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

Optimal. Leaf size=178 \[ -\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}+\frac{(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac{f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac{(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}+\frac{(d g+e f) (d g+5 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^6 e^3} \]

[Out]

(e*f + d*g)^2/(32*d^5*e^3*(d - e*x)) - (e*f - d*g)^2/(16*d^2*e^3*(d + e*x)^4) - (e^2*f^2 - d^2*g^2)/(12*d^3*e^
3*(d + e*x)^3) - ((3*e*f - d*g)*(e*f + d*g))/(32*d^4*e^3*(d + e*x)^2) - (f*(e*f + d*g))/(8*d^5*e^2*(d + e*x))
+ ((e*f + d*g)*(5*e*f + d*g)*ArcTanh[(e*x)/d])/(32*d^6*e^3)

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Rubi [A]  time = 0.199561, antiderivative size = 178, 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} \[ -\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}+\frac{(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac{f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac{(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}+\frac{(d g+e f) (d g+5 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^6 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*f + d*g)^2/(32*d^5*e^3*(d - e*x)) - (e*f - d*g)^2/(16*d^2*e^3*(d + e*x)^4) - (e^2*f^2 - d^2*g^2)/(12*d^3*e^
3*(d + e*x)^3) - ((3*e*f - d*g)*(e*f + d*g))/(32*d^4*e^3*(d + e*x)^2) - (f*(e*f + d*g))/(8*d^5*e^2*(d + e*x))
+ ((e*f + d*g)*(5*e*f + d*g)*ArcTanh[(e*x)/d])/(32*d^6*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)^3 \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^2 (d+e x)^5} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{32 d^5 e^2 (d-e x)^2}+\frac{(-e f+d g)^2}{4 d^2 e^2 (d+e x)^5}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^4}+\frac{(3 e f-d g) (e f+d g)}{16 d^4 e^2 (d+e x)^3}+\frac{f (e f+d g)}{8 d^5 e (d+e x)^2}+\frac{(e f+d g) (5 e f+d g)}{32 d^5 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac{(3 e f-d g) (e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac{f (e f+d g)}{8 d^5 e^2 (d+e x)}+\frac{((e f+d g) (5 e f+d g)) \int \frac{1}{d^2-e^2 x^2} \, dx}{32 d^5 e^2}\\ &=\frac{(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac{(3 e f-d g) (e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac{f (e f+d g)}{8 d^5 e^2 (d+e x)}+\frac{(e f+d g) (5 e f+d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^6 e^3}\\ \end{align*}

Mathematica [A]  time = 0.149004, size = 195, normalized size = 1.1 \[ \frac{\frac{16 d^3 \left (d^2 g^2-e^2 f^2\right )}{(d+e x)^3}+\frac{6 d^2 \left (d^2 g^2-2 d e f g-3 e^2 f^2\right )}{(d+e x)^2}-3 \left (d^2 g^2+6 d e f g+5 e^2 f^2\right ) \log (d-e x)+3 \left (d^2 g^2+6 d e f g+5 e^2 f^2\right ) \log (d+e x)-\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{6 d (d g+e f)^2}{d-e x}-\frac{24 d e f (d g+e f)}{d+e x}}{192 d^6 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*d*(e*f + d*g)^2)/(d - e*x) - (12*d^4*(e*f - d*g)^2)/(d + e*x)^4 + (16*d^3*(-(e^2*f^2) + d^2*g^2))/(d + e*x
)^3 + (6*d^2*(-3*e^2*f^2 - 2*d*e*f*g + d^2*g^2))/(d + e*x)^2 - (24*d*e*f*(e*f + d*g))/(d + e*x) - 3*(5*e^2*f^2
+ 6*d*e*f*g + d^2*g^2)*Log[d - e*x] + 3*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2)*Log[d + e*x])/(192*d^6*e^3)

