\(\int \frac {(f+g x)^2}{(d+e x)^4 (d^2-e^2 x^2)^2} \, dx\) [568]

Optimal result

Integrand size = 29, antiderivative size = 210 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx=\frac {(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac {(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac {(e f+d g) (3 e f+d g) \text {arctanh}\left (\frac {e x}{d}\right )}{32 d^7 e^3} \]

[Out]

1/64*(d*g+e*f)^2/d^6/e^3/(-e*x+d)-1/20*(-d*g+e*f)^2/d^2/e^3/(e*x+d)^5+1/16*(d^2*g^2-e^2*f^2)/d^3/e^3/(e*x+d)^4
-1/48*(-d*g+3*e*f)*(d*g+e*f)/d^4/e^3/(e*x+d)^3-1/16*f*(d*g+e*f)/d^5/e^2/(e*x+d)^2-1/64*(d*g+e*f)*(d*g+5*e*f)/d
^6/e^3/(e*x+d)+1/32*(d*g+e*f)*(d*g+3*e*f)*arctanh(e*x/d)/d^7/e^3

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {862, 90, 214} \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx=\frac {\text {arctanh}\left (\frac {e x}{d}\right ) (d g+e f) (d g+3 e f)}{32 d^7 e^3}+\frac {(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac {(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac {f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac {(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4} \]

[In]

Int[(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)^2),x]

[Out]

(e*f + d*g)^2/(64*d^6*e^3*(d - e*x)) - (e*f - d*g)^2/(20*d^2*e^3*(d + e*x)^5) - (e^2*f^2 - d^2*g^2)/(16*d^3*e^
3*(d + e*x)^4) - ((3*e*f - d*g)*(e*f + d*g))/(48*d^4*e^3*(d + e*x)^3) - (f*(e*f + d*g))/(16*d^5*e^2*(d + e*x)^
2) - ((e*f + d*g)*(5*e*f + d*g))/(64*d^6*e^3*(d + e*x)) + ((e*f + d*g)*(3*e*f + d*g)*ArcTanh[(e*x)/d])/(32*d^7
*e^3)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 862

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rubi steps \begin{align*} \text {integral}& = \int \frac {(f+g x)^2}{(d-e x)^2 (d+e x)^6} \, dx \\ & = \int \left (\frac {(e f+d g)^2}{64 d^6 e^2 (d-e x)^2}+\frac {(-e f+d g)^2}{4 d^2 e^2 (d+e x)^6}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^5}+\frac {(3 e f-d g) (e f+d g)}{16 d^4 e^2 (d+e x)^4}+\frac {f (e f+d g)}{8 d^5 e (d+e x)^3}+\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^2 (d+e x)^2}+\frac {(e f+d g) (3 e f+d g)}{32 d^6 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx \\ & = \frac {(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac {(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac {((e f+d g) (3 e f+d g)) \int \frac {1}{d^2-e^2 x^2} \, dx}{32 d^6 e^2} \\ & = \frac {(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac {(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac {(e f+d g) (3 e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^7 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.09 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx=\frac {\frac {15 d (e f+d g)^2}{d-e x}-\frac {48 d^5 (e f-d g)^2}{(d+e x)^5}+\frac {60 d^4 \left (-e^2 f^2+d^2 g^2\right )}{(d+e x)^4}+\frac {20 d^3 \left (-3 e^2 f^2-2 d e f g+d^2 g^2\right )}{(d+e x)^3}-\frac {60 d^2 e f (e f+d g)}{(d+e x)^2}-\frac {15 d \left (5 e^2 f^2+6 d e f g+d^2 g^2\right )}{d+e x}-15 \left (3 e^2 f^2+4 d e f g+d^2 g^2\right ) \log (d-e x)+15 \left (3 e^2 f^2+4 d e f g+d^2 g^2\right ) \log (d+e x)}{960 d^7 e^3} \]

[In]

Integrate[(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)^2),x]

[Out]

