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

Optimal. Leaf size=210 \[ \frac {(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^7 e^3}+\frac {(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac {(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac {f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac {(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4} \]

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Rubi [A]  time = 0.24, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {848, 88, 208} \begin {gather*} -\frac {e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac {(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}+\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 {(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^7 e^3} \end {gather*}

Antiderivative was successfully verified.

[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 88

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 208

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

Rule 848

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*x)/e)^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*} \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx &=\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*}

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Mathematica [A]  time = 0.18, size = 229, normalized size = 1.09 \begin {gather*} \frac {-\frac {48 d^5 (e f-d g)^2}{(d+e x)^5}-\frac {15 d \left (d^2 g^2+6 d e f g+5 e^2 f^2\right )}{d+e x}-15 \left (d^2 g^2+4 d e f g+3 e^2 f^2\right ) \log (d-e x)+15 \left (d^2 g^2+4 d e f g+3 e^2 f^2\right ) \log (d+e x)-\frac {60 d^2 e f (d g+e f)}{(d+e x)^2}+\frac {60 d^4 \left (d^2 g^2-e^2 f^2\right )}{(d+e x)^4}+\frac {20 d^3 \left (d^2 g^2-2 d e f g-3 e^2 f^2\right )}{(d+e x)^3}+\frac {15 d (d g+e f)^2}{d-e x}}{960 d^7 e^3} \end {gather*}

Antiderivative was successfully verified.

