3.569 \(\int \frac{(d+e x)^7 (f+g x)^2}{(d^2-e^2 x^2)^3} \, dx\)

Optimal. Leaf size=179 \[ -\frac{d x \left (56 d^2 g^2+48 d e f g+7 e^2 f^2\right )}{e^2}-\frac{8 d^2 \left (13 d^2 g^2+14 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{32 d^3 (d g+e f) (2 d g+e f)}{e^3 (d-e x)}-\frac{1}{3} g x^3 (7 d g+2 e f)-\frac{x^2 (2 d g+e f) (12 d g+e f)}{2 e}-\frac{1}{4} e g^2 x^4 \]

[Out]

-((d*(7*e^2*f^2 + 48*d*e*f*g + 56*d^2*g^2)*x)/e^2) - ((e*f + 2*d*g)*(e*f + 12*d*g)*x^2)/(2*e) - (g*(2*e*f + 7*
d*g)*x^3)/3 - (e*g^2*x^4)/4 + (8*d^4*(e*f + d*g)^2)/(e^3*(d - e*x)^2) - (32*d^3*(e*f + d*g)*(e*f + 2*d*g))/(e^
3*(d - e*x)) - (8*d^2*(3*e^2*f^2 + 14*d*e*f*g + 13*d^2*g^2)*Log[d - e*x])/e^3

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Rubi [A]  time = 0.241137, antiderivative size = 179, 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} \[ -\frac{d x \left (56 d^2 g^2+48 d e f g+7 e^2 f^2\right )}{e^2}-\frac{8 d^2 \left (13 d^2 g^2+14 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{32 d^3 (d g+e f) (2 d g+e f)}{e^3 (d-e x)}-\frac{1}{3} g x^3 (7 d g+2 e f)-\frac{x^2 (2 d g+e f) (12 d g+e f)}{2 e}-\frac{1}{4} e g^2 x^4 \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^7*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]

[Out]

-((d*(7*e^2*f^2 + 48*d*e*f*g + 56*d^2*g^2)*x)/e^2) - ((e*f + 2*d*g)*(e*f + 12*d*g)*x^2)/(2*e) - (g*(2*e*f + 7*
d*g)*x^3)/3 - (e*g^2*x^4)/4 + (8*d^4*(e*f + d*g)^2)/(e^3*(d - e*x)^2) - (32*d^3*(e*f + d*g)*(e*f + 2*d*g))/(e^
3*(d - e*x)) - (8*d^2*(3*e^2*f^2 + 14*d*e*f*g + 13*d^2*g^2)*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 )^3} \, dx &=\int \frac{(d+e x)^4 (f+g x)^2}{(d-e x)^3} \, dx\\ &=\int \left (-\frac{d \left (7 e^2 f^2+48 d e f g+56 d^2 g^2\right )}{e^2}+\frac{(-e f-12 d g) (e f+2 d g) x}{e}-g (2 e f+7 d g) x^2-e g^2 x^3+\frac{32 d^3 (-e f-2 d g) (e f+d g)}{e^2 (d-e x)^2}-\frac{16 d^4 (e f+d g)^2}{e^2 (-d+e x)^3}-\frac{8 d^2 \left (3 e^2 f^2+14 d e f g+13 d^2 g^2\right )}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac{d \left (7 e^2 f^2+48 d e f g+56 d^2 g^2\right ) x}{e^2}-\frac{(e f+2 d g) (e f+12 d g) x^2}{2 e}-\frac{1}{3} g (2 e f+7 d g) x^3-\frac{1}{4} e g^2 x^4+\frac{8 d^4 (e f+d g)^2}{e^3 (d-e x)^2}-\frac{32 d^3 (e f+d g) (e f+2 d g)}{e^3 (d-e x)}-\frac{8 d^2 \left (3 e^2 f^2+14 d e f g+13 d^2 g^2\right ) \log (d-e x)}{e^3}\\ \end{align*}

Mathematica [A]  time = 0.0978451, size = 193, normalized size = 1.08 \[ -\frac{x^2 \left (24 d^2 g^2+14 d e f g+e^2 f^2\right )}{2 e}+\frac{32 d^3 \left (2 d^2 g^2+3 d e f g+e^2 f^2\right )}{e^3 (e x-d)}-\frac{d x \left (56 d^2 g^2+48 d e f g+7 e^2 f^2\right )}{e^2}-\frac{8 d^2 \left (13 d^2 g^2+14 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{1}{3} g x^3 (7 d g+2 e f)-\frac{1}{4} e g^2 x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^7*(f + g*x)^2)/(d^2 - e^2*x^2)^3,x]

