3.557 \(\int \frac{(d+e x)^7 (f+g x)^2}{(d^2-e^2 x^2)^2} \, dx\)

Optimal. Leaf size=218 \[ \frac{1}{4} e x^4 \left (23 d^2 g^2+14 d e f g+e^2 f^2\right )+\frac{1}{3} d x^3 \left (49 d^2 g^2+46 d e f g+7 e^2 f^2\right )+\frac{d^2 x^2 \left (80 d^2 g^2+98 d e f g+23 e^2 f^2\right )}{2 e}+\frac{d^3 x \left (112 d^2 g^2+160 d e f g+49 e^2 f^2\right )}{e^2}+\frac{32 d^5 (d g+e f)^2}{e^3 (d-e x)}+\frac{16 d^4 (d g+e f) (9 d g+5 e f) \log (d-e x)}{e^3}+\frac{1}{5} e^2 g x^5 (7 d g+2 e f)+\frac{1}{6} e^3 g^2 x^6 \]

[Out]

(d^3*(49*e^2*f^2 + 160*d*e*f*g + 112*d^2*g^2)*x)/e^2 + (d^2*(23*e^2*f^2 + 98*d*e*f*g + 80*d^2*g^2)*x^2)/(2*e)
+ (d*(7*e^2*f^2 + 46*d*e*f*g + 49*d^2*g^2)*x^3)/3 + (e*(e^2*f^2 + 14*d*e*f*g + 23*d^2*g^2)*x^4)/4 + (e^2*g*(2*
e*f + 7*d*g)*x^5)/5 + (e^3*g^2*x^6)/6 + (32*d^5*(e*f + d*g)^2)/(e^3*(d - e*x)) + (16*d^4*(e*f + d*g)*(5*e*f +
9*d*g)*Log[d - e*x])/e^3

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Rubi [A]  time = 0.28372, antiderivative size = 218, 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} \[ \frac{1}{4} e x^4 \left (23 d^2 g^2+14 d e f g+e^2 f^2\right )+\frac{1}{3} d x^3 \left (49 d^2 g^2+46 d e f g+7 e^2 f^2\right )+\frac{d^2 x^2 \left (80 d^2 g^2+98 d e f g+23 e^2 f^2\right )}{2 e}+\frac{d^3 x \left (112 d^2 g^2+160 d e f g+49 e^2 f^2\right )}{e^2}+\frac{32 d^5 (d g+e f)^2}{e^3 (d-e x)}+\frac{16 d^4 (d g+e f) (9 d g+5 e f) \log (d-e x)}{e^3}+\frac{1}{5} e^2 g x^5 (7 d g+2 e f)+\frac{1}{6} e^3 g^2 x^6 \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^7*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]

[Out]

(d^3*(49*e^2*f^2 + 160*d*e*f*g + 112*d^2*g^2)*x)/e^2 + (d^2*(23*e^2*f^2 + 98*d*e*f*g + 80*d^2*g^2)*x^2)/(2*e)
+ (d*(7*e^2*f^2 + 46*d*e*f*g + 49*d^2*g^2)*x^3)/3 + (e*(e^2*f^2 + 14*d*e*f*g + 23*d^2*g^2)*x^4)/4 + (e^2*g*(2*
e*f + 7*d*g)*x^5)/5 + (e^3*g^2*x^6)/6 + (32*d^5*(e*f + d*g)^2)/(e^3*(d - e*x)) + (16*d^4*(e*f + d*g)*(5*e*f +
9*d*g)*Log[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)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^5 (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac{d^3 \left (49 e^2 f^2+160 d e f g+112 d^2 g^2\right )}{e^2}+\frac{d^2 \left (23 e^2 f^2+98 d e f g+80 d^2 g^2\right ) x}{e}+d \left (7 e^2 f^2+46 d e f g+49 d^2 g^2\right ) x^2+e \left (e^2 f^2+14 d e f g+23 d^2 g^2\right ) x^3+e^2 g (2 e f+7 d g) x^4+e^3 g^2 x^5+\frac{16 d^4 (-5 e f-9 d g) (e f+d g)}{e^2 (d-e x)}+\frac{32 d^5 (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac{d^3 \left (49 e^2 f^2+160 d e f g+112 d^2 g^2\right ) x}{e^2}+\frac{d^2 \left (23 e^2 f^2+98 d e f g+80 d^2 g^2\right ) x^2}{2 e}+\frac{1}{3} d \left (7 e^2 f^2+46 d e f g+49 d^2 g^2\right ) x^3+\frac{1}{4} e \left (e^2 f^2+14 d e f g+23 d^2 g^2\right ) x^4+\frac{1}{5} e^2 g (2 e f+7 d g) x^5+\frac{1}{6} e^3 g^2 x^6+\frac{32 d^5 (e f+d g)^2}{e^3 (d-e x)}+\frac{16 d^4 (e f+d g) (5 e f+9 d g) \log (d-e x)}{e^3}\\ \end{align*}

