\(\int \frac {(d+e x)^7 (f+g x)^2}{(d^2-e^2 x^2)^3} \, dx\) [569]

Optimal result

Integrand size = 29, antiderivative size = 179 \[ \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=-\frac {d \left (7 e^2 f^2+48 d e f g+56 d^2 g^2\right ) x}{e^2}-\frac {(e f+2 d g) (e f+12 d g) x^2}{2 e}-\frac {1}{3} g (2 e f+7 d g) x^3-\frac {1}{4} e g^2 x^4+\frac {8 d^4 (e f+d g)^2}{e^3 (d-e x)^2}-\frac {32 d^3 (e f+d g) (e f+2 d g)}{e^3 (d-e x)}-\frac {8 d^2 \left (3 e^2 f^2+14 d e f g+13 d^2 g^2\right ) \log (d-e x)}{e^3} \]

[Out]

-d*(56*d^2*g^2+48*d*e*f*g+7*e^2*f^2)*x/e^2-1/2*(2*d*g+e*f)*(12*d*g+e*f)*x^2/e-1/3*g*(7*d*g+2*e*f)*x^3-1/4*e*g^
2*x^4+8*d^4*(d*g+e*f)^2/e^3/(-e*x+d)^2-32*d^3*(d*g+e*f)*(2*d*g+e*f)/e^3/(-e*x+d)-8*d^2*(13*d^2*g^2+14*d*e*f*g+
3*e^2*f^2)*ln(-e*x+d)/e^3

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {862, 90} \[ \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=\frac {8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac {32 d^3 (d g+e f) (2 d g+e f)}{e^3 (d-e x)}-\frac {d x \left (56 d^2 g^2+48 d e f g+7 e^2 f^2\right )}{e^2}-\frac {8 d^2 \left (13 d^2 g^2+14 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}-\frac {1}{3} g x^3 (7 d g+2 e f)-\frac {x^2 (2 d g+e f) (12 d g+e f)}{2 e}-\frac {1}{4} e g^2 x^4 \]

[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 90

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 862

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/e)*x)^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]))

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.06 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=-\frac {d \left (7 e^2 f^2+48 d e f g+56 d^2 g^2\right ) x}{e^2}-\frac {\left (e^2 f^2+14 d e f g+24 d^2 g^2\right ) x^2}{2 e}-\frac {1}{3} g (2 e f+7 d g) x^3-\frac {1}{4} e g^2 x^4+\frac {8 d^4 (e f+d g)^2}{e^3 (d-e x)^2}+\frac {32 d^3 \left (e^2 f^2+3 d e f g+2 d^2 g^2\right )}{e^3 (-d+e x)}-\frac {8 d^2 \left (3 e^2 f^2+14 d e f g+13 d^2 g^2\right ) \log (d-e x)}{e^3} \]