[Out]

-((d*(7*e^2*f^2 + 48*d*e*f*g + 56*d^2*g^2)*x)/e^2) - ((e^2*f^2 + 14*d*e*f*g + 24*d^2*g^2)*x^2)/(2*e) - (g*(2*e
*f + 7*d*g)*x^3)/3 - (e*g^2*x^4)/4 + (8*d^4*(e*f + d*g)^2)/(e^3*(d - e*x)^2) + (32*d^3*(e^2*f^2 + 3*d*e*f*g +
2*d^2*g^2))/(e^3*(-d + e*x)) - (8*d^2*(3*e^2*f^2 + 14*d*e*f*g + 13*d^2*g^2)*Log[d - e*x])/e^3

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Maple [A]  time = 0.052, size = 263, normalized size = 1.5 \begin{align*} -{\frac{e{g}^{2}{x}^{4}}{4}}-{\frac{7\,{x}^{3}d{g}^{2}}{3}}-{\frac{2\,e{x}^{3}fg}{3}}-12\,{\frac{{x}^{2}{d}^{2}{g}^{2}}{e}}-7\,{x}^{2}dfg-{\frac{e{x}^{2}{f}^{2}}{2}}-56\,{\frac{{d}^{3}{g}^{2}x}{{e}^{2}}}-48\,{\frac{{d}^{2}fgx}{e}}-7\,d{f}^{2}x-104\,{\frac{{d}^{4}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}-112\,{\frac{{d}^{3}\ln \left ( ex-d \right ) fg}{{e}^{2}}}-24\,{\frac{{d}^{2}\ln \left ( ex-d \right ){f}^{2}}{e}}+8\,{\frac{{d}^{6}{g}^{2}}{{e}^{3} \left ( ex-d \right ) ^{2}}}+16\,{\frac{{d}^{5}fg}{{e}^{2} \left ( ex-d \right ) ^{2}}}+8\,{\frac{{d}^{4}{f}^{2}}{e \left ( ex-d \right ) ^{2}}}+64\,{\frac{{d}^{5}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}+96\,{\frac{{d}^{4}fg}{{e}^{2} \left ( ex-d \right ) }}+32\,{\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)^7*(g*x+f)^2/(-e^2*x^2+d^2)^3,x)

[Out]

-1/4*e*g^2*x^4-7/3*x^3*d*g^2-2/3*e*x^3*f*g-12/e*x^2*d^2*g^2-7*x^2*d*f*g-1/2*e*x^2*f^2-56/e^2*d^3*g^2*x-48/e*d^
2*f*g*x-7*d*f^2*x-104*d^4/e^3*ln(e*x-d)*g^2-112*d^3/e^2*ln(e*x-d)*f*g-24*d^2/e*ln(e*x-d)*f^2+8*d^6/e^3/(e*x-d)
^2*g^2+16*d^5/e^2/(e*x-d)^2*f*g+8*d^4/e/(e*x-d)^2*f^2+64*d^5/e^3/(e*x-d)*g^2+96*d^4/e^2/(e*x-d)*f*g+32*d^3/e/(
e*x-d)*f^2

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Maxima [A]  time = 0.991082, size = 306, normalized size = 1.71 \begin{align*} -\frac{8 \,{\left (3 \, d^{4} e^{2} f^{2} + 10 \, d^{5} e f g + 7 \, d^{6} g^{2} - 4 \,{\left (d^{3} e^{3} f^{2} + 3 \, d^{4} e^{2} f g + 2 \, d^{5} e g^{2}\right )} x\right )}}{e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}} - \frac{3 \, e^{3} g^{2} x^{4} + 4 \,{\left (2 \, e^{3} f g + 7 \, d e^{2} g^{2}\right )} x^{3} + 6 \,{\left (e^{3} f^{2} + 14 \, d e^{2} f g + 24 \, d^{2} e g^{2}\right )} x^{2} + 12 \,{\left (7 \, d e^{2} f^{2} + 48 \, d^{2} e f g + 56 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} - \frac{8 \,{\left (3 \, d^{2} e^{2} f^{2} + 14 \, d^{3} e f g + 13 \, 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)^7*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="maxima")

