3.1875 \(\int \frac{(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=185 \[ \frac{e^3 x \left (6 a^2 e^4-15 a c d^2 e^2+10 c^2 d^4\right )}{c^5 d^5}-\frac{5 e \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^5}{2 c^6 d^6 (a e+c d x)^2}+\frac{10 e^2 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^6 d^6}+\frac{e^4 x^2 \left (5 c d^2-3 a e^2\right )}{2 c^4 d^4}+\frac{e^5 x^3}{3 c^3 d^3} \]

[Out]

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

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Rubi [A]  time = 0.429615, antiderivative size = 185, 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 \[ \frac{e^3 x \left (6 a^2 e^4-15 a c d^2 e^2+10 c^2 d^4\right )}{c^5 d^5}-\frac{5 e \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^5}{2 c^6 d^6 (a e+c d x)^2}+\frac{10 e^2 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^6 d^6}+\frac{e^4 x^2 \left (5 c d^2-3 a e^2\right )}{2 c^4 d^4}+\frac{e^5 x^3}{3 c^3 d^3} \]

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)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{e^{3} \left (6 a^{2} e^{4} - 15 a c d^{2} e^{2} + 10 c^{2} d^{4}\right ) \int \frac{1}{c^{5}}\, dx}{d^{5}} + \frac{e^{5} x^{3}}{3 c^{3} d^{3}} - \frac{e^{4} \left (3 a e^{2} - 5 c d^{2}\right ) \int x\, dx}{c^{4} d^{4}} - \frac{10 e^{2} \left (a e^{2} - c d^{2}\right )^{3} \log{\left (a e + c d x \right )}}{c^{6} d^{6}} - \frac{5 e \left (a e^{2} - c d^{2}\right )^{4}}{c^{6} d^{6} \left (a e + c d x\right )} + \frac{\left (a e^{2} - c d^{2}\right )^{5}}{2 c^{6} d^{6} \left (a e + c d x\right )^{2}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.17813, size = 262, normalized size = 1.42 \[ \frac{-27 a^5 e^{10}+3 a^4 c d e^8 (35 d+2 e x)+3 a^3 c^2 d^2 e^6 \left (-50 d^2+10 d e x+21 e^2 x^2\right )+5 a^2 c^3 d^3 e^4 \left (18 d^3-24 d^2 e x-33 d e^2 x^2+4 e^3 x^3\right )-5 a c^4 d^4 e^2 \left (3 d^4-24 d^3 e x-24 d^2 e^2 x^2+12 d e^3 x^3+e^4 x^4\right )-60 e^2 \left (a e^2-c d^2\right )^3 (a e+c d x)^2 \log (a e+c d x)+c^5 d^5 \left (-3 d^5-30 d^4 e x+60 d^2 e^3 x^3+15 d e^4 x^4+2 e^5 x^5\right )}{6 c^6 d^6 (a e+c d x)^2} \]

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)^3,x]

[Out]

(-27*a^5*e^10 + 3*a^4*c*d*e^8*(35*d + 2*e*x) + 3*a^3*c^2*d^2*e^6*(-50*d^2 + 10*d
*e*x + 21*e^2*x^2) + 5*a^2*c^3*d^3*e^4*(18*d^3 - 24*d^2*e*x - 33*d*e^2*x^2 + 4*e
^3*x^3) - 5*a*c^4*d^4*e^2*(3*d^4 - 24*d^3*e*x - 24*d^2*e^2*x^2 + 12*d*e^3*x^3 +
e^4*x^4) + c^5*d^5*(-3*d^5 - 30*d^4*e*x + 60*d^2*e^3*x^3 + 15*d*e^4*x^4 + 2*e^5*
x^5) - 60*e^2*(-(c*d^2) + a*e^2)^3*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(6*c^6*d^6*
(a*e + c*d*x)^2)