((15*d*(e*f + d*g)^2)/(d - e*x) - (48*d^5*(e*f - d*g)^2)/(d + e*x)^5 + (60*d^4*(-(e^2*f^2) + d^2*g^2))/(d + e*
x)^4 + (20*d^3*(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x)^3 - (60*d^2*e*f*(e*f + d*g))/(d + e*x)^2 - (15*d*
(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2))/(d + e*x) - 15*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*Log[d - e*x] + 15*(3*e^2*f
^2 + 4*d*e*f*g + d^2*g^2)*Log[d + e*x])/(960*d^7*e^3)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {\left (-d^{2} g^{2}-4 d e f g -3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{64 e^{3} d^{7}}+\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{64 e^{3} d^{6} \left (-e x +d \right )}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{64 e^{3} d^{7}}-\frac {d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}}{64 e^{3} d^{6} \left (e x +d \right )}-\frac {-d^{2} g^{2}+e^{2} f^{2}}{16 e^{3} d^{3} \left (e x +d \right )^{4}}-\frac {-d^{2} g^{2}+2 d e f g +3 e^{2} f^{2}}{48 e^{3} d^{4} \left (e x +d \right )^{3}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{20 d^{2} e^{3} \left (e x +d \right )^{5}}-\frac {f \left (d g +e f \right )}{16 d^{5} e^{2} \left (e x +d \right )^{2}}\)	\(278\)
norman	\(\frac {\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) x^{3}}{6 d^{4}}-\frac {\left (d^{2} g^{2}-4 d e f g -13 e^{2} f^{2}\right ) x^{2}}{8 e \,d^{3}}+\frac {e \left (7 d^{2} g^{2}+4 d e f g -27 e^{2} f^{2}\right ) x^{4}}{24 d^{5}}+\frac {e^{2} \left (79 d^{2} g^{2}-68 d e f g -531 e^{2} f^{2}\right ) x^{5}}{480 d^{6}}+\frac {e^{3} \left (d^{2} g^{2}-2 d e f g -9 e^{2} f^{2}\right ) x^{6}}{30 d^{7}}-\frac {\left (d^{2} g^{2}+4 d e f g -29 e^{2} f^{2}\right ) x}{32 e^{2} d^{2}}}{\left (e x +d \right )^{5} \left (-e x +d \right )}-\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{64 e^{3} d^{7}}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{64 e^{3} d^{7}}\)	\(285\)
risch	\(\frac {\frac {e^{2} \left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) x^{5}}{32 d^{6}}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) e \,x^{4}}{8 d^{5}}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) x^{3}}{6 d^{4}}+\frac {\left (d^{2} g^{2}+4 d e f g +3 e^{2} f^{2}\right ) x^{2}}{24 d^{3} e}+\frac {\left (49 d^{2} g^{2}-188 d e f g -141 e^{2} f^{2}\right ) x}{480 e^{2} d^{2}}+\frac {d^{2} g^{2}-2 d e f g -9 e^{2} f^{2}}{30 d \,e^{3}}}{\left (e x +d \right )^{4} \left (-e^{2} x^{2}+d^{2}\right )}-\frac {\ln \left (-e x +d \right ) g^{2}}{64 e^{3} d^{5}}-\frac {\ln \left (-e x +d \right ) f g}{16 e^{2} d^{6}}-\frac {3 \ln \left (-e x +d \right ) f^{2}}{64 e \,d^{7}}+\frac {\ln \left (e x +d \right ) g^{2}}{64 e^{3} d^{5}}+\frac {\ln \left (e x +d \right ) f g}{16 e^{2} d^{6}}+\frac {3 \ln \left (e x +d \right ) f^{2}}{64 e \,d^{7}}\)	\(317\)
