3.1863 \(\int \frac{(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=219 \[ \frac{e^6 (a e+c d x)^5}{5 c^7 d^7}-\frac{\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac{6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7}+\frac{3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac{5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac{10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac{15 e^2 x \left (c d^2-a e^2\right )^4}{c^6 d^6} \]

[Out]

(15*e^2*(c*d^2 - a*e^2)^4*x)/(c^6*d^6) - (c*d^2 - a*e^2)^6/(c^7*d^7*(a*e + c*d*x
)) + (10*e^3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)^2)/(c^7*d^7) + (5*e^4*(c*d^2 - a*e^
2)^2*(a*e + c*d*x)^3)/(c^7*d^7) + (3*e^5*(c*d^2 - a*e^2)*(a*e + c*d*x)^4)/(2*c^7
*d^7) + (e^6*(a*e + c*d*x)^5)/(5*c^7*d^7) + (6*e*(c*d^2 - a*e^2)^5*Log[a*e + c*d
*x])/(c^7*d^7)

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Rubi [A]  time = 0.603591, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{e^6 (a e+c d x)^5}{5 c^7 d^7}-\frac{\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac{6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7}+\frac{3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac{5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac{10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac{15 e^2 x \left (c d^2-a e^2\right )^4}{c^6 d^6} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(15*e^2*(c*d^2 - a*e^2)^4*x)/(c^6*d^6) - (c*d^2 - a*e^2)^6/(c^7*d^7*(a*e + c*d*x
)) + (10*e^3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)^2)/(c^7*d^7) + (5*e^4*(c*d^2 - a*e^
2)^2*(a*e + c*d*x)^3)/(c^7*d^7) + (3*e^5*(c*d^2 - a*e^2)*(a*e + c*d*x)^4)/(2*c^7
*d^7) + (e^6*(a*e + c*d*x)^5)/(5*c^7*d^7) + (6*e*(c*d^2 - a*e^2)^5*Log[a*e + c*d
*x])/(c^7*d^7)

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Rubi in Sympy [A]  time = 106.185, size = 207, normalized size = 0.95 \[ \frac{15 e^{2} x \left (a e^{2} - c d^{2}\right )^{4}}{c^{6} d^{6}} + \frac{e^{6} \left (a e + c d x\right )^{5}}{5 c^{7} d^{7}} - \frac{3 e^{5} \left (a e + c d x\right )^{4} \left (a e^{2} - c d^{2}\right )}{2 c^{7} d^{7}} + \frac{5 e^{4} \left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )^{2}}{c^{7} d^{7}} - \frac{10 e^{3} \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )^{3}}{c^{7} d^{7}} - \frac{6 e \left (a e^{2} - c d^{2}\right )^{5} \log{\left (a e + c d x \right )}}{c^{7} d^{7}} - \frac{\left (a e^{2} - c d^{2}\right )^{6}}{c^{7} d^{7} \left (a e + c d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**8/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

15*e**2*x*(a*e**2 - c*d**2)**4/(c**6*d**6) + e**6*(a*e + c*d*x)**5/(5*c**7*d**7)
- 3*e**5*(a*e + c*d*x)**4*(a*e**2 - c*d**2)/(2*c**7*d**7) + 5*e**4*(a*e + c*d*x
)**3*(a*e**2 - c*d**2)**2/(c**7*d**7) - 10*e**3*(a*e + c*d*x)**2*(a*e**2 - c*d**
2)**3/(c**7*d**7) - 6*e*(a*e**2 - c*d**2)**5*log(a*e + c*d*x)/(c**7*d**7) - (a*e
**2 - c*d**2)**6/(c**7*d**7*(a*e + c*d*x))

