\(\int \frac {(f+g x)^2}{(d+e x)^2 (d^2-e^2 x^2)^3} \, dx\) [578]

Optimal result

Integrand size = 29, antiderivative size = 235 \[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx=\frac {(e f+d g)^2}{64 d^5 e^3 (d-e x)^2}+\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}-\frac {(e f-d g) (3 e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2}-\frac {5 e^2 f^2+2 d e f g-d^2 g^2}{32 d^6 e^3 (d+e x)}+\frac {\left (15 e^2 f^2+10 d e f g-d^2 g^2\right ) \text {arctanh}\left (\frac {e x}{d}\right )}{64 d^7 e^3} \]

[Out]

1/64*(d*g+e*f)^2/d^5/e^3/(-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)^2/d^3/e^3/(e*x
+d)^4-1/48*(-d*g+e*f)*(d*g+3*e*f)/d^4/e^3/(e*x+d)^3+1/32*(d^2*g^2-3*e^2*f^2)/d^5/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/(e*x+d)+1/64*(-d^2*g^2+10*d*e*f*g+15*e^2*f^2)*arctanh(e*x/d)/d^7/e^3

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 235, 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} \[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx=\frac {\text {arctanh}\left (\frac {e x}{d}\right ) \left (-d^2 g^2+10 d e f g+15 e^2 f^2\right )}{64 d^7 e^3}+\frac {(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d-e x)}+\frac {(d g+e f)^2}{64 d^5 e^3 (d-e x)^2}-\frac {(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}-\frac {-d^2 g^2+2 d e f g+5 e^2 f^2}{32 d^6 e^3 (d+e x)}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2} \]

[In]

Int[(f + g*x)^2/((d + e*x)^2*(d^2 - e^2*x^2)^3),x]

[Out]

(e*f + d*g)^2/(64*d^5*e^3*(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)^2/(3
2*d^3*e^3*(d + e*x)^4) - ((e*f - d*g)*(3*e*f + d*g))/(48*d^4*e^3*(d + e*x)^3) - (3*e^2*f^2 - d^2*g^2)/(32*d^5*
e^3*(d + e*x)^2) - (5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)/(32*d^6*e^3*(d + e*x)) + ((15*e^2*f^2 + 10*d*e*f*g - d^2*
g^2)*ArcTanh[(e*x)/d])/(64*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)^3 (d+e x)^5} \, dx \\ & = \int \left (\frac {(e f+d g)^2}{32 d^5 e^2 (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)^2}{8 d^3 e^2 (d+e x)^5}+\frac {(e f-d g) (3 e f+d g)}{16 d^4 e^2 (d+e x)^4}+\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^2 (d+e x)^3}+\frac {5 e^2 f^2+2 d e f g-d^2 g^2}{32 d^6 e^2 (d+e x)^2}+\frac {-15 e^2 f^2-10 d e f g+d^2 g^2}{64 d^6 e^2 \left (-d^2+e^2 x^2\right )}\right ) \, dx \\ & = \frac {(e f+d g)^2}{64 d^5 e^3 (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)^2}{32 d^3 e^3 (d+e x)^4}-\frac {(e f-d g) (3 e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2}-\frac {5 e^2 f^2+2 d e f g-d^2 g^2}{32 d^6 e^3 (d+e x)}-\frac {\left (15 e^2 f^2+10 d e f g-d^2 g^2\right ) \int \frac {1}{-d^2+e^2 x^2} \, dx}{64 d^6 e^2} \\ & = \frac {(e f+d g)^2}{64 d^5 e^3 (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)^2}{32 d^3 e^3 (d+e x)^4}-\frac {(e f-d g) (3 e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2}-\frac {5 e^2 f^2+2 d e f g-d^2 g^2}{32 d^6 e^3 (d+e x)}+\frac {\left (15 e^2 f^2+10 d e f g-d^2 g^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{64 d^7 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.04 \[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx=\frac {\frac {6 d^2 (e f+d g)^2}{(d-e x)^2}+\frac {6 d \left (5 e^2 f^2+6 d e f g+d^2 g^2\right )}{d-e x}-\frac {12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac {8 d^3 \left (-3 e^2 f^2+2 d e f g+d^2 g^2\right )}{(d+e x)^3}+\frac {12 d^2 \left (-3 e^2 f^2+d^2 g^2\right )}{(d+e x)^2}+\frac {12 d \left (-5 e^2 f^2-2 d e f g+d^2 g^2\right )}{d+e x}+3 \left (-15 e^2 f^2-10 d e f g+d^2 g^2\right ) \log (d-e x)+3 \left (15 e^2 f^2+10 d e f g-d^2 g^2\right ) \log (d+e x)}{384 d^7 e^3} \]

