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

Optimal. Leaf size=188 \[ -\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac{\left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^6 e^3}-\frac{(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}-\frac{(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac{f (d g+e f)}{8 d^5 e^2 (d-e x)}+\frac{(d g+e f)^2}{32 d^4 e^3 (d-e x)^2} \]

[Out]

(e*f + d*g)^2/(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)^2/(24*d^3*e^3*(d
+ e*x)^3) - ((e*f - d*g)*(3*e*f + d*g))/(32*d^4*e^3*(d + e*x)^2) - (3*e^2*f^2 - d^2*g^2)/(16*d^5*e^3*(d + e*x)
) + ((5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*ArcTanh[(e*x)/d])/(16*d^6*e^3)

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Rubi [A]  time = 0.209689, antiderivative size = 188, 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} \[ -\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac{\left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^6 e^3}-\frac{(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}-\frac{(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac{f (d g+e f)}{8 d^5 e^2 (d-e x)}+\frac{(d g+e f)^2}{32 d^4 e^3 (d-e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*f + d*g)^2/(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)^2/(24*d^3*e^3*(d
+ e*x)^3) - ((e*f - d*g)*(3*e*f + d*g))/(32*d^4*e^3*(d + e*x)^2) - (3*e^2*f^2 - d^2*g^2)/(16*d^5*e^3*(d + e*x)
) + ((5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*ArcTanh[(e*x)/d])/(16*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) \left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^3 (d+e x)^4} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{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)^2}{8 d^3 e^2 (d+e x)^4}+\frac{(e f-d g) (3 e f+d g)}{16 d^4 e^2 (d+e x)^3}+\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^2 (d+e x)^2}+\frac{-5 e^2 f^2-2 d e f g+d^2 g^2}{16 d^5 e^2 \left (-d^2+e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{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)^2}{24 d^3 e^3 (d+e x)^3}-\frac{(e f-d g) (3 e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}-\frac{\left (5 e^2 f^2+2 d e f g-d^2 g^2\right ) \int \frac{1}{-d^2+e^2 x^2} \, dx}{16 d^5 e^2}\\ &=\frac{(e f+d g)^2}{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)^2}{24 d^3 e^3 (d+e x)^3}-\frac{(e f-d g) (3 e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac{\left (5 e^2 f^2+2 d e f g-d^2 g^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^6 e^3}\\ \end{align*}

Mathematica [A]  time = 0.157574, size = 197, normalized size = 1.05 \[ \frac{\frac{3 d^2 \left (d^2 g^2+2 d e f g-3 e^2 f^2\right )}{(d+e x)^2}+\frac{6 d \left (d^2 g^2-3 e^2 f^2\right )}{d+e x}+3 \left (d^2 g^2-2 d e f g-5 e^2 f^2\right ) \log (d-e x)+3 \left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \log (d+e x)-\frac{4 d^3 (e f-d g)^2}{(d+e x)^3}+\frac{3 d^2 (d g+e f)^2}{(d-e x)^2}+\frac{12 d e f (d g+e f)}{d-e x}}{96 d^6 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*d^2*(e*f + d*g)^2)/(d - e*x)^2 + (12*d*e*f*(e*f + d*g))/(d - e*x) - (4*d^3*(e*f - d*g)^2)/(d + e*x)^3 + (3
*d^2*(-3*e^2*f^2 + 2*d*e*f*g + d^2*g^2))/(d + e*x)^2 + (6*d*(-3*e^2*f^2 + d^2*g^2))/(d + e*x) + 3*(-5*e^2*f^2
- 2*d*e*f*g + d^2*g^2)*Log[d - e*x] + 3*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*Log[d + e*x])/(96*d^6*e^3)

