3.559 \(\int \frac{(d+e x)^5 (f+g x)^2}{(d^2-e^2 x^2)^2} \, dx\)

Optimal. Leaf size=146 \[ \frac{x^2 \left (12 d^2 g^2+10 d e f g+e^2 f^2\right )}{2 e}+\frac{d x \left (20 d^2 g^2+24 d e f g+5 e^2 f^2\right )}{e^2}+\frac{8 d^3 (d g+e f)^2}{e^3 (d-e x)}+\frac{4 d^2 (d g+e f) (7 d g+3 e f) \log (d-e x)}{e^3}+\frac{1}{3} g x^3 (5 d g+2 e f)+\frac{1}{4} e g^2 x^4 \]

[Out]

(d*(5*e^2*f^2 + 24*d*e*f*g + 20*d^2*g^2)*x)/e^2 + ((e^2*f^2 + 10*d*e*f*g + 12*d^2*g^2)*x^2)/(2*e) + (g*(2*e*f
+ 5*d*g)*x^3)/3 + (e*g^2*x^4)/4 + (8*d^3*(e*f + d*g)^2)/(e^3*(d - e*x)) + (4*d^2*(e*f + d*g)*(3*e*f + 7*d*g)*L
og[d - e*x])/e^3

________________________________________________________________________________________

Rubi [A]  time = 0.181323, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {848, 88} \[ \frac{x^2 \left (12 d^2 g^2+10 d e f g+e^2 f^2\right )}{2 e}+\frac{d x \left (20 d^2 g^2+24 d e f g+5 e^2 f^2\right )}{e^2}+\frac{8 d^3 (d g+e f)^2}{e^3 (d-e x)}+\frac{4 d^2 (d g+e f) (7 d g+3 e f) \log (d-e x)}{e^3}+\frac{1}{3} g x^3 (5 d g+2 e f)+\frac{1}{4} e g^2 x^4 \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^5*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]

[Out]

(d*(5*e^2*f^2 + 24*d*e*f*g + 20*d^2*g^2)*x)/e^2 + ((e^2*f^2 + 10*d*e*f*g + 12*d^2*g^2)*x^2)/(2*e) + (g*(2*e*f
+ 5*d*g)*x^3)/3 + (e*g^2*x^4)/4 + (8*d^3*(e*f + d*g)^2)/(e^3*(d - e*x)) + (4*d^2*(e*f + d*g)*(3*e*f + 7*d*g)*L
og[d - e*x])/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]))

Rubi steps

\begin{align*} \int \frac{(d+e x)^5 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^3 (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac{d \left (5 e^2 f^2+24 d e f g+20 d^2 g^2\right )}{e^2}+\frac{\left (e^2 f^2+10 d e f g+12 d^2 g^2\right ) x}{e}+g (2 e f+5 d g) x^2+e g^2 x^3+\frac{4 d^2 (-3 e f-7 d g) (e f+d g)}{e^2 (d-e x)}+\frac{8 d^3 (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac{d \left (5 e^2 f^2+24 d e f g+20 d^2 g^2\right ) x}{e^2}+\frac{\left (e^2 f^2+10 d e f g+12 d^2 g^2\right ) x^2}{2 e}+\frac{1}{3} g (2 e f+5 d g) x^3+\frac{1}{4} e g^2 x^4+\frac{8 d^3 (e f+d g)^2}{e^3 (d-e x)}+\frac{4 d^2 (e f+d g) (3 e f+7 d g) \log (d-e x)}{e^3}\\ \end{align*}

Mathematica [A]  time = 0.0918917, size = 154, normalized size = 1.05 \[ \frac{x^2 \left (12 d^2 g^2+10 d e f g+e^2 f^2\right )}{2 e}+\frac{d x \left (20 d^2 g^2+24 d e f g+5 e^2 f^2\right )}{e^2}+\frac{4 d^2 \left (7 d^2 g^2+10 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}-\frac{8 d^3 (d g+e f)^2}{e^3 (e x-d)}+\frac{1}{3} g x^3 (5 d g+2 e f)+\frac{1}{4} e g^2 x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^5*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]

[Out]

(d*(5*e^2*f^2 + 24*d*e*f*g + 20*d^2*g^2)*x)/e^2 + ((e^2*f^2 + 10*d*e*f*g + 12*d^2*g^2)*x^2)/(2*e) + (g*(2*e*f
+ 5*d*g)*x^3)/3 + (e*g^2*x^4)/4 - (8*d^3*(e*f + d*g)^2)/(e^3*(-d + e*x)) + (4*d^2*(3*e^2*f^2 + 10*d*e*f*g + 7*
d^2*g^2)*Log[d - e*x])/e^3