Mathematica [A]  time = 0.126223, size = 226, normalized size = 1.04 \[ \frac{1}{4} e x^4 \left (23 d^2 g^2+14 d e f g+e^2 f^2\right )+\frac{1}{3} d x^3 \left (49 d^2 g^2+46 d e f g+7 e^2 f^2\right )+\frac{d^2 x^2 \left (80 d^2 g^2+98 d e f g+23 e^2 f^2\right )}{2 e}+\frac{d^3 x \left (112 d^2 g^2+160 d e f g+49 e^2 f^2\right )}{e^2}+\frac{16 d^4 \left (9 d^2 g^2+14 d e f g+5 e^2 f^2\right ) \log (d-e x)}{e^3}-\frac{32 d^5 (d g+e f)^2}{e^3 (e x-d)}+\frac{1}{5} e^2 g x^5 (7 d g+2 e f)+\frac{1}{6} e^3 g^2 x^6 \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^7*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]

[Out]

(d^3*(49*e^2*f^2 + 160*d*e*f*g + 112*d^2*g^2)*x)/e^2 + (d^2*(23*e^2*f^2 + 98*d*e*f*g + 80*d^2*g^2)*x^2)/(2*e)
+ (d*(7*e^2*f^2 + 46*d*e*f*g + 49*d^2*g^2)*x^3)/3 + (e*(e^2*f^2 + 14*d*e*f*g + 23*d^2*g^2)*x^4)/4 + (e^2*g*(2*
e*f + 7*d*g)*x^5)/5 + (e^3*g^2*x^6)/6 - (32*d^5*(e*f + d*g)^2)/(e^3*(-d + e*x)) + (16*d^4*(5*e^2*f^2 + 14*d*e*
f*g + 9*d^2*g^2)*Log[d - e*x])/e^3

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Maple [A]  time = 0.053, size = 286, normalized size = 1.3 \begin{align*}{\frac{{e}^{3}{g}^{2}{x}^{6}}{6}}+{\frac{7\,{e}^{2}{x}^{5}d{g}^{2}}{5}}+{\frac{2\,{e}^{3}{x}^{5}fg}{5}}+{\frac{23\,e{x}^{4}{d}^{2}{g}^{2}}{4}}+{\frac{7\,{e}^{2}{x}^{4}dfg}{2}}+{\frac{{e}^{3}{x}^{4}{f}^{2}}{4}}+{\frac{49\,{x}^{3}{d}^{3}{g}^{2}}{3}}+{\frac{46\,e{x}^{3}{d}^{2}fg}{3}}+{\frac{7\,{e}^{2}{x}^{3}d{f}^{2}}{3}}+40\,{\frac{{x}^{2}{d}^{4}{g}^{2}}{e}}+49\,{x}^{2}{d}^{3}fg+{\frac{23\,e{x}^{2}{d}^{2}{f}^{2}}{2}}+112\,{\frac{{d}^{5}{g}^{2}x}{{e}^{2}}}+160\,{\frac{{d}^{4}fgx}{e}}+49\,{d}^{3}{f}^{2}x+144\,{\frac{{d}^{6}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+224\,{\frac{{d}^{5}\ln \left ( ex-d \right ) fg}{{e}^{2}}}+80\,{\frac{{d}^{4}\ln \left ( ex-d \right ){f}^{2}}{e}}-32\,{\frac{{d}^{7}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}-64\,{\frac{{d}^{6}fg}{{e}^{2} \left ( ex-d \right ) }}-32\,{\frac{{d}^{5}{f}^{2}}{e \left ( ex-d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^7*(g*x+f)^2/(-e^2*x^2+d^2)^2,x)