[In]

Integrate[(f + g*x)^2/((d + e*x)^2*(d^2 - e^2*x^2)^3),x]

[Out]

((6*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (6*d*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2))/(d - e*x) - (12*d^4*(e*f - d*g)^2
)/(d + e*x)^4 + (8*d^3*(-3*e^2*f^2 + 2*d*e*f*g + d^2*g^2))/(d + e*x)^3 + (12*d^2*(-3*e^2*f^2 + d^2*g^2))/(d +
e*x)^2 + (12*d*(-5*e^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x) + 3*(-15*e^2*f^2 - 10*d*e*f*g + d^2*g^2)*Log[d -
e*x] + 3*(15*e^2*f^2 + 10*d*e*f*g - d^2*g^2)*Log[d + e*x])/(384*d^7*e^3)

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.22

method	result	size
norman	\(\frac {\frac {\left (31 d^{2} g^{2}+74 d e f g -81 e^{2} f^{2}\right ) x^{3}}{96 d^{4}}+\frac {\left (d^{2} g^{2}+22 d e f g +17 e^{2} f^{2}\right ) x^{2}}{32 e \,d^{3}}+\frac {e \left (11 d^{2} g^{2}-14 d e f g -69 e^{2} f^{2}\right ) x^{4}}{96 d^{5}}-\frac {e^{2} \left (29 d^{2} g^{2}+94 d e f g -51 e^{2} f^{2}\right ) x^{5}}{192 d^{6}}-\frac {e^{3} \left (d^{2} g^{2}+2 d e f g -3 e^{2} f^{2}\right ) x^{6}}{12 d^{7}}+\frac {\left (d^{2} g^{2}-10 d e f g +49 e^{2} f^{2}\right ) x}{64 d^{2} e^{2}}}{\left (e x +d \right )^{4} \left (-e x +d \right )^{2}}+\frac {\left (d^{2} g^{2}-10 d e f g -15 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{128 d^{7} e^{3}}-\frac {\left (d^{2} g^{2}-10 d e f g -15 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{128 d^{7} e^{3}}\)	\(286\)
default	\(\frac {d^{2} g^{2}+6 d e f g +5 e^{2} f^{2}}{64 e^{3} d^{6} \left (-e x +d \right )}-\frac {-d^{2} g^{2}-2 d e f g -e^{2} f^{2}}{64 e^{3} d^{5} \left (-e x +d \right )^{2}}+\frac {\left (d^{2} g^{2}-10 d e f g -15 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{128 d^{7} e^{3}}-\frac {-d^{2} g^{2}+3 e^{2} f^{2}}{32 e^{3} d^{5} \left (e x +d \right )^{2}}-\frac {-d^{2} g^{2}-2 d e f g +3 e^{2} f^{2}}{48 e^{3} d^{4} \left (e x +d \right )^{3}}-\frac {-d^{2} g^{2}+2 d e f g +5 e^{2} f^{2}}{32 e^{3} d^{6} \left (e x +d \right )}+\frac {\left (-d^{2} g^{2}+10 d e f g +15 e^{2} f^{2}\right ) \ln \left (e x +d \right )}{128 e^{3} d^{7}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{32 e^{3} d^{3} \left (e x +d \right )^{4}}\)	\(297\)
risch	\(\frac {\frac {\left (d^{2} g^{2}-10 d e f g -15 e^{2} f^{2}\right ) e^{2} x^{5}}{64 d^{6}}+\frac {\left (d^{2} g^{2}-10 d e f g -15 e^{2} f^{2}\right ) e \,x^{4}}{32 d^{5}}-\frac {\left (d^{2} g^{2}-10 d e f g -15 e^{2} f^{2}\right ) x^{3}}{96 d^{4}}-\frac {5 \left (d^{2} g^{2}-10 d e f g -15 e^{2} f^{2}\right ) x^{2}}{96 d^{3} e}+\frac {\left (35 d^{2} g^{2}+34 d e f g +51 e^{2} f^{2}\right ) x}{192 d^{2} e^{2}}+\frac {d^{2} g^{2}+2 d e f g -3 e^{2} f^{2}}{12 d \,e^{3}}}{\left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{2}}-\frac {\ln \left (-e x -d \right ) g^{2}}{128 d^{5} e^{3}}+\frac {5 \ln \left (-e x -d \right ) f g}{64 d^{6} e^{2}}+\frac {15 \ln \left (-e x -d \right ) f^{2}}{128 d^{7} e}+\frac {\ln \left (e x -d \right ) g^{2}}{128 d^{5} e^{3}}-\frac {5 \ln \left (e x -d \right ) f g}{64 d^{6} e^{2}}-\frac {15 \ln \left (e x -d \right ) f^{2}}{128 d^{7} e}\)	\(329\)
parallelrisch	\(\text {Expression too large to display}\)	\(1073\)