[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

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.21

method	result	size
default	\(-\frac {\frac {1}{4} g^{2} e^{3} x^{4}+\frac {7}{3} x^{3} d \,e^{2} g^{2}+\frac {2}{3} x^{3} e^{3} f g +12 x^{2} d^{2} e \,g^{2}+7 x^{2} d \,e^{2} f g +\frac {1}{2} x^{2} e^{3} f^{2}+56 d^{3} g^{2} x +48 d^{2} e f g x +7 d \,e^{2} f^{2} x}{e^{2}}-\frac {8 d^{2} \left (13 d^{2} g^{2}+14 d e f g +3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}-\frac {32 d^{3} \left (2 d^{2} g^{2}+3 d e f g +e^{2} f^{2}\right )}{e^{3} \left (-e x +d \right )}+\frac {8 d^{4} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right )}{e^{3} \left (-e x +d \right )^{2}}\)	\(216\)
risch	\(-\frac {e \,g^{2} x^{4}}{4}-\frac {7 x^{3} d \,g^{2}}{3}-\frac {2 e \,x^{3} f g}{3}-\frac {12 x^{2} g^{2} d^{2}}{e}-7 x^{2} f g d -\frac {e \,x^{2} f^{2}}{2}-\frac {56 d^{3} g^{2} x}{e^{2}}-\frac {48 d^{2} f g x}{e}-7 d \,f^{2} x +\frac {\left (64 g^{2} d^{5}+96 f g \,d^{4} e +32 f^{2} d^{3} e^{2}\right ) x -\frac {8 d^{4} \left (7 d^{2} g^{2}+10 d e f g +3 e^{2} f^{2}\right )}{e}}{e^{2} \left (-e x +d \right )^{2}}-\frac {104 d^{4} \ln \left (-e x +d \right ) g^{2}}{e^{3}}-\frac {112 d^{3} \ln \left (-e x +d \right ) f g}{e^{2}}-\frac {24 d^{2} \ln \left (-e x +d \right ) f^{2}}{e}\)	\(216\)
norman	\(\frac {\left (\frac {521}{3} g^{2} d^{5}+\frac {574}{3} f g \,d^{4} e +46 f^{2} d^{3} e^{2}\right ) x^{3}+\left (-\frac {23}{2} g^{2} d^{2} e^{3}-7 f g d \,e^{4}-\frac {1}{2} f^{2} e^{5}\right ) x^{6}+\left (-\frac {154}{3} g^{2} d^{3} e^{2}-\frac {140}{3} f g \,d^{2} e^{3}-7 f^{2} d \,e^{4}\right ) x^{5}-\frac {d^{4} \left (319 g^{2} d^{4} e +376 f g \,d^{3} e^{2}+100 f^{2} d^{2} e^{3}\right )}{4 e^{4}}-\frac {g^{2} e^{5} x^{8}}{4}+\frac {d^{2} \left (215 g^{2} d^{4} e +266 f g \,d^{3} e^{2}+83 f^{2} d^{2} e^{3}\right ) x^{2}}{2 e^{2}}-\frac {d^{5} \left (104 d^{2} g^{2}+112 d e f g +23 e^{2} f^{2}\right ) x}{e^{2}}-\frac {e^{4} g \left (7 d g +2 e f \right ) x^{7}}{3}}{\left (-e^{2} x^{2}+d^{2}\right )^{2}}-\frac {8 d^{2} \left (13 d^{2} g^{2}+14 d e f g +3 e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(297\)
parallelrisch	\(-\frac {1344 \ln \left (e x -d \right ) d^{5} e f g +1248 \ln \left (e x -d \right ) x^{2} d^{4} e^{2} g^{2}+288 \ln \left (e x -d \right ) x^{2} d^{2} e^{4} f^{2}-2496 \ln \left (e x -d \right ) x \,d^{5} e \,g^{2}-576 \ln \left (e x -d \right ) x \,d^{3} e^{3} f^{2}+6 e^{6} f^{2} x^{4}+288 \ln \left (e x -d \right ) d^{4} e^{2} f^{2}+8 x^{5} e^{6} f g +91 x^{4} d^{2} e^{4} g^{2}+22 x^{5} d \,e^{5} g^{2}+2028 f g e \,d^{5}-2496 d^{5} e \,g^{2} x -624 d^{3} e^{3} f^{2} x +412 d^{3} e^{3} g^{2} x^{3}+72 d \,e^{5} f^{2} x^{3}+416 d^{2} e^{4} f g \,x^{3}-2712 d^{4} e^{2} f g x +68 d \,e^{5} f g \,x^{4}+450 e^{2} f^{2} d^{4}+1248 \ln \left (e x -d \right ) d^{6} g^{2}+1872 g^{2} d^{6}+3 g^{2} e^{6} x^{6}-2688 \ln \left (e x -d \right ) x \,d^{4} e^{2} f g +1344 \ln \left (e x -d \right ) x^{2} d^{3} e^{3} f g}{12 e^{3} \left (e x -d \right )^{2}}\)	\(357\)

[In]

int((e*x+d)^7*(g*x+f)^2/(-e^2*x^2+d^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/e^2*(1/4*g^2*e^3*x^4+7/3*x^3*d*e^2*g^2+2/3*x^3*e^3*f*g+12*x^2*d^2*e*g^2+7*x^2*d*e^2*f*g+1/2*x^2*e^3*f^2+56*
d^3*g^2*x+48*d^2*e*f*g*x+7*d*e^2*f^2*x)-8*d^2*(13*d^2*g^2+14*d*e*f*g+3*e^2*f^2)*ln(-e*x+d)/e^3-32*d^3/e^3*(2*d
^2*g^2+3*d*e*f*g+e^2*f^2)/(-e*x+d)+8*d^4*(d^2*g^2+2*d*e*f*g+e^2*f^2)/e^3/(-e*x+d)^2