[Out]

1/6*e^3*g^2*x^6+7/5*e^2*x^5*d*g^2+2/5*e^3*x^5*f*g+23/4*e*x^4*d^2*g^2+7/2*e^2*x^4*d*f*g+1/4*e^3*x^4*f^2+49/3*x^
3*d^3*g^2+46/3*e*x^3*d^2*f*g+7/3*e^2*x^3*d*f^2+40/e*x^2*d^4*g^2+49*x^2*d^3*f*g+23/2*e*x^2*d^2*f^2+112/e^2*d^5*
g^2*x+160/e*d^4*f*g*x+49*d^3*f^2*x+144*d^6/e^3*ln(e*x-d)*g^2+224*d^5/e^2*ln(e*x-d)*f*g+80*d^4/e*ln(e*x-d)*f^2-
32*d^7/e^3/(e*x-d)*g^2-64*d^6/e^2/(e*x-d)*f*g-32*d^5/e/(e*x-d)*f^2

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Maxima [A]  time = 0.990392, size = 348, normalized size = 1.6 \begin{align*} -\frac{32 \,{\left (d^{5} e^{2} f^{2} + 2 \, d^{6} e f g + d^{7} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac{10 \, e^{5} g^{2} x^{6} + 12 \,{\left (2 \, e^{5} f g + 7 \, d e^{4} g^{2}\right )} x^{5} + 15 \,{\left (e^{5} f^{2} + 14 \, d e^{4} f g + 23 \, d^{2} e^{3} g^{2}\right )} x^{4} + 20 \,{\left (7 \, d e^{4} f^{2} + 46 \, d^{2} e^{3} f g + 49 \, d^{3} e^{2} g^{2}\right )} x^{3} + 30 \,{\left (23 \, d^{2} e^{3} f^{2} + 98 \, d^{3} e^{2} f g + 80 \, d^{4} e g^{2}\right )} x^{2} + 60 \,{\left (49 \, d^{3} e^{2} f^{2} + 160 \, d^{4} e f g + 112 \, d^{5} g^{2}\right )} x}{60 \, e^{2}} + \frac{16 \,{\left (5 \, d^{4} e^{2} f^{2} + 14 \, d^{5} e f g + 9 \, d^{6} 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)^7*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="maxima")

[Out]

-32*(d^5*e^2*f^2 + 2*d^6*e*f*g + d^7*g^2)/(e^4*x - d*e^3) + 1/60*(10*e^5*g^2*x^6 + 12*(2*e^5*f*g + 7*d*e^4*g^2
)*x^5 + 15*(e^5*f^2 + 14*d*e^4*f*g + 23*d^2*e^3*g^2)*x^4 + 20*(7*d*e^4*f^2 + 46*d^2*e^3*f*g + 49*d^3*e^2*g^2)*
x^3 + 30*(23*d^2*e^3*f^2 + 98*d^3*e^2*f*g + 80*d^4*e*g^2)*x^2 + 60*(49*d^3*e^2*f^2 + 160*d^4*e*f*g + 112*d^5*g
^2)*x)/e^2 + 16*(5*d^4*e^2*f^2 + 14*d^5*e*f*g + 9*d^6*g^2)*log(e*x - d)/e^3