[In]

int((g*x+f)^2/(e*x+d)^2/(-e^2*x^2+d^2)^3,x,method=_RETURNVERBOSE)

[Out]

(1/96*(31*d^2*g^2+74*d*e*f*g-81*e^2*f^2)/d^4*x^3+1/32/e*(d^2*g^2+22*d*e*f*g+17*e^2*f^2)/d^3*x^2+1/96*e*(11*d^2
*g^2-14*d*e*f*g-69*e^2*f^2)/d^5*x^4-1/192*e^2*(29*d^2*g^2+94*d*e*f*g-51*e^2*f^2)/d^6*x^5-1/12*e^3*(d^2*g^2+2*d
*e*f*g-3*e^2*f^2)/d^7*x^6+1/64*(d^2*g^2-10*d*e*f*g+49*e^2*f^2)/d^2/e^2*x)/(e*x+d)^4/(-e*x+d)^2+1/128*(d^2*g^2-
10*d*e*f*g-15*e^2*f^2)/d^7/e^3*ln(-e*x+d)-1/128*(d^2*g^2-10*d*e*f*g-15*e^2*f^2)/d^7/e^3*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (223) = 446\).

Time = 0.35 (sec) , antiderivative size = 793, normalized size of antiderivative = 3.37 \[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx=-\frac {96 \, d^{6} e^{2} f^{2} - 64 \, d^{7} e f g - 32 \, d^{8} g^{2} + 6 \, {\left (15 \, d e^{7} f^{2} + 10 \, d^{2} e^{6} f g - d^{3} e^{5} g^{2}\right )} x^{5} + 12 \, {\left (15 \, d^{2} e^{6} f^{2} + 10 \, d^{3} e^{5} f g - d^{4} e^{4} g^{2}\right )} x^{4} - 4 \, {\left (15 \, d^{3} e^{5} f^{2} + 10 \, d^{4} e^{4} f g - d^{5} e^{3} g^{2}\right )} x^{3} - 20 \, {\left (15 \, d^{4} e^{4} f^{2} + 10 \, d^{5} e^{3} f g - d^{6} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (51 \, d^{5} e^{3} f^{2} + 34 \, d^{6} e^{2} f g + 35 \, d^{7} e g^{2}\right )} x - 3 \, {\left (15 \, d^{6} e^{2} f^{2} + 10 \, d^{7} e f g - d^{8} g^{2} + {\left (15 \, e^{8} f^{2} + 10 \, d e^{7} f g - d^{2} e^{6} g^{2}\right )} x^{6} + 2 \, {\left (15 \, d e^{7} f^{2} + 10 \, d^{2} e^{6} f g - d^{3} e^{5} g^{2}\right )} x^{5} - {\left (15 \, d^{2} e^{6} f^{2} + 10 \, d^{3} e^{5} f g - d^{4} e^{4} g^{2}\right )} x^{4} - 4 \, {\left (15 \, d^{3} e^{5} f^{2} + 10 \, d^{4} e^{4} f g - d^{5} e^{3} g^{2}\right )} x^{3} - {\left (15 \, d^{4} e^{4} f^{2} + 10 \, d^{5} e^{3} f g - d^{6} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (15 \, d^{5} e^{3} f^{2} + 10 \, d^{6} e^{2} f g - d^{7} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (15 \, d^{6} e^{2} f^{2} + 10 \, d^{7} e f g - d^{8} g^{2} + {\left (15 \, e^{8} f^{2} + 10 \, d e^{7} f g - d^{2} e^{6} g^{2}\right )} x^{6} + 2 \, {\left (15 \, d e^{7} f^{2} + 10 \, d^{2} e^{6} f g - d^{3} e^{5} g^{2}\right )} x^{5} - {\left (15 \, d^{2} e^{6} f^{2} + 10 \, d^{3} e^{5} f g - d^{4} e^{4} g^{2}\right )} x^{4} - 4 \, {\left (15 \, d^{3} e^{5} f^{2} + 10 \, d^{4} e^{4} f g - d^{5} e^{3} g^{2}\right )} x^{3} - {\left (15 \, d^{4} e^{4} f^{2} + 10 \, d^{5} e^{3} f g - d^{6} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (15 \, d^{5} e^{3} f^{2} + 10 \, d^{6} e^{2} f g - d^{7} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{384 \, {\left (d^{7} e^{9} x^{6} + 2 \, d^{8} e^{8} x^{5} - d^{9} e^{7} x^{4} - 4 \, d^{10} e^{6} x^{3} - d^{11} e^{5} x^{2} + 2 \, d^{12} e^{4} x + d^{13} e^{3}\right )}} \]

[In]

integrate((g*x+f)^2/(e*x+d)^2/(-e^2*x^2+d^2)^3,x, algorithm="fricas")

[Out]

-1/384*(96*d^6*e^2*f^2 - 64*d^7*e*f*g - 32*d^8*g^2 + 6*(15*d*e^7*f^2 + 10*d^2*e^6*f*g - d^3*e^5*g^2)*x^5 + 12*
(15*d^2*e^6*f^2 + 10*d^3*e^5*f*g - d^4*e^4*g^2)*x^4 - 4*(15*d^3*e^5*f^2 + 10*d^4*e^4*f*g - d^5*e^3*g^2)*x^3 -
20*(15*d^4*e^4*f^2 + 10*d^5*e^3*f*g - d^6*e^2*g^2)*x^2 - 2*(51*d^5*e^3*f^2 + 34*d^6*e^2*f*g + 35*d^7*e*g^2)*x
- 3*(15*d^6*e^2*f^2 + 10*d^7*e*f*g - d^8*g^2 + (15*e^8*f^2 + 10*d*e^7*f*g - d^2*e^6*g^2)*x^6 + 2*(15*d*e^7*f^2
+ 10*d^2*e^6*f*g - d^3*e^5*g^2)*x^5 - (15*d^2*e^6*f^2 + 10*d^3*e^5*f*g - d^4*e^4*g^2)*x^4 - 4*(15*d^3*e^5*f^2
+ 10*d^4*e^4*f*g - d^5*e^3*g^2)*x^3 - (15*d^4*e^4*f^2 + 10*d^5*e^3*f*g - d^6*e^2*g^2)*x^2 + 2*(15*d^5*e^3*f^2
+ 10*d^6*e^2*f*g - d^7*e*g^2)*x)*log(e*x + d) + 3*(15*d^6*e^2*f^2 + 10*d^7*e*f*g - d^8*g^2 + (15*e^8*f^2 + 10
*d*e^7*f*g - d^2*e^6*g^2)*x^6 + 2*(15*d*e^7*f^2 + 10*d^2*e^6*f*g - d^3*e^5*g^2)*x^5 - (15*d^2*e^6*f^2 + 10*d^3
*e^5*f*g - d^4*e^4*g^2)*x^4 - 4*(15*d^3*e^5*f^2 + 10*d^4*e^4*f*g - d^5*e^3*g^2)*x^3 - (15*d^4*e^4*f^2 + 10*d^5
*e^3*f*g - d^6*e^2*g^2)*x^2 + 2*(15*d^5*e^3*f^2 + 10*d^6*e^2*f*g - d^7*e*g^2)*x)*log(e*x - d))/(d^7*e^9*x^6 +
2*d^8*e^8*x^5 - d^9*e^7*x^4 - 4*d^10*e^6*x^3 - d^11*e^5*x^2 + 2*d^12*e^4*x + d^13*e^3)

