e^{7} f^{6} g}{{\left (d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x\right )} {\left (e f - d g\right )}} - \frac {6 \, d e^{6} f^{5} g^{2}}{{\left (d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x\right )} {\left (e f - d g\right )}} + \frac {15 \, d^{2} e^{5} f^{4} g^{3}}{{\left (d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x\right )} {\left (e f - d g\right )}} - \frac {20 \, d^{3} e^{4} f^{3} g^{4}}{{\left (d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x\right )} {\left (e f - d g\right )}} + \frac {15 \, d^{4} e^{3} f^{2} g^{5}}{{\left (d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x\right )} {\left (e f - d g\right )}} - \frac {6 \, d^{5} e^{2} f g^{6}}{{\left (d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x\right )} {\left (e f - d g\right )}} + \frac {d^{6} e g^{7}}{{\left (d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x\right )} {\left (e f - d g\right )}}}{4 \, {\left (e^{8} f^{4} g^{2} - 4 \, d e^{7} f^{3} g^{3} + 6 \, d^{2} e^{6} f^{2} g^{4} - 4 \, d^{3} e^{5} f g^{5} + d^{4} e^{4} g^{6}\right )}} - \frac {\log \left (\frac {{\left | d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x \right |}}{{\left (d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x\right )}^{2} {\left | e f - d g \right |} {\left | e \right |} {\left | g \right |}}\right )}{2 \, {\left (e^{4} f g - d e^{3} g^{2}\right )}} + \frac {d e f g - d^{2} g^{2} + {\left (e f - d g\right )} e g x}{4 \, {\left (e f - d g\right )}^{3} e^{3} g} \] Input:
integrate((g*x+f)^2/(-2*d*e*f*g+2*d^2*g^2-2*e*g*(-d*g+e*f)*x)^2,x, algorit hm="giac")
Output:
-1/4*(e^7*f^6*g/((d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)*(e*f - d*g)) - 6* d*e^6*f^5*g^2/((d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)*(e*f - d*g)) + 15*d ^2*e^5*f^4*g^3/((d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)*(e*f - d*g)) - 20* d^3*e^4*f^3*g^4/((d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)*(e*f - d*g)) + 15 *d^4*e^3*f^2*g^5/((d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)*(e*f - d*g)) - 6 *d^5*e^2*f*g^6/((d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)*(e*f - d*g)) + d^6 *e*g^7/((d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)*(e*f - d*g)))/(e^8*f^4*g^2 - 4*d*e^7*f^3*g^3 + 6*d^2*e^6*f^2*g^4 - 4*d^3*e^5*f*g^5 + d^4*e^4*g^6) - 1/2*log(abs(d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)/((d*e*f*g - d^2*g^2 + ( e*f - d*g)*e*g*x)^2*abs(e*f - d*g)*abs(e)*abs(g)))/(e^4*f*g - d*e^3*g^2) + 1/4*(d*e*f*g - d^2*g^2 + (e*f - d*g)*e*g*x)/((e*f - d*g)^3*e^3*g)
Time = 0.15 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.65 \[ \int \frac {(f+g x)^2}{\left (-2 d e f g+2 d^2 g^2-2 e g (e f-d g) x\right )^2} \, dx=\frac {x}{4\,e^2\,{\left (d\,g-e\,f\right )}^2}-\frac {\ln \left (d+e\,x\right )}{2\,d\,e^3\,g^2-2\,e^4\,f\,g}-\frac {d\,g-e\,f}{e\,\left (x\,\left (4\,d\,e^3\,g^3-4\,e^4\,f\,g^2\right )+4\,d^2\,e^2\,g^3-4\,d\,e^3\,f\,g^2\right )} \] Input:
int((f + g*x)^2/(2*d^2*g^2 + 2*e*g*x*(d*g - e*f) - 2*d*e*f*g)^2,x)
Output:
x/(4*e^2*(d*g - e*f)^2) - log(d + e*x)/(2*d*e^3*g^2 - 2*e^4*f*g) - (d*g - e*f)/(e*(x*(4*d*e^3*g^3 - 4*e^4*f*g^2) + 4*d^2*e^2*g^3 - 4*d*e^3*f*g^2))
Time = 0.17 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.60 \[ \int \frac {(f+g x)^2}{\left (-2 d e f g+2 d^2 g^2-2 e g (e f-d g) x\right )^2} \, dx=\frac {-2 \,\mathrm {log}\left (e x +d \right ) d^{3} g^{2}+2 \,\mathrm {log}\left (e x +d \right ) d^{2} e f g -2 \,\mathrm {log}\left (e x +d \right ) d^{2} e \,g^{2} x +2 \,\mathrm {log}\left (e x +d \right ) d \,e^{2} f g x +2 d^{2} e \,g^{2} x -2 d \,e^{2} f g x +d \,e^{2} g^{2} x^{2}+e^{3} f^{2} x}{4 d \,e^{3} g^{2} \left (d^{2} e \,g^{2} x -2 d \,e^{2} f g x +e^{3} f^{2} x +d^{3} g^{2}-2 d^{2} e f g +d \,e^{2} f^{2}\right )} \] Input:
int((g*x+f)^2/(-2*d*e*f*g+2*d^2*g^2-2*e*g*(-d*g+e*f)*x)^2,x)
Output:
( - 2*log(d + e*x)*d**3*g**2 + 2*log(d + e*x)*d**2*e*f*g - 2*log(d + e*x)* d**2*e*g**2*x + 2*log(d + e*x)*d*e**2*f*g*x + 2*d**2*e*g**2*x - 2*d*e**2*f *g*x + d*e**2*g**2*x**2 + e**3*f**2*x)/(4*d*e**3*g**2*(d**3*g**2 - 2*d**2* e*f*g + d**2*e*g**2*x + d*e**2*f**2 - 2*d*e**2*f*g*x + e**3*f**2*x))