Integrand size = 29, antiderivative size = 188 \[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx=\frac {(e f+d g)^2}{32 d^4 e^3 (d-e x)^2}+\frac {f (e f+d g)}{8 d^5 e^2 (d-e x)}-\frac {(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}-\frac {(e f-d g) (3 e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac {\left (5 e^2 f^2+2 d e f g-d^2 g^2\right ) \text {arctanh}\left (\frac {e x}{d}\right )}{16 d^6 e^3} \] Output:
1/32*(d*g+e*f)^2/d^4/e^3/(-e*x+d)^2+1/8*f*(d*g+e*f)/d^5/e^2/(-e*x+d)-1/24* (-d*g+e*f)^2/d^3/e^3/(e*x+d)^3-1/32*(-d*g+e*f)*(d*g+3*e*f)/d^4/e^3/(e*x+d) ^2-1/16*(-d^2*g^2+3*e^2*f^2)/d^5/e^3/(e*x+d)+1/16*(-d^2*g^2+2*d*e*f*g+5*e^ 2*f^2)*arctanh(e*x/d)/d^6/e^3
Time = 0.10 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.05 \[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx=\frac {\frac {3 d^2 (e f+d g)^2}{(d-e x)^2}+\frac {12 d e f (e f+d g)}{d-e x}-\frac {4 d^3 (e f-d g)^2}{(d+e x)^3}+\frac {3 d^2 \left (-3 e^2 f^2+2 d e f g+d^2 g^2\right )}{(d+e x)^2}+\frac {6 d \left (-3 e^2 f^2+d^2 g^2\right )}{d+e x}+3 \left (-5 e^2 f^2-2 d e f g+d^2 g^2\right ) \log (d-e x)+3 \left (5 e^2 f^2+2 d e f g-d^2 g^2\right ) \log (d+e x)}{96 d^6 e^3} \] Input:
Integrate[(f + g*x)^2/((d + e*x)*(d^2 - e^2*x^2)^3),x]
Output:
((3*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (12*d*e*f*(e*f + d*g))/(d - e*x) - (4 *d^3*(e*f - d*g)^2)/(d + e*x)^3 + (3*d^2*(-3*e^2*f^2 + 2*d*e*f*g + d^2*g^2 ))/(d + e*x)^2 + (6*d*(-3*e^2*f^2 + d^2*g^2))/(d + e*x) + 3*(-5*e^2*f^2 - 2*d*e*f*g + d^2*g^2)*Log[d - e*x] + 3*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*Lo g[d + e*x])/(96*d^6*e^3)
Time = 0.41 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {639, 99, 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.
| \(\displaystyle \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 639 |
| \(\displaystyle \int \frac {(f+g x)^2}{(d-e x)^3 (d+e x)^4}dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {f (d g+e f)}{8 d^5 e (d-e x)^2}+\frac {(d g+e f)^2}{16 d^4 e^2 (d-e x)^3}+\frac {(e f-d g) (d g+3 e f)}{16 d^4 e^2 (d+e x)^3}+\frac {(d g-e f)^2}{8 d^3 e^2 (d+e x)^4}+\frac {d^2 g^2-2 d e f g-5 e^2 f^2}{16 d^5 e^2 \left (e^2 x^2-d^2\right )}+\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^2 (d+e x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}\left (\frac {e x}{d}\right ) \left (-d^2 g^2+2 d e f g+5 e^2 f^2\right )}{16 d^6 e^3}+\frac {f (d g+e f)}{8 d^5 e^2 (d-e x)}-\frac {(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac {(d g+e f)^2}{32 d^4 e^3 (d-e x)^2}-\frac {(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}-\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}\) |
Input:
Int[(f + g*x)^2/((d + e*x)*(d^2 - e^2*x^2)^3),x]
Output:
