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

Optimal. Leaf size=235 \[ \frac {(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d-e x)}+\frac {(d g+e f)^2}{64 d^5 e^3 (d-e x)^2}-\frac {(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}+\frac {\left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{64 d^7 e^3}-\frac {-d^2 g^2+2 d e f g+5 e^2 f^2}{32 d^6 e^3 (d+e x)}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2} \]

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Rubi [A]  time = 0.27, antiderivative size = 235, 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 {-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}+\frac {\left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{64 d^7 e^3}-\frac {(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}-\frac {(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^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} \end {gather*}

Antiderivative was successfully verified.

[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 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)^2 \left (d^2-e^2 x^2\right )^3} \, dx &=\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*}

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Mathematica [A]  time = 0.17, size = 244, normalized size = 1.04 \begin {gather*} \frac {-\frac {12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac {12 d^2 \left (d^2 g^2-3 e^2 f^2\right )}{(d+e x)^2}+\frac {6 d \left (d^2 g^2+6 d e f g+5 e^2 f^2\right )}{d-e x}+\frac {12 d \left (d^2 g^2-2 d e f g-5 e^2 f^2\right )}{d+e x}+3 \left (d^2 g^2-10 d e f g-15 e^2 f^2\right ) \log (d-e x)+3 \left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \log (d+e x)+\frac {6 d^2 (d g+e f)^2}{(d-e x)^2}+\frac {8 d^3 \left (d^2 g^2+2 d e f g-3 e^2 f^2\right )}{(d+e x)^3}}{384 d^7 e^3} \end {gather*}

Antiderivative was successfully verified.

[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)

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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)^2 \left (d^2-e^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 793, normalized size = 3.37 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -((exp(1)*x+d)^-1/exp(1)*g^2*d^2*exp(1)^
10-2*(exp(1)*x+d)^-1/exp(1)*g*d*exp(1)^11*f+(exp(1)*x+d)^-1/exp(1)*exp(1)^12*f^2)/(d^6*exp(1)^12-3*d^6*exp(1)^
10*exp(2)+3*d^6*exp(1)^8*exp(2)^2-d^6*exp(1)^6*exp(2)^3)-((5*g^2*d^5*exp(1)^12+50*g^2*d^5*exp(1)^10*exp(2)-20*
g^2*d^5*exp(1)^8*exp(2)^2-34*g^2*d^5*exp(1)^6*exp(2)^3-g^2*d^5*exp(1)^4*exp(2)^4-68*g*d^4*exp(1)^11*exp(2)*f-5
2*g*d^4*exp(1)^9*exp(2)^2*f+116*g*d^4*exp(1)^7*exp(2)^3*f+4*g*d^4*exp(1)^5*exp(2)^4*f+9*d^3*exp(1)^12*exp(2)*f
^2+66*d^3*exp(1)^10*exp(2)^2*f^2-60*d^3*exp(1)^8*exp(2)^3*f^2-18*d^3*exp(1)^6*exp(2)^4*f^2+3*d^3*exp(1)^4*exp(
2)^5*f^2)*(-(exp(1)*x+d)^-1/exp(1))^3+(17*g^2*d^4*exp(1)^9*exp(2)-85*g^2*d^4*exp(1)^7*exp(2)^2-89*g^2*d^4*exp(
1)^5*exp(2)^3-3*g^2*d^4*exp(1)^3*exp(2)^4-16*g*d^3*exp(1)^10*exp(2)*f+44*g*d^3*exp(1)^8*exp(2)^2*f+280*g*d^3*e
xp(1)^6*exp(2)^3*f+12*g*d^3*exp(1)^4*exp(2)^4*f+21*d^2*exp(1)^9*exp(2)^2*f^2-145*d^2*exp(1)^7*exp(2)^3*f^2-45*
d^2*exp(1)^5*exp(2)^4*f^2+9*d^2*exp(1)^3*exp(2)^5*f^2)*(-(exp(1)*x+d)^-1/exp(1))^2-(-3*g^2*d^3*exp(1)^8*exp(2)
-77*g^2*d^3*exp(1)^6*exp(2)^2-77*g^2*d^3*exp(1)^4*exp(2)^3-3*g^2*d^3*exp(1)^2*exp(2)^4+76*g*d^2*exp(1)^7*exp(2
)^2*f+232*g*d^2*exp(1)^5*exp(2)^3*f+12*g*d^2*exp(1)^3*exp(2)^4*f-7*d*exp(1)^8*exp(2)^2*f^2-121*d*exp(1)^6*exp(
2)^3*f^2-41*d*exp(1)^4*exp(2)^4*f^2+9*d*exp(1)^2*exp(2)^5*f^2)*(exp(1)*x+d)^-1/exp(1)-17*g^2*d^2*exp(1)^5*exp(
2)^2-22*g^2*d^2*exp(1)^3*exp(2)^3-g^2*d^2*exp(1)*exp(2)^4+12*g*d*exp(1)^6*exp(2)^2*f+64*g*d*exp(1)^4*exp(2)^3*
f+4*g*d*exp(1)^2*exp(2)^4*f-29*exp(1)^5*exp(2)^3*f^2-14*exp(1)^3*exp(2)^4*f^2+3*exp(1)*exp(2)^5*f^2)/8/d^7/(ex
p(2)-exp(1)^2)^4/((-(exp(1)*x+d)^-1/exp(1))^2*d^2*exp(1)^4-(-(exp(1)*x+d)^-1/exp(1))^2*d^2*exp(1)^2*exp(2)+2*(
exp(1)*x+d)^-1/exp(1)*d*exp(1)*exp(2)-exp(2))^2-(g^2*d^2*exp(1)^5+2*g^2*d^2*exp(1)^3*exp(2)-g*d*exp(1)^6*f-5*g
*d*exp(1)^4*f*exp(2)+3*exp(1)^5*f^2*exp(2))/(-d^7*exp(1)^8+4*d^7*exp(1)^6*exp(2)-6*d^7*exp(1)^4*exp(2)^2+4*d^7
*exp(1)^2*exp(2)^3-d^7*exp(2)^4)*ln(abs((-(exp(1)*x+d)^-1/exp(1))^2*d^2*exp(1)^4-(-(exp(1)*x+d)^-1/exp(1))^2*d
^2*exp(1)^2*exp(2)+2*(exp(1)*x+d)^-1/exp(1)*d*exp(1)*exp(2)-exp(2)))-(3*g^2*d^2*exp(1)^8+33*g^2*d^2*exp(1)^6*e
xp(2)+13*g^2*d^2*exp(1)^4*exp(2)^2-g^2*d^2*exp(1)^2*exp(2)^3-60*g*d*exp(1)^7*f*exp(2)-40*g*d*exp(1)^5*f*exp(2)
^2+4*g*d*exp(1)^3*f*exp(2)^3+15*exp(1)^8*f^2*exp(2)+45*exp(1)^6*f^2*exp(2)^2-15*exp(1)^4*f^2*exp(2)^3+3*exp(1)
^2*f^2*exp(2)^4)/2/(-8*d^6*exp(1)^8+32*d^6*exp(1)^6*exp(2)-48*d^6*exp(1)^4*exp(2)^2+32*d^6*exp(1)^2*exp(2)^3-8
*d^6*exp(2)^4)/exp(1)/abs(d)/exp(1)^2*ln(abs(-2*(exp(1)*x+d)^-1/exp(1)*d^2*exp(1)^4+2*(exp(1)*x+d)^-1/exp(1)*d
^2*exp(1)^2*exp(2)-2*d*exp(1)*exp(2)-2*exp(1)*abs(d)*exp(1)^2)/abs(-2*(exp(1)*x+d)^-1/exp(1)*d^2*exp(1)^4+2*(e
xp(1)*x+d)^-1/exp(1)*d^2*exp(1)^2*exp(2)-2*d*exp(1)*exp(2)+2*exp(1)*abs(d)*exp(1)^2))