Fricas [A] (verification not implemented)

none

Time = 0.62 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=-\frac {3 \, e^{6} g^{2} x^{6} + 288 \, d^{4} e^{2} f^{2} + 960 \, d^{5} e f g + 672 \, d^{6} g^{2} + 2 \, {\left (4 \, e^{6} f g + 11 \, d e^{5} g^{2}\right )} x^{5} + {\left (6 \, e^{6} f^{2} + 68 \, d e^{5} f g + 91 \, d^{2} e^{4} g^{2}\right )} x^{4} + 4 \, {\left (18 \, d e^{5} f^{2} + 104 \, d^{2} e^{4} f g + 103 \, d^{3} e^{3} g^{2}\right )} x^{3} - 6 \, {\left (27 \, d^{2} e^{4} f^{2} + 178 \, d^{3} e^{3} f g + 200 \, d^{4} e^{2} g^{2}\right )} x^{2} - 12 \, {\left (25 \, d^{3} e^{3} f^{2} + 48 \, d^{4} e^{2} f g + 8 \, d^{5} e g^{2}\right )} x + 96 \, {\left (3 \, d^{4} e^{2} f^{2} + 14 \, d^{5} e f g + 13 \, d^{6} g^{2} + {\left (3 \, d^{2} e^{4} f^{2} + 14 \, d^{3} e^{3} f g + 13 \, d^{4} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (3 \, d^{3} e^{3} f^{2} + 14 \, d^{4} e^{2} f g + 13 \, d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{12 \, {\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=- \frac {8 d^{2} \cdot \left (13 d^{2} g^{2} + 14 d e f g + 3 e^{2} f^{2}\right ) \log {\left (- d + e x \right )}}{e^{3}} - \frac {e g^{2} x^{4}}{4} - x^{3} \cdot \left (\frac {7 d g^{2}}{3} + \frac {2 e f g}{3}\right ) - x^{2} \cdot \left (\frac {12 d^{2} g^{2}}{e} + 7 d f g + \frac {e f^{2}}{2}\right ) - x \left (\frac {56 d^{3} g^{2}}{e^{2}} + \frac {48 d^{2} f g}{e} + 7 d f^{2}\right ) - \frac {56 d^{6} g^{2} + 80 d^{5} e f g + 24 d^{4} e^{2} f^{2} + x \left (- 64 d^{5} e g^{2} - 96 d^{4} e^{2} f g - 32 d^{3} e^{3} f^{2}\right )}{d^{2} e^{3} - 2 d e^{4} x + e^{5} x^{2}} \]

[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) - x**2*(12*d**2*g**2/e + 7*d*f*g + e*f**2/2) - x*(56*d**3*g**2/e**2 + 48*d**2*f*g/e + 7*d*f**2) - (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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.27 \[ \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=-\frac {8 \, {\left (3 \, d^{4} e^{2} f^{2} + 10 \, d^{5} e f g + 7 \, d^{6} g^{2} - 4 \, {\left (d^{3} e^{3} f^{2} + 3 \, d^{4} e^{2} f g + 2 \, d^{5} e g^{2}\right )} x\right )}}{e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}} - \frac {3 \, e^{3} g^{2} x^{4} + 4 \, {\left (2 \, e^{3} f g + 7 \, d e^{2} g^{2}\right )} x^{3} + 6 \, {\left (e^{3} f^{2} + 14 \, d e^{2} f g + 24 \, d^{2} e g^{2}\right )} x^{2} + 12 \, {\left (7 \, d e^{2} f^{2} + 48 \, d^{2} e f g + 56 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} - \frac {8 \, {\left (3 \, d^{2} e^{2} f^{2} + 14 \, d^{3} e f g + 13 \, d^{4} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=-\frac {8 \, {\left (3 \, d^{2} e^{2} f^{2} + 14 \, d^{3} e f g + 13 \, d^{4} g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{e^{3}} - \frac {8 \, {\left (3 \, d^{4} e^{2} f^{2} + 10 \, d^{5} e f g + 7 \, d^{6} g^{2} - 4 \, {\left (d^{3} e^{3} f^{2} + 3 \, d^{4} e^{2} f g + 2 \, d^{5} e g^{2}\right )} x\right )}}{{\left (e x - d\right )}^{2} e^{3}} - \frac {3 \, e^{13} g^{2} x^{4} + 8 \, e^{13} f g x^{3} + 28 \, d e^{12} g^{2} x^{3} + 6 \, e^{13} f^{2} x^{2} + 84 \, d e^{12} f g x^{2} + 144 \, d^{2} e^{11} g^{2} x^{2} + 84 \, d e^{12} f^{2} x + 576 \, d^{2} e^{11} f g x + 672 \, d^{3} e^{10} g^{2} x}{12 \, e^{12}} \]

