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

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

[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)

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Rubi [A]  time = 0.273199, antiderivative size = 235, normalized size of antiderivative = 1., 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} \[ -\frac{-d^2 g^2+2 d e f g+5 e^2 f^2}{32 d^6 e^3 (d+e x)}-\frac{3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2}+\frac{\left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{64 d^7 e^3}-\frac{(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}-\frac{(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^3}+\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d-e x)}+\frac{(d g+e f)^2}{64 d^5 e^3 (d-e x)^2} \]

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 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]))

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]

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*}

Mathematica [A]  time = 0.173548, size = 244, normalized size = 1.04 \[ \frac{\frac{8 d^3 \left (d^2 g^2+2 d e f g-3 e^2 f^2\right )}{(d+e x)^3}+\frac{12 d^2 \left (d^2 g^2-3 e^2 f^2\right )}{(d+e x)^2}+\frac{6 d \left (d^2 g^2+6 d e f g+5 e^2 f^2\right )}{d-e x}+\frac{12 d \left (d^2 g^2-2 d e f g-5 e^2 f^2\right )}{d+e x}+3 \left (d^2 g^2-10 d e f g-15 e^2 f^2\right ) \log (d-e x)+3 \left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \log (d+e x)-\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{6 d^2 (d g+e f)^2}{(d-e x)^2}}{384 d^7 e^3} \]

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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Maple [A]  time = 0.059, size = 421, normalized size = 1.8 \begin{align*} -{\frac{{g}^{2}}{64\,{e}^{3}{d}^{4} \left ( ex-d \right ) }}-{\frac{3\,fg}{32\,{e}^{2}{d}^{5} \left ( ex-d \right ) }}-{\frac{5\,{f}^{2}}{64\,e{d}^{6} \left ( ex-d \right ) }}+{\frac{{g}^{2}}{64\,{d}^{3}{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{32\,{e}^{2}{d}^{4} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{64\,e{d}^{5} \left ( ex-d \right ) ^{2}}}+{\frac{\ln \left ( ex-d \right ){g}^{2}}{128\,{e}^{3}{d}^{5}}}-{\frac{5\,\ln \left ( ex-d \right ) fg}{64\,{e}^{2}{d}^{6}}}-{\frac{15\,\ln \left ( ex-d \right ){f}^{2}}{128\,e{d}^{7}}}+{\frac{{g}^{2}}{32\,{d}^{3}{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{f}^{2}}{32\,e{d}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{{g}^{2}}{48\,{e}^{3}{d}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{fg}{24\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{16\,e{d}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{{g}^{2}}{32\,{e}^{3}{d}^{4} \left ( ex+d \right ) }}-{\frac{fg}{16\,{e}^{2}{d}^{5} \left ( ex+d \right ) }}-{\frac{5\,{f}^{2}}{32\,e{d}^{6} \left ( ex+d \right ) }}-{\frac{\ln \left ( ex+d \right ){g}^{2}}{128\,{e}^{3}{d}^{5}}}+{\frac{5\,\ln \left ( ex+d \right ) fg}{64\,{e}^{2}{d}^{6}}}+{\frac{15\,\ln \left ( ex+d \right ){f}^{2}}{128\,e{d}^{7}}}-{\frac{{g}^{2}}{32\,d{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{fg}{16\,{d}^{2}{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{32\,e{d}^{3} \left ( ex+d \right ) ^{4}}} \end{align*}

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

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Maxima [A]  time = 1.09176, size = 485, normalized size = 2.06 \begin{align*} -\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{align*}

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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Fricas [B]  time = 1.85028, size = 1604, normalized size = 6.83 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 2.40839, size = 371, normalized size = 1.58 \begin{align*} \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{align*}

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 + 30*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 + 34*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)*l
og(-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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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