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maple [A]  time = 0.02, size = 421, normalized size = 1.79 \begin {gather*} -\frac {g^{2}}{32 \left (e x +d \right )^{4} d \,e^{3}}+\frac {f g}{16 \left (e x +d \right )^{4} d^{2} e^{2}}-\frac {f^{2}}{32 \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 {g^{2}}{64 \left (e x -d \right )^{2} d^{3} e^{3}}+\frac {g^{2}}{32 \left (e x +d \right )^{2} d^{3} e^{3}}+\frac {f g}{32 \left (e x -d \right )^{2} d^{4} e^{2}}+\frac {f^{2}}{64 \left (e x -d \right )^{2} d^{5} e}-\frac {3 f^{2}}{32 \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}}{32 \left (e x +d \right ) d^{4} e^{3}}-\frac {3 f g}{32 \left (e x -d \right ) d^{5} e^{2}}-\frac {f g}{16 \left (e x +d \right ) d^{5} e^{2}}+\frac {g^{2} \ln \left (e x -d \right )}{128 d^{5} e^{3}}-\frac {g^{2} \ln \left (e x +d \right )}{128 d^{5} e^{3}}-\frac {5 f^{2}}{64 \left (e x -d \right ) d^{6} e}-\frac {5 f^{2}}{32 \left (e x +d \right ) d^{6} e}-\frac {5 f g \ln \left (e x -d \right )}{64 d^{6} e^{2}}+\frac {5 f g \ln \left (e x +d \right )}{64 d^{6} e^{2}}-\frac {15 f^{2} \ln \left (e x -d \right )}{128 d^{7} e}+\frac {15 f^{2} \ln \left (e x +d \right )}{128 d^{7} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64/(e*x-d)/d^4/e^3*g^2-3/32/(e*x-d)/d^5/e^2*f*g-5/64/(e*x-d)/d^6/e*f^2+1/64/e^3/d^3/(e*x-d)^2*g^2+1/32/e^2/
d^4/(e*x-d)^2*f*g+1/64/e/d^5/(e*x-d)^2*f^2+1/128/d^5/e^3*g^2*ln(e*x-d)-5/64/d^6/e^2*f*g*ln(e*x-d)-15/128/d^7/e
*f^2*ln(e*x-d)+1/32/e^3/d^3/(e*x+d)^2*g^2-3/32/(e*x+d)^2/d^5/e*f^2+1/48/(e*x+d)^3/d^2/e^3*g^2+1/24/(e*x+d)^3/d
^3/e^2*f*g-1/16/(e*x+d)^3/d^4/e*f^2+1/32/(e*x+d)/d^4/e^3*g^2-1/16/(e*x+d)/d^5/e^2*f*g-5/32/(e*x+d)/d^6/e*f^2-1
/128/d^5/e^3*g^2*ln(e*x+d)+5/64/d^6/e^2*f*g*ln(e*x+d)+15/128/d^7/e*f^2*ln(e*x+d)-1/32/(e*x+d)^4/d/e^3*g^2+1/16
/e^2/d^2/(e*x+d)^4*f*g-1/32/(e*x+d)^4/d^3/e*f^2

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maxima [A]  time = 0.51, size = 359, normalized size = 1.53 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 2.64, size = 296, normalized size = 1.26 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 2.15, size = 372, normalized size = 1.58 \begin {gather*} - \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} \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} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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