[In]

integrate((e*x+d)^7*(g*x+f)^2/(-e^2*x^2+d^2)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.09 \[ \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx=\frac {x\,\left (64\,d^5\,g^2+96\,d^4\,e\,f\,g+32\,d^3\,e^2\,f^2\right )-\frac {8\,\left (7\,d^6\,g^2+10\,d^5\,e\,f\,g+3\,d^4\,e^2\,f^2\right )}{e}}{d^2\,e^2-2\,d\,e^3\,x+e^4\,x^2}-x^2\,\left (\frac {6\,d^2\,e^2\,g^2+8\,d\,e^3\,f\,g+e^4\,f^2}{2\,e^3}-\frac {3\,d^2\,g^2}{2\,e}+\frac {3\,d\,\left (2\,g\,\left (2\,d\,g+e\,f\right )+3\,d\,g^2\right )}{2\,e}\right )-x\,\left (\frac {d^3\,g^2}{e^2}-\frac {3\,d^2\,\left (2\,g\,\left (2\,d\,g+e\,f\right )+3\,d\,g^2\right )}{e^2}+\frac {4\,d\,\left (d^2\,g^2+3\,d\,e\,f\,g+e^2\,f^2\right )}{e^2}+\frac {3\,d\,\left (\frac {6\,d^2\,e^2\,g^2+8\,d\,e^3\,f\,g+e^4\,f^2}{e^3}-\frac {3\,d^2\,g^2}{e}+\frac {3\,d\,\left (2\,g\,\left (2\,d\,g+e\,f\right )+3\,d\,g^2\right )}{e}\right )}{e}\right )-x^3\,\left (\frac {2\,g\,\left (2\,d\,g+e\,f\right )}{3}+d\,g^2\right )-\frac {\ln \left (e\,x-d\right )\,\left (104\,d^4\,g^2+112\,d^3\,e\,f\,g+24\,d^2\,e^2\,f^2\right )}{e^3}-\frac {e\,g^2\,x^4}{4} \]

[In]

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

[Out]

(x*(64*d^5*g^2 + 32*d^3*e^2*f^2 + 96*d^4*e*f*g) - (8*(7*d^6*g^2 + 3*d^4*e^2*f^2 + 10*d^5*e*f*g))/e)/(d^2*e^2 +
e^4*x^2 - 2*d*e^3*x) - x^2*((e^4*f^2 + 6*d^2*e^2*g^2 + 8*d*e^3*f*g)/(2*e^3) - (3*d^2*g^2)/(2*e) + (3*d*(2*g*(
2*d*g + e*f) + 3*d*g^2))/(2*e)) - x*((d^3*g^2)/e^2 - (3*d^2*(2*g*(2*d*g + e*f) + 3*d*g^2))/e^2 + (4*d*(d^2*g^2
+ e^2*f^2 + 3*d*e*f*g))/e^2 + (3*d*((e^4*f^2 + 6*d^2*e^2*g^2 + 8*d*e^3*f*g)/e^3 - (3*d^2*g^2)/e + (3*d*(2*g*(
2*d*g + e*f) + 3*d*g^2))/e))/e) - x^3*((2*g*(2*d*g + e*f))/3 + d*g^2) - (log(e*x - d)*(104*d^4*g^2 + 24*d^2*e^
2*f^2 + 112*d^3*e*f*g))/e^3 - (e*g^2*x^4)/4