(e*f + d*g)^2/(32*d^4*e^3*(d - e*x)^2) + (f*(e*f + d*g))/(8*d^5*e^2*(d - e *x)) - (e*f - d*g)^2/(24*d^3*e^3*(d + e*x)^3) - ((e*f - d*g)*(3*e*f + d*g) )/(32*d^4*e^3*(d + e*x)^2) - (3*e^2*f^2 - d^2*g^2)/(16*d^5*e^3*(d + e*x)) + ((5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*ArcTanh[(e*x)/d])/(16*d^6*e^3)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^ 2)^(p_.), x_Symbol] :> Int[(c + d*x)^(m + p)*(e + f*x)^n*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (I ntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[m]))
Time = 0.96 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.30
| method | result | size |
| default | \(-\frac {-d^{2} g^{2}+3 e^{2} f^{2}}{16 d^{5} e^{3} \left (e x +d \right )}-\frac {-d^{2} g^{2}-2 d e f g +3 e^{2} f^{2}}{32 d^{4} e^{3} \left (e x +d \right )^{2}}+\frac {\left (-d^{2} g^{2}+2 d e f g +5 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{32 e^{3} d^{6}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{24 d^{3} e^{3} \left (e x +d \right )^{3}}-\frac {-d^{2} g^{2}-2 d e f g -e^{2} f^{2}}{32 d^{4} e^{3} \left (-e x +d \right )^{2}}+\frac {\left (d^{2} g^{2}-2 d e f g -5 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{32 d^{6} e^{3}}+\frac {f \left (d g +e f \right )}{8 d^{5} e^{2} \left (-e x +d \right )}\) | \(245\) |
| norman | \(\frac {\frac {\left (11 d^{2} g^{2}+26 d e f g -31 e^{2} f^{2}\right ) x^{3}}{48 d^{4}}+\frac {\left (d^{2} g^{2}+14 d e f g +3 e^{2} f^{2}\right ) x^{2}}{16 e \,d^{3}}-\frac {e \left (d^{2} g^{2}+22 d e f g +7 e^{2} f^{2}\right ) x^{4}}{48 d^{5}}-\frac {e^{2} \left (d^{2} g^{2}+4 d e f g -2 e^{2} f^{2}\right ) x^{5}}{12 d^{6}}+\frac {\left (d^{2} g^{2}-2 d e f g +11 e^{2} f^{2}\right ) x}{16 d^{2} e^{2}}}{\left (e x +d \right )^{3} \left (-e x +d \right )^{2}}+\frac {\left (d^{2} g^{2}-2 d e f g -5 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{32 d^{6} e^{3}}-\frac {\left (d^{2} g^{2}-2 d e f g -5 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{32 d^{6} e^{3}}\) | \(251\) |
| risch | \(\frac {\frac {\left (d^{2} g^{2}-2 d e f g -5 e^{2} f^{2}\right ) e \,x^{4}}{16 d^{5}}+\frac {\left (d^{2} g^{2}-2 d e f g -5 e^{2} f^{2}\right ) x^{3}}{16 d^{4}}-\frac {5 \left (d^{2} g^{2}-2 d e f g -5 e^{2} f^{2}\right ) x^{2}}{48 d^{3} e}+\frac {\left (7 d^{2} g^{2}+10 d e f g +25 e^{2} f^{2}\right ) x}{48 d^{2} e^{2}}+\frac {d^{2} g^{2}+4 d e f g -2 e^{2} f^{2}}{12 d \,e^{3}}}{\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{2}}+\frac {\ln \left (e x -d \right ) g^{2}}{32 d^{4} e^{3}}-\frac {\ln \left (e x -d \right ) f g}{16 d^{5} e^{2}}-\frac {5 \ln \left (e x -d \right ) f^{2}}{32 d^{6} e}-\frac {\ln \left (-e x -d \right ) g^{2}}{32 d^{4} e^{3}}+\frac {\ln \left (-e x -d \right ) f g}{16 d^{5} e^{2}}+\frac {5 \ln \left (-e x -d \right ) f^{2}}{32 d^{6} e}\) | \(296\) |
| parallelrisch | \(\frac {-15 \ln \left (e x -d \right ) x^{5} e^{7} f^{2}+15 \ln \left (e x +d \right ) x^{5} e^{7} f^{2}+15 \ln \left (e x +d \right ) d^{5} e^{2} f^{2}+3 \ln \left (e x -d \right ) x \,d^{6} e \,g^{2}-15 \ln \left (e x -d \right ) x \,d^{4} e^{3} f^{2}-6 \ln \left (e x -d \right ) d^{6} e f g +84 x^{2} d^{4} e^{3} f g +52 x^{3} d^{3} e^{4} f g -44 x^{4} d^{2} e^{5} f g -32 x^{5} d \,e^{6} f g +66 x \,d^{4} e^{3} f^{2}+6 x \,d^{6} e \,g^{2}-6 \ln \left (e x -d \right ) x \,d^{5} e^{2} f g -30 \ln \left (e x +d \right ) x^{2} d^{3} e^{4} f^{2}-12 x \,d^{5} e^{2} f g -3 \ln \left (e x +d \right ) x \,d^{6} e \,g^{2}+15 \ln \left (e x +d \right ) x \,d^{4} e^{3} f^{2}+6 \ln \left (e x +d \right ) d^{6} e f g -6 \ln \left (e x -d \right ) x^{2} d^{5} e^{2} g^{2}+30 \ln \left (e x -d \right ) x^{2} d^{3} e^{4} f^{2}+6 \ln \left (e x +d \right ) x^{2} d^{5} e^{2} g^{2}+3 \ln \left (e x -d \right ) x^{5} d^{2} e^{5} g^{2}-3 \ln \left (e x +d \right ) x^{5} d^{2} e^{5} g^{2}+3 \ln \left (e x -d \right ) x^{4} d^{3} e^{4} g^{2}-15 \ln \left (e x -d \right ) x^{4} d \,e^{6} f^{2}-3 \ln \left (e x +d \right ) x^{4} d^{3} e^{4} g^{2}+15 \ln \left (e x +d \right ) x^{4} d \,e^{6} f^{2}-6 \ln \left (e x -d \right ) x^{3} d^{4} e^{3} g^{2}+30 \ln \left (e x -d \right ) x^{3} d^{2} e^{5} f^{2}+6 \ln \left (e x +d \right ) x^{3} d^{4} e^{3} g^{2}-30 \ln \left (e x +d \right ) x^{3} d^{2} e^{5} f^{2}+6 \ln \left (e x +d \right ) x \,d^{5} e^{2} f g -6 \ln \left (e x -d \right ) x^{5} d \,e^{6} f g +6 \ln \left (e x +d \right ) x^{5} d \,e^{6} f g -6 \ln \left (e x -d \right ) x^{4} d^{2} e^{5} f g +6 \ln \left (e x +d \right ) x^{4} d^{2} e^{5} f g +12 \ln \left (e x -d \right ) x^{3} d^{3} e^{4} f g -12 \ln \left (e x +d \right ) x^{3} d^{3} e^{4} f g +12 \ln \left (e x -d \right ) x^{2} d^{4} e^{3} f g -12 \ln \left (e x +d \right ) x^{2} d^{4} e^{3} f g -8 x^{5} d^{2} e^{5} g^{2}-15 \ln \left (e x -d \right ) d^{5} e^{2} f^{2}+6 x^{2} d^{5} e^{2} g^{2}+18 x^{2} d^{3} e^{4} f^{2}+22 x^{3} d^{4} e^{3} g^{2}-62 x^{3} d^{2} e^{5} f^{2}-2 x^{4} d^{3} e^{4} g^{2}-14 x^{4} d \,e^{6} f^{2}+16 x^{5} e^{7} f^{2}+3 \ln \left (e x -d \right ) d^{7} g^{2}-3 \ln \left (e x +d \right ) d^{7} g^{2}}{96 e^{3} d^{6} \left (e x +d \right ) \left (e^{2} x^{2}-d^{2}\right )^{2}}\) | \(908\) |
Input:
int((g*x+f)^2/(e*x+d)/(-e^2*x^2+d^2)^3,x,method=_RETURNVERBOSE)
Output:
-1/16*(-d^2*g^2+3*e^2*f^2)/d^5/e^3/(e*x+d)-1/32*(-d^2*g^2-2*d*e*f*g+3*e^2* f^2)/d^4/e^3/(e*x+d)^2+1/32*(-d^2*g^2+2*d*e*f*g+5*e^2*f^2)/e^3/d^6*ln(e*x+ d)-1/24*(d^2*g^2-2*d*e*f*g+e^2*f^2)/d^3/e^3/(e*x+d)^3-1/32*(-d^2*g^2-2*d*e *f*g-e^2*f^2)/d^4/e^3/(-e*x+d)^2+1/32*(d^2*g^2-2*d*e*f*g-5*e^2*f^2)/d^6/e^ 3*ln(-e*x+d)+1/8*f*(d*g+e*f)/d^5/e^2/(-e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (178) = 356\).
Time = 0.09 (sec) , antiderivative size = 662, normalized size of antiderivative = 3.52 \[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx=-\frac {16 \, d^{5} e^{2} f^{2} - 32 \, d^{6} e f g - 8 \, d^{7} g^{2} + 6 \, {\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} + 6 \, {\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 10 \, {\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (25 \, d^{4} e^{3} f^{2} + 10 \, d^{5} e^{2} f g + 7 \, d^{6} e g^{2}\right )} x - 3 \, {\left (5 \, d^{5} e^{2} f^{2} + 2 \, d^{6} e f g - d^{7} g^{2} + {\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} + {\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \, {\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} + {\left (5 \, d^{4} e^{3} f^{2} + 2 \, d^{5} e^{2} f g - d^{6} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (5 \, d^{5} e^{2} f^{2} + 2 \, d^{6} e f g - d^{7} g^{2} + {\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} + {\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \, {\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} + {\left (5 \, d^{4} e^{3} f^{2} + 2 \, d^{5} e^{2} f g - d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \, {\left (d^{6} e^{8} x^{5} + d^{7} e^{7} x^{4} - 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} + d^{10} e^{4} x + d^{11} e^{3}\right )}} \] Input:
integrate((g*x+f)^2/(e*x+d)/(-e^2*x^2+d^2)^3,x, algorithm="fricas")
Output:
-1/96*(16*d^5*e^2*f^2 - 32*d^6*e*f*g - 8*d^7*g^2 + 6*(5*d*e^6*f^2 + 2*d^2* e^5*f*g - d^3*e^4*g^2)*x^4 + 6*(5*d^2*e^5*f^2 + 2*d^3*e^4*f*g - d^4*e^3*g^ 2)*x^3 - 10*(5*d^3*e^4*f^2 + 2*d^4*e^3*f*g - d^5*e^2*g^2)*x^2 - 2*(25*d^4* e^3*f^2 + 10*d^5*e^2*f*g + 7*d^6*e*g^2)*x - 3*(5*d^5*e^2*f^2 + 2*d^6*e*f*g - d^7*g^2 + (5*e^7*f^2 + 2*d*e^6*f*g - d^2*e^5*g^2)*x^5 + (5*d*e^6*f^2 + 2*d^2*e^5*f*g - d^3*e^4*g^2)*x^4 - 2*(5*d^2*e^5*f^2 + 2*d^3*e^4*f*g - d^4* e^3*g^2)*x^3 - 2*(5*d^3*e^4*f^2 + 2*d^4*e^3*f*g - d^5*e^2*g^2)*x^2 + (5*d^ 4*e^3*f^2 + 2*d^5*e^2*f*g - d^6*e*g^2)*x)*log(e*x + d) + 3*(5*d^5*e^2*f^2 + 2*d^6*e*f*g - d^7*g^2 + (5*e^7*f^2 + 2*d*e^6*f*g - d^2*e^5*g^2)*x^5 + (5 *d*e^6*f^2 + 2*d^2*e^5*f*g - d^3*e^4*g^2)*x^4 - 2*(5*d^2*e^5*f^2 + 2*d^3*e ^4*f*g - d^4*e^3*g^2)*x^3 - 2*(5*d^3*e^4*f^2 + 2*d^4*e^3*f*g - d^5*e^2*g^2 )*x^2 + (5*d^4*e^3*f^2 + 2*d^5*e^2*f*g - d^6*e*g^2)*x)*log(e*x - d))/(d^6* e^8*x^5 + d^7*e^7*x^4 - 2*d^8*e^6*x^3 - 2*d^9*e^5*x^2 + d^10*e^4*x + d^11* e^3)
Time = 1.01 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.71 \[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx=- \frac {- 4 d^{6} g^{2} - 16 d^{5} e f g + 8 d^{4} e^{2} f^{2} + x^{4} \left (- 3 d^{2} e^{4} g^{2} + 6 d e^{5} f g + 15 e^{6} f^{2}\right ) + x^{3} \left (- 3 d^{3} e^{3} g^{2} + 6 d^{2} e^{4} f g + 15 d e^{5} f^{2}\right ) + x^{2} \cdot \left (5 d^{4} e^{2} g^{2} - 10 d^{3} e^{3} f g - 25 d^{2} e^{4} f^{2}\right ) + x \left (- 7 d^{5} e g^{2} - 10 d^{4} e^{2} f g - 25 d^{3} e^{3} f^{2}\right )}{48 d^{10} e^{3} + 48 d^{9} e^{4} x - 96 d^{8} e^{5} x^{2} - 96 d^{7} e^{6} x^{3} + 48 d^{6} e^{7} x^{4} + 48 d^{5} e^{8} x^{5}} + \frac {\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log {\left (- \frac {d}{e} + x \right )}}{32 d^{6} e^{3}} - \frac {\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log {\left (\frac {d}{e} + x \right )}}{32 d^{6} e^{3}} \] Input:
integrate((g*x+f)**2/(e*x+d)/(-e**2*x**2+d**2)**3,x)
Output:
-(-4*d**6*g**2 - 16*d**5*e*f*g + 8*d**4*e**2*f**2 + x**4*(-3*d**2*e**4*g** 2 + 6*d*e**5*f*g + 15*e**6*f**2) + x**3*(-3*d**3*e**3*g**2 + 6*d**2*e**4*f *g + 15*d*e**5*f**2) + x**2*(5*d**4*e**2*g**2 - 10*d**3*e**3*f*g - 25*d**2 *e**4*f**2) + x*(-7*d**5*e*g**2 - 10*d**4*e**2*f*g - 25*d**3*e**3*f**2))/( 48*d**10*e**3 + 48*d**9*e**4*x - 96*d**8*e**5*x**2 - 96*d**7*e**6*x**3 + 4 8*d**6*e**7*x**4 + 48*d**5*e**8*x**5) + (d**2*g**2 - 2*d*e*f*g - 5*e**2*f* *2)*log(-d/e + x)/(32*d**6*e**3) - (d**2*g**2 - 2*d*e*f*g - 5*e**2*f**2)*l og(d/e + x)/(32*d**6*e**3)
Time = 0.05 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.64 \[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx=-\frac {8 \, d^{4} e^{2} f^{2} - 16 \, d^{5} e f g - 4 \, d^{6} g^{2} + 3 \, {\left (5 \, e^{6} f^{2} + 2 \, d e^{5} f g - d^{2} e^{4} g^{2}\right )} x^{4} + 3 \, {\left (5 \, d e^{5} f^{2} + 2 \, d^{2} e^{4} f g - d^{3} e^{3} g^{2}\right )} x^{3} - 5 \, {\left (5 \, d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g - d^{4} e^{2} g^{2}\right )} x^{2} - {\left (25 \, d^{3} e^{3} f^{2} + 10 \, d^{4} e^{2} f g + 7 \, d^{5} e g^{2}\right )} x}{48 \, {\left (d^{5} e^{8} x^{5} + d^{6} e^{7} x^{4} - 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} + d^{9} e^{4} x + d^{10} e^{3}\right )}} + \frac {{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{6} e^{3}} - \frac {{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, d^{6} e^{3}} \] Input:
integrate((g*x+f)^2/(e*x+d)/(-e^2*x^2+d^2)^3,x, algorithm="maxima")
Output:
-1/48*(8*d^4*e^2*f^2 - 16*d^5*e*f*g - 4*d^6*g^2 + 3*(5*e^6*f^2 + 2*d*e^5*f *g - d^2*e^4*g^2)*x^4 + 3*(5*d*e^5*f^2 + 2*d^2*e^4*f*g - d^3*e^3*g^2)*x^3 - 5*(5*d^2*e^4*f^2 + 2*d^3*e^3*f*g - d^4*e^2*g^2)*x^2 - (25*d^3*e^3*f^2 + 10*d^4*e^2*f*g + 7*d^5*e*g^2)*x)/(d^5*e^8*x^5 + d^6*e^7*x^4 - 2*d^7*e^6*x^ 3 - 2*d^8*e^5*x^2 + d^9*e^4*x + d^10*e^3) + 1/32*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*log(e*x + d)/(d^6*e^3) - 1/32*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*l og(e*x - d)/(d^6*e^3)
Time = 0.16 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.47 \[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx=\frac {{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{32 \, d^{6} e^{3}} - \frac {{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{32 \, d^{6} e^{3}} - \frac {8 \, d^{5} e^{2} f^{2} - 16 \, d^{6} e f g - 4 \, d^{7} g^{2} + 3 \, {\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} + 3 \, {\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 5 \, {\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} - {\left (25 \, d^{4} e^{3} f^{2} + 10 \, d^{5} e^{2} f g + 7 \, d^{6} e g^{2}\right )} x}{48 \, {\left (e x + d\right )}^{3} {\left (e x - d\right )}^{2} d^{6} e^{3}} \] Input:
integrate((g*x+f)^2/(e*x+d)/(-e^2*x^2+d^2)^3,x, algorithm="giac")
Output:
1/32*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*log(abs(e*x + d))/(d^6*e^3) - 1/32* (5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*log(abs(e*x - d))/(d^6*e^3) - 1/48*(8*d^ 5*e^2*f^2 - 16*d^6*e*f*g - 4*d^7*g^2 + 3*(5*d*e^6*f^2 + 2*d^2*e^5*f*g - d^ 3*e^4*g^2)*x^4 + 3*(5*d^2*e^5*f^2 + 2*d^3*e^4*f*g - d^4*e^3*g^2)*x^3 - 5*( 5*d^3*e^4*f^2 + 2*d^4*e^3*f*g - d^5*e^2*g^2)*x^2 - (25*d^4*e^3*f^2 + 10*d^ 5*e^2*f*g + 7*d^6*e*g^2)*x)/((e*x + d)^3*(e*x - d)^2*d^6*e^3)
Time = 0.10 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.32 \[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx=\frac {\frac {d^2\,g^2+4\,d\,e\,f\,g-2\,e^2\,f^2}{12\,d\,e^3}-\frac {x^3\,\left (-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2\right )}{16\,d^4}-\frac {e\,x^4\,\left (-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2\right )}{16\,d^5}+\frac {x\,\left (7\,d^2\,g^2+10\,d\,e\,f\,g+25\,e^2\,f^2\right )}{48\,d^2\,e^2}+\frac {5\,x^2\,\left (-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2\right )}{48\,d^3\,e}}{d^5+d^4\,e\,x-2\,d^3\,e^2\,x^2-2\,d^2\,e^3\,x^3+d\,e^4\,x^4+e^5\,x^5}+\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,\left (-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2\right )}{16\,d^6\,e^3} \] Input:
int((f + g*x)^2/((d^2 - e^2*x^2)^3*(d + e*x)),x)
Output:
((d^2*g^2 - 2*e^2*f^2 + 4*d*e*f*g)/(12*d*e^3) - (x^3*(5*e^2*f^2 - d^2*g^2 + 2*d*e*f*g))/(16*d^4) - (e*x^4*(5*e^2*f^2 - d^2*g^2 + 2*d*e*f*g))/(16*d^5 ) + (x*(7*d^2*g^2 + 25*e^2*f^2 + 10*d*e*f*g))/(48*d^2*e^2) + (5*x^2*(5*e^2 *f^2 - d^2*g^2 + 2*d*e*f*g))/(48*d^3*e))/(d^5 + e^5*x^5 + d*e^4*x^4 - 2*d^ 3*e^2*x^2 - 2*d^2*e^3*x^3 + d^4*e*x) + (atanh((e*x)/d)*(5*e^2*f^2 - d^2*g^ 2 + 2*d*e*f*g))/(16*d^6*e^3)
Time = 0.21 (sec) , antiderivative size = 904, normalized size of antiderivative = 4.81 \[ \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((g*x+f)^2/(e*x+d)/(-e^2*x^2+d^2)^3,x)
Output:
(3*log(d - e*x)*d**7*g**2 - 6*log(d - e*x)*d**6*e*f*g + 3*log(d - e*x)*d** 6*e*g**2*x - 15*log(d - e*x)*d**5*e**2*f**2 - 6*log(d - e*x)*d**5*e**2*f*g *x - 6*log(d - e*x)*d**5*e**2*g**2*x**2 - 15*log(d - e*x)*d**4*e**3*f**2*x + 12*log(d - e*x)*d**4*e**3*f*g*x**2 - 6*log(d - e*x)*d**4*e**3*g**2*x**3 + 30*log(d - e*x)*d**3*e**4*f**2*x**2 + 12*log(d - e*x)*d**3*e**4*f*g*x** 3 + 3*log(d - e*x)*d**3*e**4*g**2*x**4 + 30*log(d - e*x)*d**2*e**5*f**2*x* *3 - 6*log(d - e*x)*d**2*e**5*f*g*x**4 + 3*log(d - e*x)*d**2*e**5*g**2*x** 5 - 15*log(d - e*x)*d*e**6*f**2*x**4 - 6*log(d - e*x)*d*e**6*f*g*x**5 - 15 *log(d - e*x)*e**7*f**2*x**5 - 3*log(d + e*x)*d**7*g**2 + 6*log(d + e*x)*d **6*e*f*g - 3*log(d + e*x)*d**6*e*g**2*x + 15*log(d + e*x)*d**5*e**2*f**2 + 6*log(d + e*x)*d**5*e**2*f*g*x + 6*log(d + e*x)*d**5*e**2*g**2*x**2 + 15 *log(d + e*x)*d**4*e**3*f**2*x - 12*log(d + e*x)*d**4*e**3*f*g*x**2 + 6*lo g(d + e*x)*d**4*e**3*g**2*x**3 - 30*log(d + e*x)*d**3*e**4*f**2*x**2 - 12* log(d + e*x)*d**3*e**4*f*g*x**3 - 3*log(d + e*x)*d**3*e**4*g**2*x**4 - 30* log(d + e*x)*d**2*e**5*f**2*x**3 + 6*log(d + e*x)*d**2*e**5*f*g*x**4 - 3*l og(d + e*x)*d**2*e**5*g**2*x**5 + 15*log(d + e*x)*d*e**6*f**2*x**4 + 6*log (d + e*x)*d*e**6*f*g*x**5 + 15*log(d + e*x)*e**7*f**2*x**5 + 2*d**7*g**2 + 44*d**6*e*f*g + 8*d**6*e*g**2*x + 14*d**5*e**2*f**2 + 32*d**5*e**2*f*g*x + 2*d**5*e**2*g**2*x**2 + 80*d**4*e**3*f**2*x - 4*d**4*e**3*f*g*x**2 + 18* d**4*e**3*g**2*x**3 - 10*d**3*e**4*f**2*x**2 - 36*d**3*e**4*f*g*x**3 - ...