parallelrisch	\(-\frac {-300 \ln \left (e x -d \right ) x^{2} d^{5} e^{3} f g -64 x^{6} d \,e^{7} f g -136 x^{5} d^{2} e^{6} f g +160 x^{4} d^{3} e^{5} f g +640 x^{3} d^{4} e^{4} f g -180 \ln \left (e x -d \right ) x \,d^{5} e^{3} f^{2}+60 \ln \left (e x +d \right ) x \,d^{7} e \,g^{2}+180 \ln \left (e x +d \right ) x \,d^{5} e^{3} f^{2}-60 \ln \left (e x -d \right ) d^{7} e f g +60 \ln \left (e x +d \right ) d^{7} e f g +15 \ln \left (e x -d \right ) x^{6} d^{2} e^{6} g^{2}-15 \ln \left (e x +d \right ) x^{6} d^{2} e^{6} g^{2}+60 \ln \left (e x -d \right ) x^{5} d^{3} e^{5} g^{2}+300 \ln \left (e x -d \right ) x^{4} d^{3} e^{5} f g +300 \ln \left (e x +d \right ) x^{2} d^{5} e^{3} f g -240 \ln \left (e x -d \right ) x \,d^{6} e^{2} f g -45 \ln \left (e x +d \right ) x^{6} e^{8} f^{2}-45 \ln \left (e x -d \right ) d^{6} e^{2} f^{2}+45 \ln \left (e x +d \right ) d^{6} e^{2} f^{2}+45 \ln \left (e x -d \right ) x^{6} e^{8} f^{2}-120 x^{2} d^{6} e^{2} g^{2}+1560 x^{2} d^{4} e^{4} f^{2}-30 x \,d^{7} e \,g^{2}+870 x \,d^{5} e^{3} f^{2}+32 x^{6} d^{2} e^{6} g^{2}+158 x^{5} d^{3} e^{5} g^{2}-1062 x^{5} d \,e^{7} f^{2}+280 x^{4} d^{4} e^{4} g^{2}-1080 x^{4} d^{2} e^{6} f^{2}+160 x^{3} d^{5} e^{3} g^{2}+480 x^{3} d^{3} e^{5} f^{2}-225 \ln \left (e x -d \right ) x^{2} d^{4} e^{4} f^{2}+75 \ln \left (e x +d \right ) x^{2} d^{6} e^{2} g^{2}+225 \ln \left (e x +d \right ) x^{2} d^{4} e^{4} f^{2}-60 \ln \left (e x -d \right ) x \,d^{7} e \,g^{2}+480 x^{2} d^{5} e^{3} f g -120 x \,d^{6} e^{2} f g +180 \ln \left (e x -d \right ) x^{5} d \,e^{7} f^{2}-60 \ln \left (e x +d \right ) x^{5} d^{3} e^{5} g^{2}-180 \ln \left (e x +d \right ) x^{5} d \,e^{7} f^{2}+75 \ln \left (e x -d \right ) x^{4} d^{4} e^{4} g^{2}+225 \ln \left (e x -d \right ) x^{4} d^{2} e^{6} f^{2}-75 \ln \left (e x +d \right ) x^{4} d^{4} e^{4} g^{2}-225 \ln \left (e x +d \right ) x^{4} d^{2} e^{6} f^{2}-75 \ln \left (e x -d \right ) x^{2} d^{6} e^{2} g^{2}+240 \ln \left (e x +d \right ) x \,d^{6} e^{2} f g +60 \ln \left (e x -d \right ) x^{6} d \,e^{7} f g -288 x^{6} e^{8} f^{2}-15 \ln \left (e x -d \right ) d^{8} g^{2}+240 \ln \left (e x -d \right ) x^{5} d^{2} e^{6} f g -240 \ln \left (e x +d \right ) x^{5} d^{2} e^{6} f g -60 \ln \left (e x +d \right ) x^{6} d \,e^{7} f g +15 \ln \left (e x +d \right ) d^{8} g^{2}-300 \ln \left (e x +d \right ) x^{4} d^{3} e^{5} f g}{960 e^{3} d^{7} \left (e^{2} x^{2}-d^{2}\right ) \left (e x +d \right )^{4}}\)	\(949\)