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Maple [B]  time = 0.059, size = 341, normalized size = 1.9 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{64\,{e}^{3}{d}^{4}}}-{\frac{3\,\ln \left ( ex-d \right ) fg}{32\,{e}^{2}{d}^{5}}}-{\frac{5\,\ln \left ( ex-d \right ){f}^{2}}{64\,e{d}^{6}}}-{\frac{{g}^{2}}{32\,{d}^{3}{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{16\,{e}^{2}{d}^{4} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{32\,e{d}^{5} \left ( ex-d \right ) }}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{64\,{e}^{3}{d}^{4}}}+{\frac{3\,\ln \left ( ex+d \right ) fg}{32\,{e}^{2}{d}^{5}}}+{\frac{5\,\ln \left ( ex+d \right ){f}^{2}}{64\,e{d}^{6}}}+{\frac{{g}^{2}}{12\,{e}^{3}d \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{12\,e{d}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{{g}^{2}}{32\,{e}^{3}{d}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{fg}{16\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{f}^{2}}{32\,e{d}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{g}^{2}}{16\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{fg}{8\,d{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{16\,e{d}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{fg}{8\,{e}^{2}{d}^{4} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{5} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64/e^3/d^4*ln(e*x-d)*g^2-3/32/e^2/d^5*ln(e*x-d)*f*g-5/64/e/d^6*ln(e*x-d)*f^2-1/32/e^3/d^3/(e*x-d)*g^2-1/16/
e^2/d^4/(e*x-d)*f*g-1/32/e/d^5/(e*x-d)*f^2+1/64/e^3/d^4*ln(e*x+d)*g^2+3/32/e^2/d^5*ln(e*x+d)*f*g+5/64/e/d^6*ln
(e*x+d)*f^2+1/12/e^3/d/(e*x+d)^3*g^2-1/12/e/d^3/(e*x+d)^3*f^2+1/32/e^3/d^2/(e*x+d)^2*g^2-1/16/e^2/d^3/(e*x+d)^
2*f*g-3/32/e/d^4/(e*x+d)^2*f^2-1/16/e^3/(e*x+d)^4*g^2+1/8/d/e^2/(e*x+d)^4*f*g-1/16/d^2/e/(e*x+d)^4*f^2-1/8/e^2
*f/d^4/(e*x+d)*g-1/8/e*f^2/d^5/(e*x+d)

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Maxima [A]  time = 1.03643, size = 402, normalized size = 2.26 \begin{align*} \frac{32 \, d^{4} e^{2} f^{2} - 8 \, d^{6} g^{2} - 3 \,{\left (5 \, e^{6} f^{2} + 6 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} - 9 \,{\left (5 \, d e^{5} f^{2} + 6 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} - 7 \,{\left (5 \, d^{2} e^{4} f^{2} + 6 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{3} e^{3} f^{2} + 6 \, d^{4} e^{2} f g - 7 \, d^{5} e g^{2}\right )} x}{96 \,{\left (d^{5} e^{8} x^{5} + 3 \, d^{6} e^{7} x^{4} + 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} - 3 \, d^{9} e^{4} x - d^{10} e^{3}\right )}} + \frac{{\left (5 \, e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{64 \, d^{6} e^{3}} - \frac{{\left (5 \, e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{64 \, d^{6} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(32*d^4*e^2*f^2 - 8*d^6*g^2 - 3*(5*e^6*f^2 + 6*d*e^5*f*g + d^2*e^4*g^2)*x^4 - 9*(5*d*e^5*f^2 + 6*d^2*e^4*
f*g + d^3*e^3*g^2)*x^3 - 7*(5*d^2*e^4*f^2 + 6*d^3*e^3*f*g + d^4*e^2*g^2)*x^2 + 3*(5*d^3*e^3*f^2 + 6*d^4*e^2*f*
g - 7*d^5*e*g^2)*x)/(d^5*e^8*x^5 + 3*d^6*e^7*x^4 + 2*d^7*e^6*x^3 - 2*d^8*e^5*x^2 - 3*d^9*e^4*x - d^10*e^3) + 1
/64*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2)*log(e*x + d)/(d^6*e^3) - 1/64*(5*e^2*f^2 + 6*d*e*f*g + d^2*g^2)*log(e*x
- d)/(d^6*e^3)