[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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 693, normalized size = 3.30 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (-2*exp(2)^3*d^2*exp(1)*g^2+10*exp(2)^3*
d*exp(1)^2*g*f-10*exp(2)^3*exp(1)^3*f^2-10*exp(2)^2*d^2*exp(1)^3*g^2+20*exp(2)^2*d*exp(1)^4*g*f-6*exp(2)^2*exp
(1)^5*f^2-4*exp(2)*d^2*exp(1)^5*g^2+2*exp(2)*d*exp(1)^6*g*f)/(exp(2)^5*d^7-5*exp(2)^4*d^7*exp(1)^2+10*exp(2)^3
*d^7*exp(1)^4-10*exp(2)^2*d^7*exp(1)^6+5*exp(2)*d^7*exp(1)^8-d^7*exp(1)^10)*ln(abs(-x^2*exp(2)+d^2))+(exp(2)^5
*f^2-exp(2)^4*d^2*g^2+8*exp(2)^4*d*exp(1)*g*f-15*exp(2)^4*exp(1)^2*f^2-25*exp(2)^3*d^2*exp(1)^2*g^2+80*exp(2)^
3*d*exp(1)^3*g*f-45*exp(2)^3*exp(1)^4*f^2-35*exp(2)^2*d^2*exp(1)^4*g^2+40*exp(2)^2*d*exp(1)^5*g*f-5*exp(2)^2*e
xp(1)^6*f^2-3*exp(2)*d^2*exp(1)^6*g^2)*1/2/(2*exp(2)^5*d^6-10*exp(2)^4*d^6*exp(1)^2+20*exp(2)^3*d^6*exp(1)^4-2
0*exp(2)^2*d^6*exp(1)^6+10*exp(2)*d^6*exp(1)^8-2*d^6*exp(1)^10)/exp(1)/abs(d)*ln(abs(-2*x*exp(2)-2*exp(1)*abs(
d))/abs(-2*x*exp(2)+2*exp(1)*abs(d)))+(4*exp(2)^3*d^2*exp(1)^2*g^2-20*exp(2)^3*d*exp(1)^3*g*f+20*exp(2)^3*exp(
1)^4*f^2+20*exp(2)^2*d^2*exp(1)^4*g^2-40*exp(2)^2*d*exp(1)^5*g*f+12*exp(2)^2*exp(1)^6*f^2+8*exp(2)*d^2*exp(1)^
6*g^2-4*exp(2)*d*exp(1)^7*g*f)/(exp(2)^5*d^7*exp(1)-5*exp(2)^4*d^7*exp(1)^3+10*exp(2)^3*d^7*exp(1)^5-10*exp(2)
^2*d^7*exp(1)^7+5*exp(2)*d^7*exp(1)^9-d^7*exp(1)^11)*ln(abs(x*exp(1)+d))-((-3*exp(2)^5*d*exp(1)^3*f^2-21*exp(2
)^4*d^3*exp(1)^3*g^2+96*exp(2)^4*d^2*exp(1)^4*g*f-75*exp(2)^4*d*exp(1)^5*f^2-45*exp(2)^3*d^3*exp(1)^5*g^2+63*e
xp(2)^3*d*exp(1)^7*f^2+57*exp(2)^2*d^3*exp(1)^7*g^2-96*exp(2)^2*d^2*exp(1)^8*g*f+15*exp(2)^2*d*exp(1)^9*f^2+9*
exp(2)*d^3*exp(1)^9*g^2)*x^4+(-9*exp(2)^5*d^2*exp(1)^2*f^2-51*exp(2)^4*d^4*exp(1)^2*g^2+228*exp(2)^4*d^3*exp(1
)^3*g*f-165*exp(2)^4*d^2*exp(1)^4*f^2-87*exp(2)^3*d^4*exp(1)^4*g^2-60*exp(2)^3*d^3*exp(1)^5*g*f+165*exp(2)^3*d
^2*exp(1)^6*f^2+135*exp(2)^2*d^4*exp(1)^6*g^2-180*exp(2)^2*d^3*exp(1)^7*g*f+9*exp(2)^2*d^2*exp(1)^8*f^2+3*exp(
2)*d^4*exp(1)^8*g^2+12*exp(2)*d^3*exp(1)^9*g*f)*x^3+(-9*exp(2)^5*d^3*exp(1)*f^2-35*exp(2)^4*d^5*exp(1)*g^2+148
*exp(2)^4*d^4*exp(1)^2*g*f-83*exp(2)^4*d^3*exp(1)^3*f^2-9*exp(2)^3*d^5*exp(1)^3*g^2-204*exp(2)^3*d^4*exp(1)^4*
g*f+183*exp(2)^3*d^3*exp(1)^5*f^2+117*exp(2)^2*d^5*exp(1)^5*g^2-36*exp(2)^2*d^4*exp(1)^6*g*f-81*exp(2)^2*d^3*e
xp(1)^7*f^2-67*exp(2)*d^5*exp(1)^7*g^2+92*exp(2)*d^4*exp(1)^8*g*f-10*exp(2)*d^3*exp(1)^9*f^2-6*d^5*exp(1)^9*g^
2)*x^2+(-3*exp(2)^5*d^4*f^2-3*exp(2)^4*d^6*g^2+6*exp(2)^4*d^5*exp(1)*g*f+21*exp(2)^4*d^4*exp(1)^2*f^2+63*exp(2
)^3*d^6*exp(1)^2*g^2-252*exp(2)^3*d^5*exp(1)^3*g*f+147*exp(2)^3*d^4*exp(1)^4*f^2+69*exp(2)^2*d^6*exp(1)^4*g^2+
96*exp(2)^2*d^5*exp(1)^5*g*f-153*exp(2)^2*d^4*exp(1)^6*f^2-123*exp(2)*d^6*exp(1)^6*g^2+156*exp(2)*d^5*exp(1)^7
*g*f-12*exp(2)*d^4*exp(1)^8*f^2-6*d^6*exp(1)^8*g^2-6*d^5*exp(1)^9*g*f)*x-6*exp(2)^4*d^6*g*f+12*exp(2)^4*d^5*ex
p(1)*f^2+38*exp(2)^3*d^7*exp(1)*g^2-124*exp(2)^3*d^6*exp(1)^2*g*f+74*exp(2)^3*d^5*exp(1)^3*f^2+18*exp(2)^2*d^7
*exp(1)^3*g^2+72*exp(2)^2*d^6*exp(1)^4*g*f-90*exp(2)^2*d^5*exp(1)^5*f^2-54*exp(2)*d^7*exp(1)^5*g^2+60*exp(2)*d
^6*exp(1)^6*g*f+6*exp(2)*d^5*exp(1)^7*f^2-2*d^7*exp(1)^7*g^2-2*d^6*exp(1)^8*g*f-2*d^5*exp(1)^9*f^2)/6/d^7/(exp
(2)-exp(1)^2)^5/(-x*exp(1)-d)^3/(x^2*exp(2)-d^2)