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Maple [A]  time = 0.058, size = 348, normalized size = 1.9 \begin{align*}{\frac{{g}^{2}}{32\,{d}^{2}{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{16\,{e}^{2}{d}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{32\,e{d}^{4} \left ( ex-d \right ) ^{2}}}+{\frac{\ln \left ( ex-d \right ){g}^{2}}{32\,{e}^{3}{d}^{4}}}-{\frac{\ln \left ( ex-d \right ) fg}{16\,{e}^{2}{d}^{5}}}-{\frac{5\,\ln \left ( ex-d \right ){f}^{2}}{32\,e{d}^{6}}}-{\frac{fg}{8\,{e}^{2}{d}^{4} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{5} \left ( ex-d \right ) }}+{\frac{{g}^{2}}{16\,{e}^{3}{d}^{3} \left ( ex+d \right ) }}-{\frac{3\,{f}^{2}}{16\,e{d}^{5} \left ( ex+d \right ) }}+{\frac{{g}^{2}}{32\,{d}^{2}{e}^{3} \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{\ln \left ( ex+d \right ){g}^{2}}{32\,{e}^{3}{d}^{4}}}+{\frac{\ln \left ( ex+d \right ) fg}{16\,{e}^{2}{d}^{5}}}+{\frac{5\,\ln \left ( ex+d \right ){f}^{2}}{32\,e{d}^{6}}}-{\frac{{g}^{2}}{24\,d{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{fg}{12\,{d}^{2}{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{24\,e{d}^{3} \left ( ex+d \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32/e^3/d^2/(e*x-d)^2*g^2+1/16/e^2/d^3/(e*x-d)^2*f*g+1/32/e/d^4/(e*x-d)^2*f^2+1/32/e^3/d^4*ln(e*x-d)*g^2-1/16
/e^2/d^5*ln(e*x-d)*f*g-5/32/e/d^6*ln(e*x-d)*f^2-1/8/e^2/d^4/(e*x-d)*f*g-1/8/e/d^5/(e*x-d)*f^2+1/16/e^3/d^3/(e*
x+d)*g^2-3/16/e*f^2/d^5/(e*x+d)+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
-1/32/e^3/d^4*ln(e*x+d)*g^2+1/16/e^2/d^5*ln(e*x+d)*f*g+5/32/e/d^6*ln(e*x+d)*f^2-1/24/e^3/d/(e*x+d)^3*g^2+1/12/
e^2/d^2/(e*x+d)^3*f*g-1/24/e/d^3/(e*x+d)^3*f^2

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Maxima [A]  time = 1.07283, size = 416, normalized size = 2.21 \begin{align*} -\frac{8 \, d^{4} e^{2} f^{2} - 16 \, d^{5} e f g - 4 \, d^{6} g^{2} + 3 \,{\left (5 \, e^{6} f^{2} + 2 \, d e^{5} f g - d^{2} e^{4} g^{2}\right )} x^{4} + 3 \,{\left (5 \, d e^{5} f^{2} + 2 \, d^{2} e^{4} f g - d^{3} e^{3} g^{2}\right )} x^{3} - 5 \,{\left (5 \, d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g - d^{4} e^{2} g^{2}\right )} x^{2} -{\left (25 \, d^{3} e^{3} f^{2} + 10 \, d^{4} e^{2} f g + 7 \, d^{5} e g^{2}\right )} x}{48 \,{\left (d^{5} e^{8} x^{5} + d^{6} e^{7} x^{4} - 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} + d^{9} e^{4} x + d^{10} e^{3}\right )}} + \frac{{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{6} e^{3}} - \frac{{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, 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)/(-e^2*x^2+d^2)^3,x, algorithm="maxima")

[Out]

-1/48*(8*d^4*e^2*f^2 - 16*d^5*e*f*g - 4*d^6*g^2 + 3*(5*e^6*f^2 + 2*d*e^5*f*g - d^2*e^4*g^2)*x^4 + 3*(5*d*e^5*f
^2 + 2*d^2*e^4*f*g - d^3*e^3*g^2)*x^3 - 5*(5*d^2*e^4*f^2 + 2*d^3*e^3*f*g - d^4*e^2*g^2)*x^2 - (25*d^3*e^3*f^2
+ 10*d^4*e^2*f*g + 7*d^5*e*g^2)*x)/(d^5*e^8*x^5 + d^6*e^7*x^4 - 2*d^7*e^6*x^3 - 2*d^8*e^5*x^2 + d^9*e^4*x + d^
10*e^3) + 1/32*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*log(e*x + d)/(d^6*e^3) - 1/32*(5*e^2*f^2 + 2*d*e*f*g - d^2*g^
2)*log(e*x - d)/(d^6*e^3)

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Fricas [B]  time = 1.79197, size = 1307, normalized size = 6.95 \begin{align*} -\frac{16 \, d^{5} e^{2} f^{2} - 32 \, d^{6} e f g - 8 \, d^{7} g^{2} + 6 \,{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} + 6 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 10 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (25 \, d^{4} e^{3} f^{2} + 10 \, d^{5} e^{2} f g + 7 \, d^{6} e g^{2}\right )} x - 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 2 \, d^{6} e f g - d^{7} g^{2} +{\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} +{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} +{\left (5 \, d^{4} e^{3} f^{2} + 2 \, 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} + 2 \, d^{6} e f g - d^{7} g^{2} +{\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} +{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} +{\left (5 \, d^{4} e^{3} f^{2} + 2 \, d^{5} e^{2} f g - d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \,{\left (d^{6} e^{8} x^{5} + d^{7} e^{7} x^{4} - 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} + 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)/(-e^2*x^2+d^2)^3,x, algorithm="fricas")

[Out]