[Out]

-8*(3*d^4*e^2*f^2 + 10*d^5*e*f*g + 7*d^6*g^2 - 4*(d^3*e^3*f^2 + 3*d^4*e^2*f*g + 2*d^5*e*g^2)*x)/(e^5*x^2 - 2*d
*e^4*x + d^2*e^3) - 1/12*(3*e^3*g^2*x^4 + 4*(2*e^3*f*g + 7*d*e^2*g^2)*x^3 + 6*(e^3*f^2 + 14*d*e^2*f*g + 24*d^2
*e*g^2)*x^2 + 12*(7*d*e^2*f^2 + 48*d^2*e*f*g + 56*d^3*g^2)*x)/e^2 - 8*(3*d^2*e^2*f^2 + 14*d^3*e*f*g + 13*d^4*g
^2)*log(e*x - d)/e^3

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Fricas [A]  time = 1.77242, size = 722, normalized size = 4.03 \begin{align*} -\frac{3 \, e^{6} g^{2} x^{6} + 288 \, d^{4} e^{2} f^{2} + 960 \, d^{5} e f g + 672 \, d^{6} g^{2} + 2 \,{\left (4 \, e^{6} f g + 11 \, d e^{5} g^{2}\right )} x^{5} +{\left (6 \, e^{6} f^{2} + 68 \, d e^{5} f g + 91 \, d^{2} e^{4} g^{2}\right )} x^{4} + 4 \,{\left (18 \, d e^{5} f^{2} + 104 \, d^{2} e^{4} f g + 103 \, d^{3} e^{3} g^{2}\right )} x^{3} - 6 \,{\left (27 \, d^{2} e^{4} f^{2} + 178 \, d^{3} e^{3} f g + 200 \, d^{4} e^{2} g^{2}\right )} x^{2} - 12 \,{\left (25 \, d^{3} e^{3} f^{2} + 48 \, d^{4} e^{2} f g + 8 \, d^{5} e g^{2}\right )} x + 96 \,{\left (3 \, d^{4} e^{2} f^{2} + 14 \, d^{5} e f g + 13 \, d^{6} g^{2} +{\left (3 \, d^{2} e^{4} f^{2} + 14 \, d^{3} e^{3} f g + 13 \, d^{4} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (3 \, d^{3} e^{3} f^{2} + 14 \, d^{4} e^{2} f g + 13 \, d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{12 \,{\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

-1/12*(3*e^6*g^2*x^6 + 288*d^4*e^2*f^2 + 960*d^5*e*f*g + 672*d^6*g^2 + 2*(4*e^6*f*g + 11*d*e^5*g^2)*x^5 + (6*e
^6*f^2 + 68*d*e^5*f*g + 91*d^2*e^4*g^2)*x^4 + 4*(18*d*e^5*f^2 + 104*d^2*e^4*f*g + 103*d^3*e^3*g^2)*x^3 - 6*(27
*d^2*e^4*f^2 + 178*d^3*e^3*f*g + 200*d^4*e^2*g^2)*x^2 - 12*(25*d^3*e^3*f^2 + 48*d^4*e^2*f*g + 8*d^5*e*g^2)*x +
96*(3*d^4*e^2*f^2 + 14*d^5*e*f*g + 13*d^6*g^2 + (3*d^2*e^4*f^2 + 14*d^3*e^3*f*g + 13*d^4*e^2*g^2)*x^2 - 2*(3*
d^3*e^3*f^2 + 14*d^4*e^2*f*g + 13*d^5*e*g^2)*x)*log(e*x - d))/(e^5*x^2 - 2*d*e^4*x + d^2*e^3)

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Sympy [A]  time = 1.73791, size = 223, normalized size = 1.25 \begin{align*} - \frac{8 d^{2} \left (13 d^{2} g^{2} + 14 d e f g + 3 e^{2} f^{2}\right ) \log{\left (- d + e x \right )}}{e^{3}} - \frac{e g^{2} x^{4}}{4} - x^{3} \left (\frac{7 d g^{2}}{3} + \frac{2 e f g}{3}\right ) + \frac{- 56 d^{6} g^{2} - 80 d^{5} e f g - 24 d^{4} e^{2} f^{2} + x \left (64 d^{5} e g^{2} + 96 d^{4} e^{2} f g + 32 d^{3} e^{3} f^{2}\right )}{d^{2} e^{3} - 2 d e^{4} x + e^{5} x^{2}} - \frac{x^{2} \left (24 d^{2} g^{2} + 14 d e f g + e^{2} f^{2}\right )}{2 e} - \frac{x \left (56 d^{3} g^{2} + 48 d^{2} e f g + 7 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)**7*(g*x+f)**2/(-e**2*x**2+d**2)**3,x)