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.58 \[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx=- \frac {- 16 d^{7} g^{2} - 32 d^{6} e f g + 48 d^{5} e^{2} f^{2} + x^{5} \left (- 3 d^{2} e^{5} g^{2} + 30 d e^{6} f g + 45 e^{7} f^{2}\right ) + x^{4} \left (- 6 d^{3} e^{4} g^{2} + 60 d^{2} e^{5} f g + 90 d e^{6} f^{2}\right ) + x^{3} \cdot \left (2 d^{4} e^{3} g^{2} - 20 d^{3} e^{4} f g - 30 d^{2} e^{5} f^{2}\right ) + x^{2} \cdot \left (10 d^{5} e^{2} g^{2} - 100 d^{4} e^{3} f g - 150 d^{3} e^{4} f^{2}\right ) + x \left (- 35 d^{6} e g^{2} - 34 d^{5} e^{2} f g - 51 d^{4} e^{3} f^{2}\right )}{192 d^{12} e^{3} + 384 d^{11} e^{4} x - 192 d^{10} e^{5} x^{2} - 768 d^{9} e^{6} x^{3} - 192 d^{8} e^{7} x^{4} + 384 d^{7} e^{8} x^{5} + 192 d^{6} e^{9} x^{6}} + \frac {\left (d^{2} g^{2} - 10 d e f g - 15 e^{2} f^{2}\right ) \log {\left (- \frac {d}{e} + x \right )}}{128 d^{7} e^{3}} - \frac {\left (d^{2} g^{2} - 10 d e f g - 15 e^{2} f^{2}\right ) \log {\left (\frac {d}{e} + x \right )}}{128 d^{7} e^{3}} \]

[In]

integrate((g*x+f)**2/(e*x+d)**2/(-e**2*x**2+d**2)**3,x)

[Out]

-(-16*d**7*g**2 - 32*d**6*e*f*g + 48*d**5*e**2*f**2 + x**5*(-3*d**2*e**5*g**2 + 30*d*e**6*f*g + 45*e**7*f**2)
+ x**4*(-6*d**3*e**4*g**2 + 60*d**2*e**5*f*g + 90*d*e**6*f**2) + x**3*(2*d**4*e**3*g**2 - 20*d**3*e**4*f*g - 3
0*d**2*e**5*f**2) + x**2*(10*d**5*e**2*g**2 - 100*d**4*e**3*f*g - 150*d**3*e**4*f**2) + x*(-35*d**6*e*g**2 - 3
4*d**5*e**2*f*g - 51*d**4*e**3*f**2))/(192*d**12*e**3 + 384*d**11*e**4*x - 192*d**10*e**5*x**2 - 768*d**9*e**6
*x**3 - 192*d**8*e**7*x**4 + 384*d**7*e**8*x**5 + 192*d**6*e**9*x**6) + (d**2*g**2 - 10*d*e*f*g - 15*e**2*f**2
)*log(-d/e + x)/(128*d**7*e**3) - (d**2*g**2 - 10*d*e*f*g - 15*e**2*f**2)*log(d/e + x)/(128*d**7*e**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.53 \[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx=-\frac {48 \, d^{5} e^{2} f^{2} - 32 \, d^{6} e f g - 16 \, d^{7} g^{2} + 3 \, {\left (15 \, e^{7} f^{2} + 10 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} + 6 \, {\left (15 \, d e^{6} f^{2} + 10 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (15 \, d^{2} e^{5} f^{2} + 10 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 10 \, {\left (15 \, d^{3} e^{4} f^{2} + 10 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} - {\left (51 \, d^{4} e^{3} f^{2} + 34 \, d^{5} e^{2} f g + 35 \, d^{6} e g^{2}\right )} x}{192 \, {\left (d^{6} e^{9} x^{6} + 2 \, d^{7} e^{8} x^{5} - d^{8} e^{7} x^{4} - 4 \, d^{9} e^{6} x^{3} - d^{10} e^{5} x^{2} + 2 \, d^{11} e^{4} x + d^{12} e^{3}\right )}} + \frac {{\left (15 \, e^{2} f^{2} + 10 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{128 \, d^{7} e^{3}} - \frac {{\left (15 \, e^{2} f^{2} + 10 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{128 \, d^{7} e^{3}} \]

[In]

integrate((g*x+f)^2/(e*x+d)^2/(-e^2*x^2+d^2)^3,x, algorithm="maxima")

[Out]

-1/192*(48*d^5*e^2*f^2 - 32*d^6*e*f*g - 16*d^7*g^2 + 3*(15*e^7*f^2 + 10*d*e^6*f*g - d^2*e^5*g^2)*x^5 + 6*(15*d
*e^6*f^2 + 10*d^2*e^5*f*g - d^3*e^4*g^2)*x^4 - 2*(15*d^2*e^5*f^2 + 10*d^3*e^4*f*g - d^4*e^3*g^2)*x^3 - 10*(15*
d^3*e^4*f^2 + 10*d^4*e^3*f*g - d^5*e^2*g^2)*x^2 - (51*d^4*e^3*f^2 + 34*d^5*e^2*f*g + 35*d^6*e*g^2)*x)/(d^6*e^9
*x^6 + 2*d^7*e^8*x^5 - d^8*e^7*x^4 - 4*d^9*e^6*x^3 - d^10*e^5*x^2 + 2*d^11*e^4*x + d^12*e^3) + 1/128*(15*e^2*f
^2 + 10*d*e*f*g - d^2*g^2)*log(e*x + d)/(d^7*e^3) - 1/128*(15*e^2*f^2 + 10*d*e*f*g - d^2*g^2)*log(e*x - d)/(d^
7*e^3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.43 \[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx=-\frac {{\left (15 \, e^{2} f^{2} + 10 \, d e f g - d^{2} g^{2}\right )} \log \left ({\left | -\frac {2 \, d}{e x + d} + 1 \right |}\right )}{128 \, d^{7} e^{3}} - \frac {11 \, e^{2} f^{2} + 14 \, d e f g + 3 \, d^{2} g^{2} - \frac {8 \, {\left (3 \, d e^{3} f^{2} + 4 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )}}{{\left (e x + d\right )} e}}{256 \, d^{7} e^{3} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{2}} - \frac {\frac {15 \, d^{6} e^{11} f^{2}}{e x + d} + \frac {9 \, d^{7} e^{11} f^{2}}{{\left (e x + d\right )}^{2}} + \frac {6 \, d^{8} e^{11} f^{2}}{{\left (e x + d\right )}^{3}} + \frac {3 \, d^{9} e^{11} f^{2}}{{\left (e x + d\right )}^{4}} + \frac {6 \, d^{7} e^{10} f g}{e x + d} - \frac {4 \, d^{9} e^{10} f g}{{\left (e x + d\right )}^{3}} - \frac {6 \, d^{10} e^{10} f g}{{\left (e x + d\right )}^{4}} - \frac {3 \, d^{8} e^{9} g^{2}}{e x + d} - \frac {3 \, d^{9} e^{9} g^{2}}{{\left (e x + d\right )}^{2}} - \frac {2 \, d^{10} e^{9} g^{2}}{{\left (e x + d\right )}^{3}} + \frac {3 \, d^{11} e^{9} g^{2}}{{\left (e x + d\right )}^{4}}}{96 \, d^{12} e^{12}} \]