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Fricas [A]  time = 1.70984, size = 716, normalized size = 3.28 \begin{align*} \frac{10 \, e^{7} g^{2} x^{7} - 1920 \, d^{5} e^{2} f^{2} - 3840 \, d^{6} e f g - 1920 \, d^{7} g^{2} + 2 \,{\left (12 \, e^{7} f g + 37 \, d e^{6} g^{2}\right )} x^{6} + 3 \,{\left (5 \, e^{7} f^{2} + 62 \, d e^{6} f g + 87 \, d^{2} e^{5} g^{2}\right )} x^{5} + 5 \,{\left (25 \, d e^{6} f^{2} + 142 \, d^{2} e^{5} f g + 127 \, d^{3} e^{4} g^{2}\right )} x^{4} + 10 \,{\left (55 \, d^{2} e^{5} f^{2} + 202 \, d^{3} e^{4} f g + 142 \, d^{4} e^{3} g^{2}\right )} x^{3} + 90 \,{\left (25 \, d^{3} e^{4} f^{2} + 74 \, d^{4} e^{3} f g + 48 \, d^{5} e^{2} g^{2}\right )} x^{2} - 60 \,{\left (49 \, d^{4} e^{3} f^{2} + 160 \, d^{5} e^{2} f g + 112 \, d^{6} e g^{2}\right )} x - 960 \,{\left (5 \, d^{5} e^{2} f^{2} + 14 \, d^{6} e f g + 9 \, d^{7} g^{2} -{\left (5 \, d^{4} e^{3} f^{2} + 14 \, d^{5} e^{2} f g + 9 \, d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{60 \,{\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)^7*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="fricas")

[Out]

1/60*(10*e^7*g^2*x^7 - 1920*d^5*e^2*f^2 - 3840*d^6*e*f*g - 1920*d^7*g^2 + 2*(12*e^7*f*g + 37*d*e^6*g^2)*x^6 +
3*(5*e^7*f^2 + 62*d*e^6*f*g + 87*d^2*e^5*g^2)*x^5 + 5*(25*d*e^6*f^2 + 142*d^2*e^5*f*g + 127*d^3*e^4*g^2)*x^4 +
10*(55*d^2*e^5*f^2 + 202*d^3*e^4*f*g + 142*d^4*e^3*g^2)*x^3 + 90*(25*d^3*e^4*f^2 + 74*d^4*e^3*f*g + 48*d^5*e^
2*g^2)*x^2 - 60*(49*d^4*e^3*f^2 + 160*d^5*e^2*f*g + 112*d^6*e*g^2)*x - 960*(5*d^5*e^2*f^2 + 14*d^6*e*f*g + 9*d
^7*g^2 - (5*d^4*e^3*f^2 + 14*d^5*e^2*f*g + 9*d^6*e*g^2)*x)*log(e*x - d))/(e^4*x - d*e^3)

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Sympy [A]  time = 1.20428, size = 255, normalized size = 1.17 \begin{align*} \frac{16 d^{4} \left (d g + e f\right ) \left (9 d g + 5 e f\right ) \log{\left (- d + e x \right )}}{e^{3}} + \frac{e^{3} g^{2} x^{6}}{6} + x^{5} \left (\frac{7 d e^{2} g^{2}}{5} + \frac{2 e^{3} f g}{5}\right ) + x^{4} \left (\frac{23 d^{2} e g^{2}}{4} + \frac{7 d e^{2} f g}{2} + \frac{e^{3} f^{2}}{4}\right ) + x^{3} \left (\frac{49 d^{3} g^{2}}{3} + \frac{46 d^{2} e f g}{3} + \frac{7 d e^{2} f^{2}}{3}\right ) - \frac{32 d^{7} g^{2} + 64 d^{6} e f g + 32 d^{5} e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac{x^{2} \left (80 d^{4} g^{2} + 98 d^{3} e f g + 23 d^{2} e^{2} f^{2}\right )}{2 e} + \frac{x \left (112 d^{5} g^{2} + 160 d^{4} e f g + 49 d^{3} e^{2} f^{2}\right )}{e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**7*(g*x+f)**2/(-e**2*x**2+d**2)**2,x)

[Out]