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maple [B]  time = 0.02, size = 394, normalized size = 1.88 \begin {gather*} \frac {f g}{10 \left (e x +d \right )^{5} d \,e^{2}}-\frac {f^{2}}{20 \left (e x +d \right )^{5} d^{2} e}-\frac {g^{2}}{20 \left (e x +d \right )^{5} e^{3}}+\frac {g^{2}}{16 \left (e x +d \right )^{4} d \,e^{3}}-\frac {f^{2}}{16 \left (e x +d \right )^{4} d^{3} e}+\frac {g^{2}}{48 \left (e x +d \right )^{3} d^{2} e^{3}}-\frac {f g}{24 \left (e x +d \right )^{3} d^{3} e^{2}}-\frac {f^{2}}{16 \left (e x +d \right )^{3} d^{4} e}-\frac {f g}{16 \left (e x +d \right )^{2} d^{4} e^{2}}-\frac {f^{2}}{16 \left (e x +d \right )^{2} d^{5} e}-\frac {g^{2}}{64 \left (e x -d \right ) d^{4} e^{3}}-\frac {g^{2}}{64 \left (e x +d \right ) d^{4} e^{3}}-\frac {f g}{32 \left (e x -d \right ) d^{5} e^{2}}-\frac {3 f g}{32 \left (e x +d \right ) d^{5} e^{2}}-\frac {g^{2} \ln \left (e x -d \right )}{64 d^{5} e^{3}}+\frac {g^{2} \ln \left (e x +d \right )}{64 d^{5} e^{3}}-\frac {f^{2}}{64 \left (e x -d \right ) d^{6} e}-\frac {5 f^{2}}{64 \left (e x +d \right ) d^{6} e}-\frac {f g \ln \left (e x -d \right )}{16 d^{6} e^{2}}+\frac {f g \ln \left (e x +d \right )}{16 d^{6} e^{2}}-\frac {3 f^{2} \ln \left (e x -d \right )}{64 d^{7} e}+\frac {3 f^{2} \ln \left (e x +d \right )}{64 d^{7} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64/e^3/d^5*ln(e*x-d)*g^2-1/16/e^2/d^6*ln(e*x-d)*f*g-3/64/e/d^7*ln(e*x-d)*f^2-1/64/e^3/d^4/(e*x-d)*g^2-1/32/
e^2/d^5/(e*x-d)*f*g-1/64/e/d^6/(e*x-d)*f^2+1/64/e^3/d^5*ln(e*x+d)*g^2+1/16/e^2/d^6*ln(e*x+d)*f*g+3/64/e/d^7*ln
(e*x+d)*f^2-1/64/e^3/d^4/(e*x+d)*g^2-3/32/e^2/d^5/(e*x+d)*f*g-5/64/e/d^6/(e*x+d)*f^2+1/16/e^3/d/(e*x+d)^4*g^2-
1/16/e/d^3/(e*x+d)^4*f^2+1/48/e^3/d^2/(e*x+d)^3*g^2-1/24/e^2/d^3/(e*x+d)^3*f*g-1/16/e/d^4/(e*x+d)^3*f^2-1/20/e
^3/(e*x+d)^5*g^2+1/10/e^2/d/(e*x+d)^5*f*g-1/20/e/d^2/(e*x+d)^5*f^2-1/16/e^2*f/d^4/(e*x+d)^2*g-1/16/e*f^2/d^5/(
e*x+d)^2

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maxima [A]  time = 0.53, size = 342, normalized size = 1.63 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 2.72, size = 314, normalized size = 1.50 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 2.15, size = 427, normalized size = 2.03 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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