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Mathematica [A]  time = 0.242795, size = 339, normalized size = 1.55 \[ \frac{-10 a^6 e^{12}+10 a^5 c d e^{10} (6 d+5 e x)-30 a^4 c^2 d^2 e^8 \left (5 d^2+8 d e x-e^2 x^2\right )+10 a^3 c^3 d^3 e^6 \left (20 d^3+45 d^2 e x-15 d e^2 x^2-e^3 x^3\right )-5 a^2 c^4 d^4 e^4 \left (30 d^4+80 d^3 e x-60 d^2 e^2 x^2-10 d e^3 x^3-e^4 x^4\right )+a c^5 d^5 e^2 \left (60 d^5+150 d^4 e x-300 d^3 e^2 x^2-100 d^2 e^3 x^3-25 d e^4 x^4-3 e^5 x^5\right )-60 e \left (a e^2-c d^2\right )^5 (a e+c d x) \log (a e+c d x)+c^6 d^6 \left (-10 d^6+150 d^4 e^2 x^2+100 d^3 e^3 x^3+50 d^2 e^4 x^4+15 d e^5 x^5+2 e^6 x^6\right )}{10 c^7 d^7 (a e+c d x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(-10*a^6*e^12 + 10*a^5*c*d*e^10*(6*d + 5*e*x) - 30*a^4*c^2*d^2*e^8*(5*d^2 + 8*d*
e*x - e^2*x^2) + 10*a^3*c^3*d^3*e^6*(20*d^3 + 45*d^2*e*x - 15*d*e^2*x^2 - e^3*x^
3) - 5*a^2*c^4*d^4*e^4*(30*d^4 + 80*d^3*e*x - 60*d^2*e^2*x^2 - 10*d*e^3*x^3 - e^
4*x^4) + a*c^5*d^5*e^2*(60*d^5 + 150*d^4*e*x - 300*d^3*e^2*x^2 - 100*d^2*e^3*x^3
- 25*d*e^4*x^4 - 3*e^5*x^5) + c^6*d^6*(-10*d^6 + 150*d^4*e^2*x^2 + 100*d^3*e^3*
x^3 + 50*d^2*e^4*x^4 + 15*d*e^5*x^5 + 2*e^6*x^6) - 60*e*(-(c*d^2) + a*e^2)^5*(a*
e + c*d*x)*Log[a*e + c*d*x])/(10*c^7*d^7*(a*e + c*d*x))

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Maple [B]  time = 0.016, size = 502, normalized size = 2.3 \[ -30\,{\frac{d{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}}}+60\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ){a}^{2}}{d{c}^{4}}}-{\frac{{d}^{5}}{c \left ( cdx+ae \right ) }}+5\,{\frac{{e}^{4}{x}^{3}}{{c}^{2}}}+6\,{\frac{{d}^{3}{e}^{2}a}{{c}^{2} \left ( cdx+ae \right ) }}-6\,{\frac{{e}^{11}\ln \left ( cdx+ae \right ){a}^{5}}{{c}^{7}{d}^{7}}}-4\,{\frac{{e}^{6}{x}^{3}a}{{c}^{3}{d}^{2}}}+45\,{\frac{{a}^{2}{e}^{6}x}{{c}^{4}{d}^{2}}}-{\frac{{a}^{6}{e}^{12}}{{c}^{7}{d}^{7} \left ( cdx+ae \right ) }}+6\,{\frac{{a}^{5}{e}^{10}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) }}-15\,{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) }}+20\,{\frac{{a}^{3}{e}^{6}}{d{c}^{4} \left ( cdx+ae \right ) }}-15\,{\frac{{a}^{2}d{e}^{4}}{{c}^{3} \left ( cdx+ae \right ) }}-24\,{\frac{{a}^{3}{e}^{8}x}{{c}^{5}{d}^{4}}}-2\,{\frac{{e}^{9}{x}^{2}{a}^{3}}{{c}^{5}{d}^{5}}}+9\,{\frac{{e}^{7}{x}^{2}{a}^{2}}{{c}^{4}{d}^{3}}}-15\,{\frac{{e}^{5}{x}^{2}a}{{c}^{3}d}}-60\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{5}{d}^{3}}}-40\,{\frac{a{e}^{4}x}{{c}^{3}}}+{\frac{{e}^{6}{x}^{5}}{5\,{c}^{2}{d}^{2}}}+{\frac{3\,{e}^{5}{x}^{4}}{2\,{c}^{2}d}}+10\,{\frac{d{e}^{3}{x}^{2}}{{c}^{2}}}+15\,{\frac{{d}^{2}{e}^{2}x}{{c}^{2}}}+6\,{\frac{{d}^{3}e\ln \left ( cdx+ae \right ) }{{c}^{2}}}+5\,{\frac{{a}^{4}{e}^{10}x}{{c}^{6}{d}^{6}}}-{\frac{{e}^{7}{x}^{4}a}{2\,{c}^{3}{d}^{3}}}+{\frac{{e}^{8}{x}^{3}{a}^{2}}{{c}^{4}{d}^{4}}}+30\,{\frac{{e}^{9}\ln \left ( cdx+ae \right ){a}^{4}}{{c}^{6}{d}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^8/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

-30*d*e^3/c^3*ln(c*d*x+a*e)*a+60/d*e^5/c^4*ln(c*d*x+a*e)*a^2-1/c*d^5/(c*d*x+a*e)
+5*e^4/c^2*x^3+6/c^2*d^3/(c*d*x+a*e)*a*e^2-6/d^7*e^11/c^7*ln(c*d*x+a*e)*a^5-4*e^
6/c^3/d^2*x^3*a+45*e^6/c^4/d^2*a^2*x-1/c^7/d^7/(c*d*x+a*e)*a^6*e^12+6/c^6/d^5/(c
*d*x+a*e)*a^5*e^10-15/c^5/d^3/(c*d*x+a*e)*a^4*e^8+20/c^4/d/(c*d*x+a*e)*a^3*e^6-1
5/c^3*d/(c*d*x+a*e)*a^2*e^4-24*e^8/c^5/d^4*a^3*x-2*e^9/c^5/d^5*x^2*a^3+9*e^7/c^4
/d^3*x^2*a^2-15*e^5/c^3/d*x^2*a-60/d^3*e^7/c^5*ln(c*d*x+a*e)*a^3-40*e^4/c^3*a*x+
1/5*e^6/c^2/d^2*x^5+3/2*e^5/c^2/d*x^4+10*e^3/c^2*d*x^2+15*e^2/c^2*d^2*x+6*d^3*e/
c^2*ln(c*d*x+a*e)+5*e^10/c^6/d^6*a^4*x-1/2*e^7/c^3/d^3*x^4*a+e^8/c^4/d^4*x^3*a^2
+30/d^5*e^9/c^6*ln(c*d*x+a*e)*a^4

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Maxima [A]  time = 0.727202, size = 537, normalized size = 2.45 \[ -\frac{c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{c^{8} d^{8} x + a c^{7} d^{7} e} + \frac{2 \, c^{4} d^{4} e^{6} x^{5} + 5 \,{\left (3 \, c^{4} d^{5} e^{5} - a c^{3} d^{3} e^{7}\right )} x^{4} + 10 \,{\left (5 \, c^{4} d^{6} e^{4} - 4 \, a c^{3} d^{4} e^{6} + a^{2} c^{2} d^{2} e^{8}\right )} x^{3} + 10 \,{\left (10 \, c^{4} d^{7} e^{3} - 15 \, a c^{3} d^{5} e^{5} + 9 \, a^{2} c^{2} d^{3} e^{7} - 2 \, a^{3} c d e^{9}\right )} x^{2} + 10 \,{\left (15 \, c^{4} d^{8} e^{2} - 40 \, a c^{3} d^{6} e^{4} + 45 \, a^{2} c^{2} d^{4} e^{6} - 24 \, a^{3} c d^{2} e^{8} + 5 \, a^{4} e^{10}\right )} x}{10 \, c^{6} d^{6}} + \frac{6 \,{\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="maxima")

[Out]

-(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4
*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)/(c^8*d^8*x + a*c^7*d^7*e) + 1/10*(2*
c^4*d^4*e^6*x^5 + 5*(3*c^4*d^5*e^5 - a*c^3*d^3*e^7)*x^4 + 10*(5*c^4*d^6*e^4 - 4*
a*c^3*d^4*e^6 + a^2*c^2*d^2*e^8)*x^3 + 10*(10*c^4*d^7*e^3 - 15*a*c^3*d^5*e^5 + 9
*a^2*c^2*d^3*e^7 - 2*a^3*c*d*e^9)*x^2 + 10*(15*c^4*d^8*e^2 - 40*a*c^3*d^6*e^4 +
45*a^2*c^2*d^4*e^6 - 24*a^3*c*d^2*e^8 + 5*a^4*e^10)*x)/(c^6*d^6) + 6*(c^5*d^10*e
- 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 - 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 -
a^5*e^11)*log(c*d*x + a*e)/(c^7*d^7)

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Fricas [A]  time = 0.222218, size = 736, normalized size = 3.36 \[ \frac{2 \, c^{6} d^{6} e^{6} x^{6} - 10 \, c^{6} d^{12} + 60 \, a c^{5} d^{10} e^{2} - 150 \, a^{2} c^{4} d^{8} e^{4} + 200 \, a^{3} c^{3} d^{6} e^{6} - 150 \, a^{4} c^{2} d^{4} e^{8} + 60 \, a^{5} c d^{2} e^{10} - 10 \, a^{6} e^{12} + 3 \,{\left (5 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 5 \,{\left (10 \, c^{6} d^{8} e^{4} - 5 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 10 \,{\left (10 \, c^{6} d^{9} e^{3} - 10 \, a c^{5} d^{7} e^{5} + 5 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 30 \,{\left (5 \, c^{6} d^{10} e^{2} - 10 \, a c^{5} d^{8} e^{4} + 10 \, a^{2} c^{4} d^{6} e^{6} - 5 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 10 \,{\left (15 \, a c^{5} d^{9} e^{3} - 40 \, a^{2} c^{4} d^{7} e^{5} + 45 \, a^{3} c^{3} d^{5} e^{7} - 24 \, a^{4} c^{2} d^{3} e^{9} + 5 \, a^{5} c d e^{11}\right )} x + 60 \,{\left (a c^{5} d^{10} e^{2} - 5 \, a^{2} c^{4} d^{8} e^{4} + 10 \, a^{3} c^{3} d^{6} e^{6} - 10 \, a^{4} c^{2} d^{4} e^{8} + 5 \, a^{5} c d^{2} e^{10} - a^{6} e^{12} +{\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{10 \,{\left (c^{8} d^{8} x + a c^{7} d^{7} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")