16*d**4*(d*g + e*f)*(9*d*g + 5*e*f)*log(-d + e*x)/e**3 + e**3*g**2*x**6/6 + x**5*(7*d*e**2*g**2/5 + 2*e**3*f*g
/5) + x**4*(23*d**2*e*g**2/4 + 7*d*e**2*f*g/2 + e**3*f**2/4) + x**3*(49*d**3*g**2/3 + 46*d**2*e*f*g/3 + 7*d*e*
*2*f**2/3) - (32*d**7*g**2 + 64*d**6*e*f*g + 32*d**5*e**2*f**2)/(-d*e**3 + e**4*x) + x**2*(80*d**4*g**2 + 98*d
**3*e*f*g + 23*d**2*e**2*f**2)/(2*e) + x*(112*d**5*g**2 + 160*d**4*e*f*g + 49*d**3*e**2*f**2)/e**2

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Giac [A]  time = 1.16513, size = 495, normalized size = 2.27 \begin{align*} 8 \,{\left (9 \, d^{6} g^{2} e^{7} + 14 \, d^{5} f g e^{8} + 5 \, d^{4} f^{2} e^{9}\right )} e^{\left (-10\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{1}{60} \,{\left (10 \, g^{2} x^{6} e^{27} + 84 \, d g^{2} x^{5} e^{26} + 345 \, d^{2} g^{2} x^{4} e^{25} + 980 \, d^{3} g^{2} x^{3} e^{24} + 2400 \, d^{4} g^{2} x^{2} e^{23} + 6720 \, d^{5} g^{2} x e^{22} + 24 \, f g x^{5} e^{27} + 210 \, d f g x^{4} e^{26} + 920 \, d^{2} f g x^{3} e^{25} + 2940 \, d^{3} f g x^{2} e^{24} + 9600 \, d^{4} f g x e^{23} + 15 \, f^{2} x^{4} e^{27} + 140 \, d f^{2} x^{3} e^{26} + 690 \, d^{2} f^{2} x^{2} e^{25} + 2940 \, d^{3} f^{2} x e^{24}\right )} e^{\left (-24\right )} + \frac{8 \,{\left (9 \, d^{7} g^{2} e^{6} + 14 \, d^{6} f g e^{7} + 5 \, d^{5} f^{2} e^{8}\right )} e^{\left (-9\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{32 \,{\left (d^{8} g^{2} e^{7} + 2 \, d^{7} f g e^{8} + d^{6} f^{2} e^{9} +{\left (d^{7} g^{2} e^{8} + 2 \, d^{6} f g e^{9} + d^{5} f^{2} e^{10}\right )} x\right )} e^{\left (-10\right )}}{x^{2} e^{2} - d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7*(g*x+f)^2/(-e^2*x^2+d^2)^2,x, algorithm="giac")

[Out]

8*(9*d^6*g^2*e^7 + 14*d^5*f*g*e^8 + 5*d^4*f^2*e^9)*e^(-10)*log(abs(x^2*e^2 - d^2)) + 1/60*(10*g^2*x^6*e^27 + 8
4*d*g^2*x^5*e^26 + 345*d^2*g^2*x^4*e^25 + 980*d^3*g^2*x^3*e^24 + 2400*d^4*g^2*x^2*e^23 + 6720*d^5*g^2*x*e^22 +
24*f*g*x^5*e^27 + 210*d*f*g*x^4*e^26 + 920*d^2*f*g*x^3*e^25 + 2940*d^3*f*g*x^2*e^24 + 9600*d^4*f*g*x*e^23 + 1
5*f^2*x^4*e^27 + 140*d*f^2*x^3*e^26 + 690*d^2*f^2*x^2*e^25 + 2940*d^3*f^2*x*e^24)*e^(-24) + 8*(9*d^7*g^2*e^6 +
14*d^6*f*g*e^7 + 5*d^5*f^2*e^8)*e^(-9)*log(abs(2*x*e^2 - 2*abs(d)*e)/abs(2*x*e^2 + 2*abs(d)*e))/abs(d) - 32*(
d^8*g^2*e^7 + 2*d^7*f*g*e^8 + d^6*f^2*e^9 + (d^7*g^2*e^8 + 2*d^6*f*g*e^9 + d^5*f^2*e^10)*x)*e^(-10)/(x^2*e^2 -
d^2)