[In]

integrate((g*x+f)^2/(e*x+d)^2/(-e^2*x^2+d^2)^3,x, algorithm="giac")

[Out]

-1/128*(15*e^2*f^2 + 10*d*e*f*g - d^2*g^2)*log(abs(-2*d/(e*x + d) + 1))/(d^7*e^3) - 1/256*(11*e^2*f^2 + 14*d*e
*f*g + 3*d^2*g^2 - 8*(3*d*e^3*f^2 + 4*d^2*e^2*f*g + d^3*e*g^2)/((e*x + d)*e))/(d^7*e^3*(2*d/(e*x + d) - 1)^2)
- 1/96*(15*d^6*e^11*f^2/(e*x + d) + 9*d^7*e^11*f^2/(e*x + d)^2 + 6*d^8*e^11*f^2/(e*x + d)^3 + 3*d^9*e^11*f^2/(
e*x + d)^4 + 6*d^7*e^10*f*g/(e*x + d) - 4*d^9*e^10*f*g/(e*x + d)^3 - 6*d^10*e^10*f*g/(e*x + d)^4 - 3*d^8*e^9*g
^2/(e*x + d) - 3*d^9*e^9*g^2/(e*x + d)^2 - 2*d^10*e^9*g^2/(e*x + d)^3 + 3*d^11*e^9*g^2/(e*x + d)^4)/(d^12*e^12
)

Mupad [B] (verification not implemented)

Time = 11.94 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.26 \[ \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^3} \, dx=\frac {\frac {d^2\,g^2+2\,d\,e\,f\,g-3\,e^2\,f^2}{12\,d\,e^3}+\frac {x^3\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{96\,d^4}-\frac {e\,x^4\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{32\,d^5}+\frac {x\,\left (35\,d^2\,g^2+34\,d\,e\,f\,g+51\,e^2\,f^2\right )}{192\,d^2\,e^2}+\frac {5\,x^2\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{96\,d^3\,e}-\frac {e^2\,x^5\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{64\,d^6}}{d^6+2\,d^5\,e\,x-d^4\,e^2\,x^2-4\,d^3\,e^3\,x^3-d^2\,e^4\,x^4+2\,d\,e^5\,x^5+e^6\,x^6}+\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{64\,d^7\,e^3} \]

[In]

int((f + g*x)^2/((d^2 - e^2*x^2)^3*(d + e*x)^2),x)

[Out]

((d^2*g^2 - 3*e^2*f^2 + 2*d*e*f*g)/(12*d*e^3) + (x^3*(15*e^2*f^2 - d^2*g^2 + 10*d*e*f*g))/(96*d^4) - (e*x^4*(1
5*e^2*f^2 - d^2*g^2 + 10*d*e*f*g))/(32*d^5) + (x*(35*d^2*g^2 + 51*e^2*f^2 + 34*d*e*f*g))/(192*d^2*e^2) + (5*x^
2*(15*e^2*f^2 - d^2*g^2 + 10*d*e*f*g))/(96*d^3*e) - (e^2*x^5*(15*e^2*f^2 - d^2*g^2 + 10*d*e*f*g))/(64*d^6))/(d
^6 + e^6*x^6 + 2*d*e^5*x^5 - d^4*e^2*x^2 - 4*d^3*e^3*x^3 - d^2*e^4*x^4 + 2*d^5*e*x) + (atanh((e*x)/d)*(15*e^2*
f^2 - d^2*g^2 + 10*d*e*f*g))/(64*d^7*e^3)