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Fricas [B]  time = 2.02957, size = 1303, normalized size = 7.32 \begin{align*} \frac{64 \, d^{5} e^{2} f^{2} - 16 \, d^{7} g^{2} - 6 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 18 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} - 14 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 6 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g - 7 \, d^{6} e g^{2}\right )} x - 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 6 \, d^{6} e f g + d^{7} g^{2} -{\left (5 \, e^{7} f^{2} + 6 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 3 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} + 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g + d^{6} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 6 \, d^{6} e f g + d^{7} g^{2} -{\left (5 \, e^{7} f^{2} + 6 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 3 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} + 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g + d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{192 \,{\left (d^{6} e^{8} x^{5} + 3 \, d^{7} e^{7} x^{4} + 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} - 3 \, d^{10} e^{4} x - d^{11} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(64*d^5*e^2*f^2 - 16*d^7*g^2 - 6*(5*d*e^6*f^2 + 6*d^2*e^5*f*g + d^3*e^4*g^2)*x^4 - 18*(5*d^2*e^5*f^2 + 6
*d^3*e^4*f*g + d^4*e^3*g^2)*x^3 - 14*(5*d^3*e^4*f^2 + 6*d^4*e^3*f*g + d^5*e^2*g^2)*x^2 + 6*(5*d^4*e^3*f^2 + 6*
d^5*e^2*f*g - 7*d^6*e*g^2)*x - 3*(5*d^5*e^2*f^2 + 6*d^6*e*f*g + d^7*g^2 - (5*e^7*f^2 + 6*d*e^6*f*g + d^2*e^5*g
^2)*x^5 - 3*(5*d*e^6*f^2 + 6*d^2*e^5*f*g + d^3*e^4*g^2)*x^4 - 2*(5*d^2*e^5*f^2 + 6*d^3*e^4*f*g + d^4*e^3*g^2)*
x^3 + 2*(5*d^3*e^4*f^2 + 6*d^4*e^3*f*g + d^5*e^2*g^2)*x^2 + 3*(5*d^4*e^3*f^2 + 6*d^5*e^2*f*g + d^6*e*g^2)*x)*l
og(e*x + d) + 3*(5*d^5*e^2*f^2 + 6*d^6*e*f*g + d^7*g^2 - (5*e^7*f^2 + 6*d*e^6*f*g + d^2*e^5*g^2)*x^5 - 3*(5*d*
e^6*f^2 + 6*d^2*e^5*f*g + d^3*e^4*g^2)*x^4 - 2*(5*d^2*e^5*f^2 + 6*d^3*e^4*f*g + d^4*e^3*g^2)*x^3 + 2*(5*d^3*e^
4*f^2 + 6*d^4*e^3*f*g + d^5*e^2*g^2)*x^2 + 3*(5*d^4*e^3*f^2 + 6*d^5*e^2*f*g + d^6*e*g^2)*x)*log(e*x - d))/(d^6
*e^8*x^5 + 3*d^7*e^7*x^4 + 2*d^8*e^6*x^3 - 2*d^9*e^5*x^2 - 3*d^10*e^4*x - d^11*e^3)

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Sympy [B]  time = 2.15653, size = 371, normalized size = 2.08 \begin{align*} - \frac{8 d^{6} g^{2} - 32 d^{4} e^{2} f^{2} + x^{4} \left (3 d^{2} e^{4} g^{2} + 18 d e^{5} f g + 15 e^{6} f^{2}\right ) + x^{3} \left (9 d^{3} e^{3} g^{2} + 54 d^{2} e^{4} f g + 45 d e^{5} f^{2}\right ) + x^{2} \left (7 d^{4} e^{2} g^{2} + 42 d^{3} e^{3} f g + 35 d^{2} e^{4} f^{2}\right ) + x \left (21 d^{5} e g^{2} - 18 d^{4} e^{2} f g - 15 d^{3} e^{3} f^{2}\right )}{- 96 d^{10} e^{3} - 288 d^{9} e^{4} x - 192 d^{8} e^{5} x^{2} + 192 d^{7} e^{6} x^{3} + 288 d^{6} e^{7} x^{4} + 96 d^{5} e^{8} x^{5}} - \frac{\left (d g + e f\right ) \left (d g + 5 e f\right ) \log{\left (- \frac{d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} + \frac{\left (d g + e f\right ) \left (d g + 5 e f\right ) \log{\left (\frac{d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*d**6*g**2 - 32*d**4*e**2*f**2 + x**4*(3*d**2*e**4*g**2 + 18*d*e**5*f*g + 15*e**6*f**2) + x**3*(9*d**3*e**3
*g**2 + 54*d**2*e**4*f*g + 45*d*e**5*f**2) + x**2*(7*d**4*e**2*g**2 + 42*d**3*e**3*f*g + 35*d**2*e**4*f**2) +
x*(21*d**5*e*g**2 - 18*d**4*e**2*f*g - 15*d**3*e**3*f**2))/(-96*d**10*e**3 - 288*d**9*e**4*x - 192*d**8*e**5*x
**2 + 192*d**7*e**6*x**3 + 288*d**6*e**7*x**4 + 96*d**5*e**8*x**5) - (d*g + e*f)*(d*g + 5*e*f)*log(-d*(d*g + e
*f)*(d*g + 5*e*f)/(e*(d**2*g**2 + 6*d*e*f*g + 5*e**2*f**2)) + x)/(64*d**6*e**3) + (d*g + e*f)*(d*g + 5*e*f)*lo
g(d*(d*g + e*f)*(d*g + 5*e*f)/(e*(d**2*g**2 + 6*d*e*f*g + 5*e**2*f**2)) + x)/(64*d**6*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)^3/(-e^2*x^2+d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError