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

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

[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

_______________________________________________________________________________________

Rubi [A]  time = 0.499029, 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 \[ \frac{8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{32 d^3 (d g+e f) (2 d g+e f)}{e^3 (d-e x)}-\frac{d x \left (56 d^2 g^2+48 d e f g+7 e^2 f^2\right )}{e^2}-\frac{8 d^2 \left (13 d^2 g^2+14 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}-\frac{1}{3} g x^3 (7 d g+2 e f)-\frac{x^2 (2 d g+e f) (12 d g+e f)}{2 e}-\frac{1}{4} e g^2 x^4 \]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{8 d^{4} \left (d g + e f\right )^{2}}{e^{3} \left (d - e x\right )^{2}} - \frac{32 d^{3} \left (d g + e f\right ) \left (2 d g + e f\right )}{e^{3} \left (d - e x\right )} - \frac{8 d^{2} \left (13 d^{2} g^{2} + 14 d e f g + 3 e^{2} f^{2}\right ) \log{\left (d - e x \right )}}{e^{3}} - \frac{d x \left (8 d g \left (7 d g + 6 e f\right ) + 7 e^{2} f^{2}\right )}{e^{2}} - \frac{e g^{2} x^{4}}{4} - \frac{g x^{3} \left (7 d g + 2 e f\right )}{3} - \frac{\left (2 d g + e f\right ) \left (12 d g + e f\right ) \int x\, dx}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*d**4*(d*g + e*f)**2/(e**3*(d - e*x)**2) - 32*d**3*(d*g + e*f)*(2*d*g + e*f)/(e
**3*(d - e*x)) - 8*d**2*(13*d**2*g**2 + 14*d*e*f*g + 3*e**2*f**2)*log(d - e*x)/e
**3 - d*x*(8*d*g*(7*d*g + 6*e*f) + 7*e**2*f**2)/e**2 - e*g**2*x**4/4 - g*x**3*(7
*d*g + 2*e*f)/3 - (2*d*g + e*f)*(12*d*g + e*f)*Integral(x, x)/e

_______________________________________________________________________________________

Mathematica [A]  time = 0.161405, size = 193, normalized size = 1.08 \[ \frac{8 d^4 (d g+e f)^2}{e^3 (d-e x)^2}-\frac{x^2 \left (24 d^2 g^2+14 d e f g+e^2 f^2\right )}{2 e}-\frac{d x \left (56 d^2 g^2+48 d e f g+7 e^2 f^2\right )}{e^2}-\frac{8 d^2 \left (13 d^2 g^2+14 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{32 d^3 \left (2 d^2 g^2+3 d e f g+e^2 f^2\right )}{e^3 (e x-d)}-\frac{1}{3} g x^3 (7 d g+2 e f)-\frac{1}{4} e g^2 x^4 \]

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 263, normalized size = 1.5 \[ -{\frac{e{g}^{2}{x}^{4}}{4}}-{\frac{7\,{x}^{3}d{g}^{2}}{3}}-{\frac{2\,e{x}^{3}fg}{3}}-12\,{\frac{{x}^{2}{d}^{2}{g}^{2}}{e}}-7\,{x}^{2}dfg-{\frac{e{x}^{2}{f}^{2}}{2}}-56\,{\frac{{d}^{3}{g}^{2}x}{{e}^{2}}}-48\,{\frac{{d}^{2}fgx}{e}}-7\,d{f}^{2}x-104\,{\frac{{d}^{4}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}-112\,{\frac{{d}^{3}\ln \left ( ex-d \right ) fg}{{e}^{2}}}-24\,{\frac{{d}^{2}\ln \left ( ex-d \right ){f}^{2}}{e}}+64\,{\frac{{d}^{5}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}+96\,{\frac{{d}^{4}fg}{{e}^{2} \left ( ex-d \right ) }}+32\,{\frac{{d}^{3}{f}^{2}}{e \left ( ex-d \right ) }}+8\,{\frac{{d}^{6}{g}^{2}}{{e}^{3} \left ( ex-d \right ) ^{2}}}+16\,{\frac{{d}^{5}fg}{{e}^{2} \left ( ex-d \right ) ^{2}}}+8\,{\frac{{d}^{4}{f}^{2}}{e \left ( ex-d \right ) ^{2}}} \]

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+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+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

_______________________________________________________________________________________

Maxima [A]  time = 0.698915, size = 306, normalized size = 1.71 \[ -\frac{8 \,{\left (3 \, d^{4} e^{2} f^{2} + 10 \, d^{5} e f g + 7 \, d^{6} g^{2} - 4 \,{\left (d^{3} e^{3} f^{2} + 3 \, d^{4} e^{2} f g + 2 \, d^{5} e g^{2}\right )} x\right )}}{e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}} - \frac{3 \, e^{3} g^{2} x^{4} + 4 \,{\left (2 \, e^{3} f g + 7 \, d e^{2} g^{2}\right )} x^{3} + 6 \,{\left (e^{3} f^{2} + 14 \, d e^{2} f g + 24 \, d^{2} e g^{2}\right )} x^{2} + 12 \,{\left (7 \, d e^{2} f^{2} + 48 \, d^{2} e f g + 56 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} - \frac{8 \,{\left (3 \, d^{2} e^{2} f^{2} + 14 \, d^{3} e f g + 13 \, d^{4} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.274277, size = 454, normalized size = 2.54 \[ -\frac{3 \, e^{6} g^{2} x^{6} + 288 \, d^{4} e^{2} f^{2} + 960 \, d^{5} e f g + 672 \, d^{6} g^{2} + 2 \,{\left (4 \, e^{6} f g + 11 \, d e^{5} g^{2}\right )} x^{5} +{\left (6 \, e^{6} f^{2} + 68 \, d e^{5} f g + 91 \, d^{2} e^{4} g^{2}\right )} x^{4} + 4 \,{\left (18 \, d e^{5} f^{2} + 104 \, d^{2} e^{4} f g + 103 \, d^{3} e^{3} g^{2}\right )} x^{3} - 6 \,{\left (27 \, d^{2} e^{4} f^{2} + 178 \, d^{3} e^{3} f g + 200 \, d^{4} e^{2} g^{2}\right )} x^{2} - 12 \,{\left (25 \, d^{3} e^{3} f^{2} + 48 \, d^{4} e^{2} f g + 8 \, d^{5} e g^{2}\right )} x + 96 \,{\left (3 \, d^{4} e^{2} f^{2} + 14 \, d^{5} e f g + 13 \, d^{6} g^{2} +{\left (3 \, d^{2} e^{4} f^{2} + 14 \, d^{3} e^{3} f g + 13 \, d^{4} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (3 \, d^{3} e^{3} f^{2} + 14 \, d^{4} e^{2} f g + 13 \, d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{12 \,{\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]

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*(1
8*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]  time = 6.18843, size = 223, normalized size = 1.25 \[ - \frac{8 d^{2} \left (13 d^{2} g^{2} + 14 d e f g + 3 e^{2} f^{2}\right ) \log{\left (- d + e x \right )}}{e^{3}} - \frac{e g^{2} x^{4}}{4} - x^{3} \left (\frac{7 d g^{2}}{3} + \frac{2 e f g}{3}\right ) + \frac{- 56 d^{6} g^{2} - 80 d^{5} e f g - 24 d^{4} e^{2} f^{2} + x \left (64 d^{5} e g^{2} + 96 d^{4} e^{2} f g + 32 d^{3} e^{3} f^{2}\right )}{d^{2} e^{3} - 2 d e^{4} x + e^{5} x^{2}} - \frac{x^{2} \left (24 d^{2} g^{2} + 14 d e f g + e^{2} f^{2}\right )}{2 e} - \frac{x \left (56 d^{3} g^{2} + 48 d^{2} e f g + 7 d e^{2} f^{2}\right )}{e^{2}} \]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268887, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

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]

Done