-1/96*(16*d^5*e^2*f^2 - 32*d^6*e*f*g - 8*d^7*g^2 + 6*(5*d*e^6*f^2 + 2*d^2*e^5*f*g - d^3*e^4*g^2)*x^4 + 6*(5*d^
2*e^5*f^2 + 2*d^3*e^4*f*g - d^4*e^3*g^2)*x^3 - 10*(5*d^3*e^4*f^2 + 2*d^4*e^3*f*g - d^5*e^2*g^2)*x^2 - 2*(25*d^
4*e^3*f^2 + 10*d^5*e^2*f*g + 7*d^6*e*g^2)*x - 3*(5*d^5*e^2*f^2 + 2*d^6*e*f*g - d^7*g^2 + (5*e^7*f^2 + 2*d*e^6*
f*g - d^2*e^5*g^2)*x^5 + (5*d*e^6*f^2 + 2*d^2*e^5*f*g - d^3*e^4*g^2)*x^4 - 2*(5*d^2*e^5*f^2 + 2*d^3*e^4*f*g -
d^4*e^3*g^2)*x^3 - 2*(5*d^3*e^4*f^2 + 2*d^4*e^3*f*g - d^5*e^2*g^2)*x^2 + (5*d^4*e^3*f^2 + 2*d^5*e^2*f*g - d^6*
e*g^2)*x)*log(e*x + d) + 3*(5*d^5*e^2*f^2 + 2*d^6*e*f*g - d^7*g^2 + (5*e^7*f^2 + 2*d*e^6*f*g - d^2*e^5*g^2)*x^
5 + (5*d*e^6*f^2 + 2*d^2*e^5*f*g - d^3*e^4*g^2)*x^4 - 2*(5*d^2*e^5*f^2 + 2*d^3*e^4*f*g - d^4*e^3*g^2)*x^3 - 2*
(5*d^3*e^4*f^2 + 2*d^4*e^3*f*g - d^5*e^2*g^2)*x^2 + (5*d^4*e^3*f^2 + 2*d^5*e^2*f*g - d^6*e*g^2)*x)*log(e*x - d
))/(d^6*e^8*x^5 + d^7*e^7*x^4 - 2*d^8*e^6*x^3 - 2*d^9*e^5*x^2 + d^10*e^4*x + d^11*e^3)

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Sympy [A]  time = 2.1155, size = 320, normalized size = 1.7 \begin{align*} \frac{4 d^{6} g^{2} + 16 d^{5} e f g - 8 d^{4} e^{2} f^{2} + x^{4} \left (3 d^{2} e^{4} g^{2} - 6 d e^{5} f g - 15 e^{6} f^{2}\right ) + x^{3} \left (3 d^{3} e^{3} g^{2} - 6 d^{2} e^{4} f g - 15 d e^{5} f^{2}\right ) + x^{2} \left (- 5 d^{4} e^{2} g^{2} + 10 d^{3} e^{3} f g + 25 d^{2} e^{4} f^{2}\right ) + x \left (7 d^{5} e g^{2} + 10 d^{4} e^{2} f g + 25 d^{3} e^{3} f^{2}\right )}{48 d^{10} e^{3} + 48 d^{9} e^{4} x - 96 d^{8} e^{5} x^{2} - 96 d^{7} e^{6} x^{3} + 48 d^{6} e^{7} x^{4} + 48 d^{5} e^{8} x^{5}} + \frac{\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log{\left (- \frac{d}{e} + x \right )}}{32 d^{6} e^{3}} - \frac{\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log{\left (\frac{d}{e} + x \right )}}{32 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)/(-e**2*x**2+d**2)**3,x)

[Out]

(4*d**6*g**2 + 16*d**5*e*f*g - 8*d**4*e**2*f**2 + x**4*(3*d**2*e**4*g**2 - 6*d*e**5*f*g - 15*e**6*f**2) + x**3
*(3*d**3*e**3*g**2 - 6*d**2*e**4*f*g - 15*d*e**5*f**2) + x**2*(-5*d**4*e**2*g**2 + 10*d**3*e**3*f*g + 25*d**2*
e**4*f**2) + x*(7*d**5*e*g**2 + 10*d**4*e**2*f*g + 25*d**3*e**3*f**2))/(48*d**10*e**3 + 48*d**9*e**4*x - 96*d*
*8*e**5*x**2 - 96*d**7*e**6*x**3 + 48*d**6*e**7*x**4 + 48*d**5*e**8*x**5) + (d**2*g**2 - 2*d*e*f*g - 5*e**2*f*
*2)*log(-d/e + x)/(32*d**6*e**3) - (d**2*g**2 - 2*d*e*f*g - 5*e**2*f**2)*log(d/e + x)/(32*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)/(-e^2*x^2+d^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError