Integrand size = 29, antiderivative size = 139 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=-\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4}-\frac {(e f-d g) (e f+3 d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f+d g)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(e f+d g)^2}{16 d^4 e^3 (d+e x)}+\frac {(e f+d g)^2 \text {arctanh}\left (\frac {e x}{d}\right )}{16 d^5 e^3} \] Output:
-1/8*(-d*g+e*f)^2/d/e^3/(e*x+d)^4-1/12*(-d*g+e*f)*(3*d*g+e*f)/d^2/e^3/(e*x +d)^3-1/16*(d*g+e*f)^2/d^3/e^3/(e*x+d)^2-1/16*(d*g+e*f)^2/d^4/e^3/(e*x+d)+ 1/16*(d*g+e*f)^2*arctanh(e*x/d)/d^5/e^3
Time = 0.06 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.02 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=-\frac {\frac {12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac {8 d^3 \left (e^2 f^2+2 d e f g-3 d^2 g^2\right )}{(d+e x)^3}+\frac {6 d^2 (e f+d g)^2}{(d+e x)^2}+\frac {6 d (e f+d g)^2}{d+e x}+3 (e f+d g)^2 \log (d-e x)-3 (e f+d g)^2 \log (d+e x)}{96 d^5 e^3} \] Input:
Integrate[(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)),x]
Output:
-1/96*((12*d^4*(e*f - d*g)^2)/(d + e*x)^4 + (8*d^3*(e^2*f^2 + 2*d*e*f*g - 3*d^2*g^2))/(d + e*x)^3 + (6*d^2*(e*f + d*g)^2)/(d + e*x)^2 + (6*d*(e*f + d*g)^2)/(d + e*x) + 3*(e*f + d*g)^2*Log[d - e*x] - 3*(e*f + d*g)^2*Log[d + e*x])/(d^5*e^3)
Time = 0.31 (sec) , antiderivative size = 139, 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.
| \(\displaystyle \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 639 |
| \(\displaystyle \int \frac {(f+g x)^2}{(d-e x) (d+e x)^5}dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {(d g+e f)^2}{16 d^4 e^2 (d+e x)^2}+\frac {(d g+e f)^2}{8 d^3 e^2 (d+e x)^3}+\frac {(e f-d g) (3 d g+e f)}{4 d^2 e^2 (d+e x)^4}+\frac {(d g+e f)^2}{16 d^4 e^2 \left (d^2-e^2 x^2\right )}+\frac {(d g-e f)^2}{2 d e^2 (d+e x)^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}\left (\frac {e x}{d}\right ) (d g+e f)^2}{16 d^5 e^3}-\frac {(d g+e f)^2}{16 d^4 e^3 (d+e x)}-\frac {(d g+e f)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(3 d g+e f) (e f-d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4}\) |
Input:
Int[(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)),x]
Output:
-1/8*(e*f - d*g)^2/(d*e^3*(d + e*x)^4) - ((e*f - d*g)*(e*f + 3*d*g))/(12*d ^2*e^3*(d + e*x)^3) - (e*f + d*g)^2/(16*d^3*e^3*(d + e*x)^2) - (e*f + d*g) ^2/(16*d^4*e^3*(d + e*x)) + ((e*f + d*g)^2*ArcTanh[(e*x)/d])/(16*d^5*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 = 0.94 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.43
| method | result | size |
| norman | \(\frac {-\frac {\left (3 d^{2} g^{2}-26 d e f g -61 e^{2} f^{2}\right ) x^{3}}{48 d^{4}}-\frac {\left (d^{2} g^{2}-2 d e f g -7 e^{2} f^{2}\right ) x^{2}}{4 e \,d^{3}}+\frac {e^{2} \left (d f g +2 e \,f^{2}\right ) x^{4}}{6 d^{5}}-\frac {\left (d^{2} g^{2}+2 d e f g -15 e^{2} f^{2}\right ) x}{16 d^{2} e^{2}}}{\left (e x +d \right )^{4}}-\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{32 e^{3} d^{5}}+\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (e x +d \right )}{32 e^{3} d^{5}}\) | \(199\) |