[Out]

1/10*(2*c^6*d^6*e^6*x^6 - 10*c^6*d^12 + 60*a*c^5*d^10*e^2 - 150*a^2*c^4*d^8*e^4
+ 200*a^3*c^3*d^6*e^6 - 150*a^4*c^2*d^4*e^8 + 60*a^5*c*d^2*e^10 - 10*a^6*e^12 +
3*(5*c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 5*(10*c^6*d^8*e^4 - 5*a*c^5*d^6*e^6 + a^
2*c^4*d^4*e^8)*x^4 + 10*(10*c^6*d^9*e^3 - 10*a*c^5*d^7*e^5 + 5*a^2*c^4*d^5*e^7 -
a^3*c^3*d^3*e^9)*x^3 + 30*(5*c^6*d^10*e^2 - 10*a*c^5*d^8*e^4 + 10*a^2*c^4*d^6*e
^6 - 5*a^3*c^3*d^4*e^8 + a^4*c^2*d^2*e^10)*x^2 + 10*(15*a*c^5*d^9*e^3 - 40*a^2*c
^4*d^7*e^5 + 45*a^3*c^3*d^5*e^7 - 24*a^4*c^2*d^3*e^9 + 5*a^5*c*d*e^11)*x + 60*(a
*c^5*d^10*e^2 - 5*a^2*c^4*d^8*e^4 + 10*a^3*c^3*d^6*e^6 - 10*a^4*c^2*d^4*e^8 + 5*
a^5*c*d^2*e^10 - a^6*e^12 + (c^6*d^11*e - 5*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5 -
10*a^3*c^3*d^5*e^7 + 5*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*x)*log(c*d*x + a*e))/(c^
8*d^8*x + a*c^7*d^7*e)

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Sympy [A]  time = 7.17159, size = 348, normalized size = 1.59 \[ - \frac{a^{6} e^{12} - 6 a^{5} c d^{2} e^{10} + 15 a^{4} c^{2} d^{4} e^{8} - 20 a^{3} c^{3} d^{6} e^{6} + 15 a^{2} c^{4} d^{8} e^{4} - 6 a c^{5} d^{10} e^{2} + c^{6} d^{12}}{a c^{7} d^{7} e + c^{8} d^{8} x} + \frac{e^{6} x^{5}}{5 c^{2} d^{2}} - \frac{x^{4} \left (a e^{7} - 3 c d^{2} e^{5}\right )}{2 c^{3} d^{3}} + \frac{x^{3} \left (a^{2} e^{8} - 4 a c d^{2} e^{6} + 5 c^{2} d^{4} e^{4}\right )}{c^{4} d^{4}} - \frac{x^{2} \left (2 a^{3} e^{9} - 9 a^{2} c d^{2} e^{7} + 15 a c^{2} d^{4} e^{5} - 10 c^{3} d^{6} e^{3}\right )}{c^{5} d^{5}} + \frac{x \left (5 a^{4} e^{10} - 24 a^{3} c d^{2} e^{8} + 45 a^{2} c^{2} d^{4} e^{6} - 40 a c^{3} d^{6} e^{4} + 15 c^{4} d^{8} e^{2}\right )}{c^{6} d^{6}} - \frac{6 e \left (a e^{2} - c d^{2}\right )^{5} \log{\left (a e + c d x \right )}}{c^{7} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**8/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

-(a**6*e**12 - 6*a**5*c*d**2*e**10 + 15*a**4*c**2*d**4*e**8 - 20*a**3*c**3*d**6*
e**6 + 15*a**2*c**4*d**8*e**4 - 6*a*c**5*d**10*e**2 + c**6*d**12)/(a*c**7*d**7*e
+ c**8*d**8*x) + e**6*x**5/(5*c**2*d**2) - x**4*(a*e**7 - 3*c*d**2*e**5)/(2*c**
3*d**3) + x**3*(a**2*e**8 - 4*a*c*d**2*e**6 + 5*c**2*d**4*e**4)/(c**4*d**4) - x*
*2*(2*a**3*e**9 - 9*a**2*c*d**2*e**7 + 15*a*c**2*d**4*e**5 - 10*c**3*d**6*e**3)/
(c**5*d**5) + x*(5*a**4*e**10 - 24*a**3*c*d**2*e**8 + 45*a**2*c**2*d**4*e**6 - 4
0*a*c**3*d**6*e**4 + 15*c**4*d**8*e**2)/(c**6*d**6) - 6*e*(a*e**2 - c*d**2)**5*l
og(a*e + c*d*x)/(c**7*d**7)

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GIAC/XCAS [A]  time = 0.459256, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^8/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")

[Out]

Done