________________________________________________________________________________________

Maple [A]  time = 0.052, size = 204, normalized size = 1.4 \begin{align*}{\frac{e{g}^{2}{x}^{4}}{4}}+{\frac{5\,{x}^{3}d{g}^{2}}{3}}+{\frac{2\,e{x}^{3}fg}{3}}+6\,{\frac{{x}^{2}{d}^{2}{g}^{2}}{e}}+5\,{x}^{2}dfg+{\frac{e{x}^{2}{f}^{2}}{2}}+20\,{\frac{{d}^{3}{g}^{2}x}{{e}^{2}}}+24\,{\frac{{d}^{2}fgx}{e}}+5\,d{f}^{2}x+28\,{\frac{{d}^{4}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+40\,{\frac{{d}^{3}\ln \left ( ex-d \right ) fg}{{e}^{2}}}+12\,{\frac{{d}^{2}\ln \left ( ex-d \right ){f}^{2}}{e}}-8\,{\frac{{d}^{5}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}-16\,{\frac{{d}^{4}fg}{{e}^{2} \left ( ex-d \right ) }}-8\,{\frac{{d}^{3}{f}^{2}}{e \left ( ex-d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5*(g*x+f)^2/(-e^2*x^2+d^2)^2,x)

[Out]

1/4*e*g^2*x^4+5/3*x^3*d*g^2+2/3*e*x^3*f*g+6/e*x^2*d^2*g^2+5*x^2*d*f*g+1/2*e*x^2*f^2+20/e^2*d^3*g^2*x+24/e*d^2*
f*g*x+5*d*f^2*x+28*d^4/e^3*ln(e*x-d)*g^2+40*d^3/e^2*ln(e*x-d)*f*g+12*d^2/e*ln(e*x-d)*f^2-8*d^5/e^3/(e*x-d)*g^2
-16*d^4/e^2/(e*x-d)*f*g-8*d^3/e/(e*x-d)*f^2

________________________________________________________________________________________

Maxima [A]  time = 0.969659, size = 246, normalized size = 1.68 \begin{align*} -\frac{8 \,{\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac{3 \, e^{3} g^{2} x^{4} + 4 \,{\left (2 \, e^{3} f g + 5 \, d e^{2} g^{2}\right )} x^{3} + 6 \,{\left (e^{3} f^{2} + 10 \, d e^{2} f g + 12 \, d^{2} e g^{2}\right )} x^{2} + 12 \,{\left (5 \, d e^{2} f^{2} + 24 \, d^{2} e f g + 20 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} + \frac{4 \,{\left (3 \, d^{2} e^{2} f^{2} + 10 \, d^{3} e f g + 7 \, d^{4} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="maxima")

[Out]

-8*(d^3*e^2*f^2 + 2*d^4*e*f*g + d^5*g^2)/(e^4*x - d*e^3) + 1/12*(3*e^3*g^2*x^4 + 4*(2*e^3*f*g + 5*d*e^2*g^2)*x
^3 + 6*(e^3*f^2 + 10*d*e^2*f*g + 12*d^2*e*g^2)*x^2 + 12*(5*d*e^2*f^2 + 24*d^2*e*f*g + 20*d^3*g^2)*x)/e^2 + 4*(
3*d^2*e^2*f^2 + 10*d^3*e*f*g + 7*d^4*g^2)*log(e*x - d)/e^3

________________________________________________________________________________________

Fricas [A]  time = 1.72244, size = 529, normalized size = 3.62 \begin{align*} \frac{3 \, e^{5} g^{2} x^{5} - 96 \, d^{3} e^{2} f^{2} - 192 \, d^{4} e f g - 96 \, d^{5} g^{2} +{\left (8 \, e^{5} f g + 17 \, d e^{4} g^{2}\right )} x^{4} + 2 \,{\left (3 \, e^{5} f^{2} + 26 \, d e^{4} f g + 26 \, d^{2} e^{3} g^{2}\right )} x^{3} + 6 \,{\left (9 \, d e^{4} f^{2} + 38 \, d^{2} e^{3} f g + 28 \, d^{3} e^{2} g^{2}\right )} x^{2} - 12 \,{\left (5 \, d^{2} e^{3} f^{2} + 24 \, d^{3} e^{2} f g + 20 \, d^{4} e g^{2}\right )} x - 48 \,{\left (3 \, d^{3} e^{2} f^{2} + 10 \, d^{4} e f g + 7 \, d^{5} g^{2} -{\left (3 \, d^{2} e^{3} f^{2} + 10 \, d^{3} e^{2} f g + 7 \, d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{12 \,{\left (e^{4} x - d e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="fricas")

[Out]