| default | \(-\frac {-3 d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{12 d^{2} e^{3} \left (e x +d \right )^{3}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{8 e^{3} d \left (e x +d \right )^{4}}+\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (e x +d \right )}{32 e^{3} d^{5}}-\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{16 d^{4} e^{3} \left (e x +d \right )}-\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{16 d^{3} e^{3} \left (e x +d \right )^{2}}+\frac {\left (-d^{2} g^{2}-2 d e f g -e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{32 e^{3} d^{5}}\) | \(220\) |
| risch | \(\frac {-\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) x^{3}}{16 d^{4}}-\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) x^{2}}{4 d^{3} e}-\frac {\left (3 d^{2} g^{2}+38 d e f g +19 e^{2} f^{2}\right ) x}{48 d^{2} e^{2}}-\frac {f \left (d g +2 e f \right )}{6 e^{2} d}}{\left (e x +d \right )^{4}}-\frac {\ln \left (-e x +d \right ) g^{2}}{32 e^{3} d^{3}}-\frac {\ln \left (-e x +d \right ) f g}{16 e^{2} d^{4}}-\frac {\ln \left (-e x +d \right ) f^{2}}{32 e \,d^{5}}+\frac {\ln \left (e x +d \right ) g^{2}}{32 e^{3} d^{3}}+\frac {\ln \left (e x +d \right ) f g}{16 e^{2} d^{4}}+\frac {\ln \left (e x +d \right ) f^{2}}{32 e \,d^{5}}\) | \(224\) |
| parallelrisch | \(-\frac {-18 \ln \left (e x +d \right ) x^{2} d^{2} e^{4} f^{2}+12 \ln \left (e x -d \right ) x \,d^{5} e \,g^{2}+12 \ln \left (e x -d \right ) x \,d^{3} e^{3} f^{2}-12 \ln \left (e x +d \right ) x \,d^{5} e \,g^{2}+18 \ln \left (e x -d \right ) x^{2} d^{2} e^{4} f^{2}-16 d \,e^{5} f g \,x^{4}-52 d^{2} e^{4} f g \,x^{3}-48 d^{3} e^{3} f g \,x^{2}+12 d^{4} e^{2} f g x +12 \ln \left (e x -d \right ) x^{3} d \,e^{5} f^{2}-12 \ln \left (e x +d \right ) x^{3} d^{3} e^{3} g^{2}-12 \ln \left (e x +d \right ) x \,d^{3} e^{3} f^{2}+6 d^{5} e \,g^{2} x -90 d^{3} e^{3} f^{2} x -168 d^{2} e^{4} f^{2} x^{2}+24 d^{4} e^{2} g^{2} x^{2}-6 \ln \left (e x +d \right ) x^{4} d \,e^{5} f g +24 \ln \left (e x -d \right ) x^{3} d^{2} e^{4} f g +18 \ln \left (e x -d \right ) x^{2} d^{4} e^{2} g^{2}+12 \ln \left (e x -d \right ) x^{3} d^{3} e^{3} g^{2}+3 \ln \left (e x -d \right ) x^{4} d^{2} e^{4} g^{2}-3 \ln \left (e x +d \right ) x^{4} e^{6} f^{2}+3 \ln \left (e x -d \right ) d^{4} e^{2} f^{2}-3 \ln \left (e x +d \right ) d^{4} e^{2} f^{2}-18 \ln \left (e x +d \right ) x^{2} d^{4} e^{2} g^{2}-3 \ln \left (e x +d \right ) x^{4} d^{2} e^{4} g^{2}+6 d^{3} e^{3} g^{2} x^{3}-122 d \,e^{5} f^{2} x^{3}+6 \ln \left (e x -d \right ) d^{5} e f g -6 \ln \left (e x +d \right ) d^{5} e f g -12 \ln \left (e x +d \right ) x^{3} d \,e^{5} f^{2}-24 \ln \left (e x +d \right ) x^{3} d^{2} e^{4} f g +36 \ln \left (e x -d \right ) x^{2} d^{3} e^{3} f g -36 \ln \left (e x +d \right ) x^{2} d^{3} e^{3} f g +24 \ln \left (e x -d \right ) x \,d^{4} e^{2} f g -24 \ln \left (e x +d \right ) x \,d^{4} e^{2} f g +3 \ln \left (e x -d \right ) x^{4} e^{6} f^{2}+6 \ln \left (e x -d \right ) x^{4} d \,e^{5} f g -32 e^{6} f^{2} x^{4}+3 \ln \left (e x -d \right ) d^{6} g^{2}-3 \ln \left (e x +d \right ) d^{6} g^{2}}{96 e^{3} d^{5} \left (e x +d \right )^{4}}\) | \(714\) |
Input:
int((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2),x,method=_RETURNVERBOSE)
Output:
(-1/48*(3*d^2*g^2-26*d*e*f*g-61*e^2*f^2)/d^4*x^3-1/4/e*(d^2*g^2-2*d*e*f*g- 7*e^2*f^2)/d^3*x^2+1/6*e^2*(d*f*g+2*e*f^2)/d^5*x^4-1/16*(d^2*g^2+2*d*e*f*g -15*e^2*f^2)/d^2/e^2*x)/(e*x+d)^4-1/32*(d^2*g^2+2*d*e*f*g+e^2*f^2)/e^3/d^5 *ln(-e*x+d)+1/32*(d^2*g^2+2*d*e*f*g+e^2*f^2)/e^3/d^5*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (129) = 258\).
Time = 0.08 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.68 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=-\frac {32 \, d^{4} e^{2} f^{2} + 16 \, d^{5} e f g + 6 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 24 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (19 \, d^{3} e^{3} f^{2} + 38 \, d^{4} e^{2} f g + 3 \, d^{5} e g^{2}\right )} x - 3 \, {\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2} + {\left (e^{6} f^{2} + 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} + 4 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 6 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 4 \, {\left (d^{3} e^{3} f^{2} + 2 \, d^{4} e^{2} f g + d^{5} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2} + {\left (e^{6} f^{2} + 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} + 4 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 6 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 4 \, {\left (d^{3} e^{3} f^{2} + 2 \, d^{4} e^{2} f g + d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \, {\left (d^{5} e^{7} x^{4} + 4 \, d^{6} e^{6} x^{3} + 6 \, d^{7} e^{5} x^{2} + 4 \, d^{8} e^{4} x + d^{9} e^{3}\right )}} \] Input:
integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2),x, algorithm="fricas")
Output:
-1/96*(32*d^4*e^2*f^2 + 16*d^5*e*f*g + 6*(d*e^5*f^2 + 2*d^2*e^4*f*g + d^3* e^3*g^2)*x^3 + 24*(d^2*e^4*f^2 + 2*d^3*e^3*f*g + d^4*e^2*g^2)*x^2 + 2*(19* d^3*e^3*f^2 + 38*d^4*e^2*f*g + 3*d^5*e*g^2)*x - 3*(d^4*e^2*f^2 + 2*d^5*e*f *g + d^6*g^2 + (e^6*f^2 + 2*d*e^5*f*g + d^2*e^4*g^2)*x^4 + 4*(d*e^5*f^2 + 2*d^2*e^4*f*g + d^3*e^3*g^2)*x^3 + 6*(d^2*e^4*f^2 + 2*d^3*e^3*f*g + d^4*e^ 2*g^2)*x^2 + 4*(d^3*e^3*f^2 + 2*d^4*e^2*f*g + d^5*e*g^2)*x)*log(e*x + d) + 3*(d^4*e^2*f^2 + 2*d^5*e*f*g + d^6*g^2 + (e^6*f^2 + 2*d*e^5*f*g + d^2*e^4 *g^2)*x^4 + 4*(d*e^5*f^2 + 2*d^2*e^4*f*g + d^3*e^3*g^2)*x^3 + 6*(d^2*e^4*f ^2 + 2*d^3*e^3*f*g + d^4*e^2*g^2)*x^2 + 4*(d^3*e^3*f^2 + 2*d^4*e^2*f*g + d ^5*e*g^2)*x)*log(e*x - d))/(d^5*e^7*x^4 + 4*d^6*e^6*x^3 + 6*d^7*e^5*x^2 + 4*d^8*e^4*x + d^9*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (122) = 244\).