[In]

int((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/64*(-d^2*g^2-4*d*e*f*g-3*e^2*f^2)/e^3/d^7*ln(-e*x+d)+1/64*(d^2*g^2+2*d*e*f*g+e^2*f^2)/e^3/d^6/(-e*x+d)+1/64/
e^3*(d^2*g^2+4*d*e*f*g+3*e^2*f^2)/d^7*ln(e*x+d)-1/64/e^3*(d^2*g^2+6*d*e*f*g+5*e^2*f^2)/d^6/(e*x+d)-1/16*(-d^2*
g^2+e^2*f^2)/e^3/d^3/(e*x+d)^4-1/48*(-d^2*g^2+2*d*e*f*g+3*e^2*f^2)/e^3/d^4/(e*x+d)^3-1/20*(d^2*g^2-2*d*e*f*g+e
^2*f^2)/d^2/e^3/(e*x+d)^5-1/16*f*(d*g+e*f)/d^5/e^2/(e*x+d)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (197) = 394\).

Time = 0.34 (sec) , antiderivative size = 693, normalized size of antiderivative = 3.30 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx=\frac {288 \, d^{6} e^{2} f^{2} + 64 \, d^{7} e f g - 32 \, d^{8} g^{2} - 30 \, {\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 120 \, {\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} - 160 \, {\left (3 \, d^{3} e^{5} f^{2} + 4 \, d^{4} e^{4} f g + d^{5} e^{3} g^{2}\right )} x^{3} - 40 \, {\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (141 \, d^{5} e^{3} f^{2} + 188 \, d^{6} e^{2} f g - 49 \, d^{7} e g^{2}\right )} x - 15 \, {\left (3 \, d^{6} e^{2} f^{2} + 4 \, d^{7} e f g + d^{8} g^{2} - {\left (3 \, e^{8} f^{2} + 4 \, d e^{7} f g + d^{2} e^{6} g^{2}\right )} x^{6} - 4 \, {\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 5 \, {\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} + 5 \, {\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 4 \, {\left (3 \, d^{5} e^{3} f^{2} + 4 \, d^{6} e^{2} f g + d^{7} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 15 \, {\left (3 \, d^{6} e^{2} f^{2} + 4 \, d^{7} e f g + d^{8} g^{2} - {\left (3 \, e^{8} f^{2} + 4 \, d e^{7} f g + d^{2} e^{6} g^{2}\right )} x^{6} - 4 \, {\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 5 \, {\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} + 5 \, {\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 4 \, {\left (3 \, d^{5} e^{3} f^{2} + 4 \, d^{6} e^{2} f g + d^{7} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{960 \, {\left (d^{7} e^{9} x^{6} + 4 \, d^{8} e^{8} x^{5} + 5 \, d^{9} e^{7} x^{4} - 5 \, d^{11} e^{5} x^{2} - 4 \, d^{12} e^{4} x - d^{13} e^{3}\right )}} \]

[In]

integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2)^2,x, algorithm="fricas")

[Out]

1/960*(288*d^6*e^2*f^2 + 64*d^7*e*f*g - 32*d^8*g^2 - 30*(3*d*e^7*f^2 + 4*d^2*e^6*f*g + d^3*e^5*g^2)*x^5 - 120*
(3*d^2*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 - 160*(3*d^3*e^5*f^2 + 4*d^4*e^4*f*g + d^5*e^3*g^2)*x^3 - 40
*(3*d^4*e^4*f^2 + 4*d^5*e^3*f*g + d^6*e^2*g^2)*x^2 + 2*(141*d^5*e^3*f^2 + 188*d^6*e^2*f*g - 49*d^7*e*g^2)*x -
15*(3*d^6*e^2*f^2 + 4*d^7*e*f*g + d^8*g^2 - (3*e^8*f^2 + 4*d*e^7*f*g + d^2*e^6*g^2)*x^6 - 4*(3*d*e^7*f^2 + 4*d
^2*e^6*f*g + d^3*e^5*g^2)*x^5 - 5*(3*d^2*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 + 5*(3*d^4*e^4*f^2 + 4*d^5
*e^3*f*g + d^6*e^2*g^2)*x^2 + 4*(3*d^5*e^3*f^2 + 4*d^6*e^2*f*g + d^7*e*g^2)*x)*log(e*x + d) + 15*(3*d^6*e^2*f^
2 + 4*d^7*e*f*g + d^8*g^2 - (3*e^8*f^2 + 4*d*e^7*f*g + d^2*e^6*g^2)*x^6 - 4*(3*d*e^7*f^2 + 4*d^2*e^6*f*g + d^3
*e^5*g^2)*x^5 - 5*(3*d^2*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 + 5*(3*d^4*e^4*f^2 + 4*d^5*e^3*f*g + d^6*e
^2*g^2)*x^2 + 4*(3*d^5*e^3*f^2 + 4*d^6*e^2*f*g + d^7*e*g^2)*x)*log(e*x - d))/(d^7*e^9*x^6 + 4*d^8*e^8*x^5 + 5*
d^9*e^7*x^4 - 5*d^11*e^5*x^2 - 4*d^12*e^4*x - d^13*e^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (192) = 384\).

Time = 0.93 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.03 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx=\frac {- 16 d^{7} g^{2} + 32 d^{6} e f g + 144 d^{5} e^{2} f^{2} + x^{5} \left (- 15 d^{2} e^{5} g^{2} - 60 d e^{6} f g - 45 e^{7} f^{2}\right ) + x^{4} \left (- 60 d^{3} e^{4} g^{2} - 240 d^{2} e^{5} f g - 180 d e^{6} f^{2}\right ) + x^{3} \left (- 80 d^{4} e^{3} g^{2} - 320 d^{3} e^{4} f g - 240 d^{2} e^{5} f^{2}\right ) + x^{2} \left (- 20 d^{5} e^{2} g^{2} - 80 d^{4} e^{3} f g - 60 d^{3} e^{4} f^{2}\right ) + x \left (- 49 d^{6} e g^{2} + 188 d^{5} e^{2} f g + 141 d^{4} e^{3} f^{2}\right )}{- 480 d^{12} e^{3} - 1920 d^{11} e^{4} x - 2400 d^{10} e^{5} x^{2} + 2400 d^{8} e^{7} x^{4} + 1920 d^{7} e^{8} x^{5} + 480 d^{6} e^{9} x^{6}} - \frac {\left (d g + e f\right ) \left (d g + 3 e f\right ) \log {\left (- \frac {d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} e^{3}} + \frac {\left (d g + e f\right ) \left (d g + 3 e f\right ) \log {\left (\frac {d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} e^{3}} \]

[In]

integrate((g*x+f)**2/(e*x+d)**4/(-e**2*x**2+d**2)**2,x)

[Out]

(-16*d**7*g**2 + 32*d**6*e*f*g + 144*d**5*e**2*f**2 + x**5*(-15*d**2*e**5*g**2 - 60*d*e**6*f*g - 45*e**7*f**2)
+ x**4*(-60*d**3*e**4*g**2 - 240*d**2*e**5*f*g - 180*d*e**6*f**2) + x**3*(-80*d**4*e**3*g**2 - 320*d**3*e**4*
f*g - 240*d**2*e**5*f**2) + x**2*(-20*d**5*e**2*g**2 - 80*d**4*e**3*f*g - 60*d**3*e**4*f**2) + x*(-49*d**6*e*g
**2 + 188*d**5*e**2*f*g + 141*d**4*e**3*f**2))/(-480*d**12*e**3 - 1920*d**11*e**4*x - 2400*d**10*e**5*x**2 + 2
400*d**8*e**7*x**4 + 1920*d**7*e**8*x**5 + 480*d**6*e**9*x**6) - (d*g + e*f)*(d*g + 3*e*f)*log(-d*(d*g + e*f)*
(d*g + 3*e*f)/(e*(d**2*g**2 + 4*d*e*f*g + 3*e**2*f**2)) + x)/(64*d**7*e**3) + (d*g + e*f)*(d*g + 3*e*f)*log(d*
(d*g + e*f)*(d*g + 3*e*f)/(e*(d**2*g**2 + 4*d*e*f*g + 3*e**2*f**2)) + x)/(64*d**7*e**3)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.63 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx=\frac {144 \, d^{5} e^{2} f^{2} + 32 \, d^{6} e f g - 16 \, d^{7} g^{2} - 15 \, {\left (3 \, e^{7} f^{2} + 4 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 60 \, {\left (3 \, d e^{6} f^{2} + 4 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 80 \, {\left (3 \, d^{2} e^{5} f^{2} + 4 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} - 20 \, {\left (3 \, d^{3} e^{4} f^{2} + 4 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + {\left (141 \, d^{4} e^{3} f^{2} + 188 \, d^{5} e^{2} f g - 49 \, d^{6} e g^{2}\right )} x}{480 \, {\left (d^{6} e^{9} x^{6} + 4 \, d^{7} e^{8} x^{5} + 5 \, d^{8} e^{7} x^{4} - 5 \, d^{10} e^{5} x^{2} - 4 \, d^{11} e^{4} x - d^{12} e^{3}\right )}} + \frac {{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{64 \, d^{7} e^{3}} - \frac {{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{64 \, d^{7} e^{3}} \]

[In]

integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2)^2,x, algorithm="maxima")

[Out]

1/480*(144*d^5*e^2*f^2 + 32*d^6*e*f*g - 16*d^7*g^2 - 15*(3*e^7*f^2 + 4*d*e^6*f*g + d^2*e^5*g^2)*x^5 - 60*(3*d*
e^6*f^2 + 4*d^2*e^5*f*g + d^3*e^4*g^2)*x^4 - 80*(3*d^2*e^5*f^2 + 4*d^3*e^4*f*g + d^4*e^3*g^2)*x^3 - 20*(3*d^3*
e^4*f^2 + 4*d^4*e^3*f*g + d^5*e^2*g^2)*x^2 + (141*d^4*e^3*f^2 + 188*d^5*e^2*f*g - 49*d^6*e*g^2)*x)/(d^6*e^9*x^
6 + 4*d^7*e^8*x^5 + 5*d^8*e^7*x^4 - 5*d^10*e^5*x^2 - 4*d^11*e^4*x - d^12*e^3) + 1/64*(3*e^2*f^2 + 4*d*e*f*g +
d^2*g^2)*log(e*x + d)/(d^7*e^3) - 1/64*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*log(e*x - d)/(d^7*e^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.47 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx=\frac {{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{64 \, d^{7} e^{3}} - \frac {{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{64 \, d^{7} e^{3}} + \frac {144 \, d^{6} e^{2} f^{2} + 32 \, d^{7} e f g - 16 \, d^{8} g^{2} - 15 \, {\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 60 \, {\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} - 80 \, {\left (3 \, d^{3} e^{5} f^{2} + 4 \, d^{4} e^{4} f g + d^{5} e^{3} g^{2}\right )} x^{3} - 20 \, {\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + {\left (141 \, d^{5} e^{3} f^{2} + 188 \, d^{6} e^{2} f g - 49 \, d^{7} e g^{2}\right )} x}{480 \, {\left (e x + d\right )}^{5} {\left (e x - d\right )} d^{7} e^{3}} \]

[In]

integrate((g*x+f)^2/(e*x+d)^4/(-e^2*x^2+d^2)^2,x, algorithm="giac")

[Out]

1/64*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*log(abs(e*x + d))/(d^7*e^3) - 1/64*(3*e^2*f^2 + 4*d*e*f*g + d^2*g^2)*lo
g(abs(e*x - d))/(d^7*e^3) + 1/480*(144*d^6*e^2*f^2 + 32*d^7*e*f*g - 16*d^8*g^2 - 15*(3*d*e^7*f^2 + 4*d^2*e^6*f
*g + d^3*e^5*g^2)*x^5 - 60*(3*d^2*e^6*f^2 + 4*d^3*e^5*f*g + d^4*e^4*g^2)*x^4 - 80*(3*d^3*e^5*f^2 + 4*d^4*e^4*f
*g + d^5*e^3*g^2)*x^3 - 20*(3*d^4*e^4*f^2 + 4*d^5*e^3*f*g + d^6*e^2*g^2)*x^2 + (141*d^5*e^3*f^2 + 188*d^6*e^2*
f*g - 49*d^7*e*g^2)*x)/((e*x + d)^5*(e*x - d)*d^7*e^3)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.50 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx=\frac {\frac {x^3\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}{6\,d^4}-\frac {-d^2\,g^2+2\,d\,e\,f\,g+9\,e^2\,f^2}{30\,d\,e^3}+\frac {e\,x^4\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}{8\,d^5}-\frac {x\,\left (-49\,d^2\,g^2+188\,d\,e\,f\,g+141\,e^2\,f^2\right )}{480\,d^2\,e^2}+\frac {x^2\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}{24\,d^3\,e}+\frac {e^2\,x^5\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}{32\,d^6}}{d^6+4\,d^5\,e\,x+5\,d^4\,e^2\,x^2-5\,d^2\,e^4\,x^4-4\,d\,e^5\,x^5-e^6\,x^6}+\frac {\mathrm {atanh}\left (\frac {e\,x\,\left (d\,g+e\,f\right )\,\left (d\,g+3\,e\,f\right )}{d\,\left (d^2\,g^2+4\,d\,e\,f\,g+3\,e^2\,f^2\right )}\right )\,\left (d\,g+e\,f\right )\,\left (d\,g+3\,e\,f\right )}{32\,d^7\,e^3} \]

[In]

int((f + g*x)^2/((d^2 - e^2*x^2)^2*(d + e*x)^4),x)

[Out]

((x^3*(d^2*g^2 + 3*e^2*f^2 + 4*d*e*f*g))/(6*d^4) - (9*e^2*f^2 - d^2*g^2 + 2*d*e*f*g)/(30*d*e^3) + (e*x^4*(d^2*
g^2 + 3*e^2*f^2 + 4*d*e*f*g))/(8*d^5) - (x*(141*e^2*f^2 - 49*d^2*g^2 + 188*d*e*f*g))/(480*d^2*e^2) + (x^2*(d^2
*g^2 + 3*e^2*f^2 + 4*d*e*f*g))/(24*d^3*e) + (e^2*x^5*(d^2*g^2 + 3*e^2*f^2 + 4*d*e*f*g))/(32*d^6))/(d^6 - e^6*x
^6 - 4*d*e^5*x^5 + 5*d^4*e^2*x^2 - 5*d^2*e^4*x^4 + 4*d^5*e*x) + (atanh((e*x*(d*g + e*f)*(d*g + 3*e*f))/(d*(d^2
*g^2 + 3*e^2*f^2 + 4*d*e*f*g)))*(d*g + e*f)*(d*g + 3*e*f))/(32*d^7*e^3)