[Out]

-8*d**2*(13*d**2*g**2 + 14*d*e*f*g + 3*e**2*f**2)*log(-d + e*x)/e**3 - e*g**2*x**4/4 - x**3*(7*d*g**2/3 + 2*e*
f*g/3) + (-56*d**6*g**2 - 80*d**5*e*f*g - 24*d**4*e**2*f**2 + x*(64*d**5*e*g**2 + 96*d**4*e**2*f*g + 32*d**3*e
**3*f**2))/(d**2*e**3 - 2*d*e**4*x + e**5*x**2) - x**2*(24*d**2*g**2 + 14*d*e*f*g + e**2*f**2)/(2*e) - x*(56*d
**3*g**2 + 48*d**2*e*f*g + 7*d*e**2*f**2)/e**2

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Giac [B]  time = 1.16236, size = 491, normalized size = 2.74 \begin{align*} -4 \,{\left (13 \, d^{4} g^{2} e^{7} + 14 \, d^{3} f g e^{8} + 3 \, d^{2} f^{2} e^{9}\right )} e^{\left (-10\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{12} \,{\left (3 \, g^{2} x^{4} e^{25} + 28 \, d g^{2} x^{3} e^{24} + 144 \, d^{2} g^{2} x^{2} e^{23} + 672 \, d^{3} g^{2} x e^{22} + 8 \, f g x^{3} e^{25} + 84 \, d f g x^{2} e^{24} + 576 \, d^{2} f g x e^{23} + 6 \, f^{2} x^{2} e^{25} + 84 \, d f^{2} x e^{24}\right )} e^{\left (-24\right )} - \frac{4 \,{\left (13 \, d^{5} g^{2} e^{6} + 14 \, d^{4} f g e^{7} + 3 \, d^{3} 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{8 \,{\left (7 \, d^{8} g^{2} e^{7} + 10 \, d^{7} f g e^{8} + 3 \, d^{6} f^{2} e^{9} - 4 \,{\left (2 \, d^{5} g^{2} e^{10} + 3 \, d^{4} f g e^{11} + d^{3} f^{2} e^{12}\right )} x^{3} -{\left (9 \, d^{6} g^{2} e^{9} + 14 \, d^{5} f g e^{10} + 5 \, d^{4} f^{2} e^{11}\right )} x^{2} + 2 \,{\left (3 \, d^{7} g^{2} e^{8} + 4 \, d^{6} f g e^{9} + d^{5} f^{2} e^{10}\right )} x\right )} e^{\left (-10\right )}}{{\left (x^{2} e^{2} - d^{2}\right )}^{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)^3,x, algorithm="giac")

[Out]

-4*(13*d^4*g^2*e^7 + 14*d^3*f*g*e^8 + 3*d^2*f^2*e^9)*e^(-10)*log(abs(x^2*e^2 - d^2)) - 1/12*(3*g^2*x^4*e^25 +
28*d*g^2*x^3*e^24 + 144*d^2*g^2*x^2*e^23 + 672*d^3*g^2*x*e^22 + 8*f*g*x^3*e^25 + 84*d*f*g*x^2*e^24 + 576*d^2*f
*g*x*e^23 + 6*f^2*x^2*e^25 + 84*d*f^2*x*e^24)*e^(-24) - 4*(13*d^5*g^2*e^6 + 14*d^4*f*g*e^7 + 3*d^3*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) - 8*(7*d^8*g^2*e^7 + 10*d^7*f*g*e^8 + 3*d
^6*f^2*e^9 - 4*(2*d^5*g^2*e^10 + 3*d^4*f*g*e^11 + d^3*f^2*e^12)*x^3 - (9*d^6*g^2*e^9 + 14*d^5*f*g*e^10 + 5*d^4
*f^2*e^11)*x^2 + 2*(3*d^7*g^2*e^8 + 4*d^6*f*g*e^9 + d^5*f^2*e^10)*x)*e^(-10)/(x^2*e^2 - d^2)^2