1/12*(3*e^5*g^2*x^5 - 96*d^3*e^2*f^2 - 192*d^4*e*f*g - 96*d^5*g^2 + (8*e^5*f*g + 17*d*e^4*g^2)*x^4 + 2*(3*e^5*
f^2 + 26*d*e^4*f*g + 26*d^2*e^3*g^2)*x^3 + 6*(9*d*e^4*f^2 + 38*d^2*e^3*f*g + 28*d^3*e^2*g^2)*x^2 - 12*(5*d^2*e
^3*f^2 + 24*d^3*e^2*f*g + 20*d^4*e*g^2)*x - 48*(3*d^3*e^2*f^2 + 10*d^4*e*f*g + 7*d^5*g^2 - (3*d^2*e^3*f^2 + 10
*d^3*e^2*f*g + 7*d^4*e*g^2)*x)*log(e*x - d))/(e^4*x - d*e^3)

________________________________________________________________________________________

Sympy [A]  time = 0.952896, size = 167, normalized size = 1.14 \begin{align*} \frac{4 d^{2} \left (d g + e f\right ) \left (7 d g + 3 e f\right ) \log{\left (- d + e x \right )}}{e^{3}} + \frac{e g^{2} x^{4}}{4} + x^{3} \left (\frac{5 d g^{2}}{3} + \frac{2 e f g}{3}\right ) - \frac{8 d^{5} g^{2} + 16 d^{4} e f g + 8 d^{3} e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac{x^{2} \left (12 d^{2} g^{2} + 10 d e f g + e^{2} f^{2}\right )}{2 e} + \frac{x \left (20 d^{3} g^{2} + 24 d^{2} e f g + 5 d e^{2} f^{2}\right )}{e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5*(g*x+f)**2/(-e**2*x**2+d**2)**2,x)

[Out]

4*d**2*(d*g + e*f)*(7*d*g + 3*e*f)*log(-d + e*x)/e**3 + e*g**2*x**4/4 + x**3*(5*d*g**2/3 + 2*e*f*g/3) - (8*d**
5*g**2 + 16*d**4*e*f*g + 8*d**3*e**2*f**2)/(-d*e**3 + e**4*x) + x**2*(12*d**2*g**2 + 10*d*e*f*g + e**2*f**2)/(
2*e) + x*(20*d**3*g**2 + 24*d**2*e*f*g + 5*d*e**2*f**2)/e**2

________________________________________________________________________________________

Giac [B]  time = 1.23424, size = 393, normalized size = 2.69 \begin{align*} 2 \,{\left (7 \, d^{4} g^{2} e^{5} + 10 \, d^{3} f g e^{6} + 3 \, d^{2} f^{2} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{1}{12} \,{\left (3 \, g^{2} x^{4} e^{17} + 20 \, d g^{2} x^{3} e^{16} + 72 \, d^{2} g^{2} x^{2} e^{15} + 240 \, d^{3} g^{2} x e^{14} + 8 \, f g x^{3} e^{17} + 60 \, d f g x^{2} e^{16} + 288 \, d^{2} f g x e^{15} + 6 \, f^{2} x^{2} e^{17} + 60 \, d f^{2} x e^{16}\right )} e^{\left (-16\right )} + \frac{2 \,{\left (7 \, d^{5} g^{2} e^{4} + 10 \, d^{4} f g e^{5} + 3 \, d^{3} f^{2} e^{6}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac{8 \,{\left (d^{6} g^{2} e^{5} + 2 \, d^{5} f g e^{6} + d^{4} f^{2} e^{7} +{\left (d^{5} g^{2} e^{6} + 2 \, d^{4} f g e^{7} + d^{3} f^{2} e^{8}\right )} x\right )} e^{\left (-8\right )}}{x^{2} e^{2} - d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="giac")

[Out]

2*(7*d^4*g^2*e^5 + 10*d^3*f*g*e^6 + 3*d^2*f^2*e^7)*e^(-8)*log(abs(x^2*e^2 - d^2)) + 1/12*(3*g^2*x^4*e^17 + 20*
d*g^2*x^3*e^16 + 72*d^2*g^2*x^2*e^15 + 240*d^3*g^2*x*e^14 + 8*f*g*x^3*e^17 + 60*d*f*g*x^2*e^16 + 288*d^2*f*g*x
*e^15 + 6*f^2*x^2*e^17 + 60*d*f^2*x*e^16)*e^(-16) + 2*(7*d^5*g^2*e^4 + 10*d^4*f*g*e^5 + 3*d^3*f^2*e^6)*e^(-7)*
log(abs(2*x*e^2 - 2*abs(d)*e)/abs(2*x*e^2 + 2*abs(d)*e))/abs(d) - 8*(d^6*g^2*e^5 + 2*d^5*f*g*e^6 + d^4*f^2*e^7
+ (d^5*g^2*e^6 + 2*d^4*f*g*e^7 + d^3*f^2*e^8)*x)*e^(-8)/(x^2*e^2 - d^2)