Time = 0.76 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.03 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=- \frac {8 d^{4} f g + 16 d^{3} e f^{2} + x^{3} \cdot \left (3 d^{2} e^{2} g^{2} + 6 d e^{3} f g + 3 e^{4} f^{2}\right ) + x^{2} \cdot \left (12 d^{3} e g^{2} + 24 d^{2} e^{2} f g + 12 d e^{3} f^{2}\right ) + x \left (3 d^{4} g^{2} + 38 d^{3} e f g + 19 d^{2} e^{2} f^{2}\right )}{48 d^{8} e^{2} + 192 d^{7} e^{3} x + 288 d^{6} e^{4} x^{2} + 192 d^{5} e^{5} x^{3} + 48 d^{4} e^{6} x^{4}} - \frac {\left (d g + e f\right )^{2} \log {\left (- \frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{32 d^{5} e^{3}} + \frac {\left (d g + e f\right )^{2} \log {\left (\frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{32 d^{5} e^{3}} \] Input:
integrate((g*x+f)**2/(e*x+d)**4/(-e**2*x**2+d**2),x)
Output:
-(8*d**4*f*g + 16*d**3*e*f**2 + x**3*(3*d**2*e**2*g**2 + 6*d*e**3*f*g + 3* e**4*f**2) + x**2*(12*d**3*e*g**2 + 24*d**2*e**2*f*g + 12*d*e**3*f**2) + x *(3*d**4*g**2 + 38*d**3*e*f*g + 19*d**2*e**2*f**2))/(48*d**8*e**2 + 192*d* *7*e**3*x + 288*d**6*e**4*x**2 + 192*d**5*e**5*x**3 + 48*d**4*e**6*x**4) - (d*g + e*f)**2*log(-d*(d*g + e*f)**2/(e*(d**2*g**2 + 2*d*e*f*g + e**2*f** 2)) + x)/(32*d**5*e**3) + (d*g + e*f)**2*log(d*(d*g + e*f)**2/(e*(d**2*g** 2 + 2*d*e*f*g + e**2*f**2)) + x)/(32*d**5*e**3)
Time = 0.05 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.70 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=-\frac {16 \, d^{3} e f^{2} + 8 \, d^{4} f g + 3 \, {\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{3} + 12 \, {\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x^{2} + {\left (19 \, d^{2} e^{2} f^{2} + 38 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} x}{48 \, {\left (d^{4} e^{6} x^{4} + 4 \, d^{5} e^{5} x^{3} + 6 \, d^{6} e^{4} x^{2} + 4 \, d^{7} e^{3} x + d^{8} e^{2}\right )}} + \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{5} e^{3}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, d^{5} e^{3}} \] Input:
integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2),x, algorithm="maxima")
Output:
-1/48*(16*d^3*e*f^2 + 8*d^4*f*g + 3*(e^4*f^2 + 2*d*e^3*f*g + d^2*e^2*g^2)* x^3 + 12*(d*e^3*f^2 + 2*d^2*e^2*f*g + d^3*e*g^2)*x^2 + (19*d^2*e^2*f^2 + 3 8*d^3*e*f*g + 3*d^4*g^2)*x)/(d^4*e^6*x^4 + 4*d^5*e^5*x^3 + 6*d^6*e^4*x^2 + 4*d^7*e^3*x + d^8*e^2) + 1/32*(e^2*f^2 + 2*d*e*f*g + d^2*g^2)*log(e*x + d )/(d^5*e^3) - 1/32*(e^2*f^2 + 2*d*e*f*g + d^2*g^2)*log(e*x - d)/(d^5*e^3)
Time = 0.14 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.53 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=\frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{32 \, d^{5} e^{3}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{32 \, d^{5} e^{3}} - \frac {16 \, d^{4} e^{2} f^{2} + 8 \, d^{5} e f g + 3 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 12 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + {\left (19 \, d^{3} e^{3} f^{2} + 38 \, d^{4} e^{2} f g + 3 \, d^{5} e g^{2}\right )} x}{48 \, {\left (e x + d\right )}^{4} d^{5} e^{3}} \] Input:
integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2),x, algorithm="giac")
Output:
1/32*(e^2*f^2 + 2*d*e*f*g + d^2*g^2)*log(abs(e*x + d))/(d^5*e^3) - 1/32*(e ^2*f^2 + 2*d*e*f*g + d^2*g^2)*log(abs(e*x - d))/(d^5*e^3) - 1/48*(16*d^4*e ^2*f^2 + 8*d^5*e*f*g + 3*(d*e^5*f^2 + 2*d^2*e^4*f*g + d^3*e^3*g^2)*x^3 + 1 2*(d^2*e^4*f^2 + 2*d^3*e^3*f*g + d^4*e^2*g^2)*x^2 + (19*d^3*e^3*f^2 + 38*d ^4*e^2*f*g + 3*d^5*e*g^2)*x)/((e*x + d)^4*d^5*e^3)
Time = 6.59 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.29 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,{\left (d\,g+e\,f\right )}^2}{16\,d^5\,e^3}-\frac {\frac {x^3\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{16\,d^4}+\frac {2\,e\,f^2+d\,g\,f}{6\,d\,e^2}+\frac {x\,\left (3\,d^2\,g^2+38\,d\,e\,f\,g+19\,e^2\,f^2\right )}{48\,d^2\,e^2}+\frac {x^2\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{4\,d^3\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \] Input:
int((f + g*x)^2/((d^2 - e^2*x^2)*(d + e*x)^4),x)
Output:
(atanh((e*x)/d)*(d*g + e*f)^2)/(16*d^5*e^3) - ((x^3*(d^2*g^2 + e^2*f^2 + 2 *d*e*f*g))/(16*d^4) + (2*e*f^2 + d*f*g)/(6*d*e^2) + (x*(3*d^2*g^2 + 19*e^2 *f^2 + 38*d*e*f*g))/(48*d^2*e^2) + (x^2*(d^2*g^2 + e^2*f^2 + 2*d*e*f*g))/( 4*d^3*e))/(d^4 + e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x)
Time = 0.23 (sec) , antiderivative size = 723, normalized size of antiderivative = 5.20 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx =\text {Too large to display} \] Input:
int((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2),x)
Output:
( - 6*log(d - e*x)*d**6*g**2 - 12*log(d - e*x)*d**5*e*f*g - 24*log(d - e*x )*d**5*e*g**2*x - 6*log(d - e*x)*d**4*e**2*f**2 - 48*log(d - e*x)*d**4*e** 2*f*g*x - 36*log(d - e*x)*d**4*e**2*g**2*x**2 - 24*log(d - e*x)*d**3*e**3* f**2*x - 72*log(d - e*x)*d**3*e**3*f*g*x**2 - 24*log(d - e*x)*d**3*e**3*g* *2*x**3 - 36*log(d - e*x)*d**2*e**4*f**2*x**2 - 48*log(d - e*x)*d**2*e**4* f*g*x**3 - 6*log(d - e*x)*d**2*e**4*g**2*x**4 - 24*log(d - e*x)*d*e**5*f** 2*x**3 - 12*log(d - e*x)*d*e**5*f*g*x**4 - 6*log(d - e*x)*e**6*f**2*x**4 + 6*log(d + e*x)*d**6*g**2 + 12*log(d + e*x)*d**5*e*f*g + 24*log(d + e*x)*d **5*e*g**2*x + 6*log(d + e*x)*d**4*e**2*f**2 + 48*log(d + e*x)*d**4*e**2*f *g*x + 36*log(d + e*x)*d**4*e**2*g**2*x**2 + 24*log(d + e*x)*d**3*e**3*f** 2*x + 72*log(d + e*x)*d**3*e**3*f*g*x**2 + 24*log(d + e*x)*d**3*e**3*g**2* x**3 + 36*log(d + e*x)*d**2*e**4*f**2*x**2 + 48*log(d + e*x)*d**2*e**4*f*g *x**3 + 6*log(d + e*x)*d**2*e**4*g**2*x**4 + 24*log(d + e*x)*d*e**5*f**2*x **3 + 12*log(d + e*x)*d*e**5*f*g*x**4 + 6*log(d + e*x)*e**6*f**2*x**4 + 3* d**6*g**2 - 26*d**5*e*f*g - 61*d**4*e**2*f**2 - 128*d**4*e**2*f*g*x - 30*d **4*e**2*g**2*x**2 - 64*d**3*e**3*f**2*x - 60*d**3*e**3*f*g*x**2 - 30*d**2 *e**4*f**2*x**2 + 3*d**2*e**4*g**2*x**4 + 6*d*e**5*f*g*x**4 + 3*e**6*f**2* x**4)/(192*d**5*e**3*(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4))