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Maple [B]  time = 0.016, size = 412, normalized size = 2.2 \[{\frac{{e}^{5}{x}^{3}}{3\,{c}^{3}{d}^{3}}}-{\frac{3\,{e}^{6}{x}^{2}a}{2\,{c}^{4}{d}^{4}}}+{\frac{5\,{e}^{4}{x}^{2}}{2\,{c}^{3}{d}^{2}}}+6\,{\frac{{a}^{2}{e}^{7}x}{{c}^{5}{d}^{5}}}-15\,{\frac{a{e}^{5}x}{{c}^{4}{d}^{3}}}+10\,{\frac{{e}^{3}x}{{c}^{3}d}}-5\,{\frac{{a}^{4}{e}^{9}}{{c}^{6}{d}^{6} \left ( cdx+ae \right ) }}+20\,{\frac{{e}^{7}{a}^{3}}{{d}^{4}{c}^{5} \left ( cdx+ae \right ) }}-30\,{\frac{{a}^{2}{e}^{5}}{{d}^{2}{c}^{4} \left ( cdx+ae \right ) }}+20\,{\frac{a{e}^{3}}{{c}^{3} \left ( cdx+ae \right ) }}-5\,{\frac{{d}^{2}e}{{c}^{2} \left ( cdx+ae \right ) }}+{\frac{{a}^{5}{e}^{10}}{2\,{c}^{6}{d}^{6} \left ( cdx+ae \right ) ^{2}}}-{\frac{5\,{a}^{4}{e}^{8}}{2\,{d}^{4}{c}^{5} \left ( cdx+ae \right ) ^{2}}}+5\,{\frac{{a}^{3}{e}^{6}}{{d}^{2}{c}^{4} \left ( cdx+ae \right ) ^{2}}}-5\,{\frac{{a}^{2}{e}^{4}}{{c}^{3} \left ( cdx+ae \right ) ^{2}}}+{\frac{5\,a{d}^{2}{e}^{2}}{2\,{c}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{{d}^{4}}{2\,c \left ( cdx+ae \right ) ^{2}}}-10\,{\frac{{e}^{8}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{6}{d}^{6}}}+30\,{\frac{{e}^{6}\ln \left ( cdx+ae \right ){a}^{2}}{{d}^{4}{c}^{5}}}-30\,{\frac{{e}^{4}\ln \left ( cdx+ae \right ) a}{{d}^{2}{c}^{4}}}+10\,{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}}} \]

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)^3,x)

[Out]

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

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Maxima [A]  time = 0.751618, size = 419, normalized size = 2.26 \[ -\frac{c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} - 30 \, a^{2} c^{3} d^{6} e^{4} + 50 \, a^{3} c^{2} d^{4} e^{6} - 35 \, a^{4} c d^{2} e^{8} + 9 \, a^{5} e^{10} + 10 \,{\left (c^{5} d^{9} e - 4 \, a c^{4} d^{7} e^{3} + 6 \, a^{2} c^{3} d^{5} e^{5} - 4 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x}{2 \,{\left (c^{8} d^{8} x^{2} + 2 \, a c^{7} d^{7} e x + a^{2} c^{6} d^{6} e^{2}\right )}} + \frac{2 \, c^{2} d^{2} e^{5} x^{3} + 3 \,{\left (5 \, c^{2} d^{3} e^{4} - 3 \, a c d e^{6}\right )} x^{2} + 6 \,{\left (10 \, c^{2} d^{4} e^{3} - 15 \, a c d^{2} e^{5} + 6 \, a^{2} e^{7}\right )} x}{6 \, c^{5} d^{5}} + \frac{10 \,{\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/2*(c^5*d^10 + 5*a*c^4*d^8*e^2 - 30*a^2*c^3*d^6*e^4 + 50*a^3*c^2*d^4*e^6 - 35*
a^4*c*d^2*e^8 + 9*a^5*e^10 + 10*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5
- 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)*x)/(c^8*d^8*x^2 + 2*a*c^7*d^7*e*x + a^2*c^6*
d^6*e^2) + 1/6*(2*c^2*d^2*e^5*x^3 + 3*(5*c^2*d^3*e^4 - 3*a*c*d*e^6)*x^2 + 6*(10*
c^2*d^4*e^3 - 15*a*c*d^2*e^5 + 6*a^2*e^7)*x)/(c^5*d^5) + 10*(c^3*d^6*e^2 - 3*a*c
^2*d^4*e^4 + 3*a^2*c*d^2*e^6 - a^3*e^8)*log(c*d*x + a*e)/(c^6*d^6)

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Fricas [A]  time = 0.205833, size = 633, normalized size = 3.42 \[ \frac{2 \, c^{5} d^{5} e^{5} x^{5} - 3 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 90 \, a^{2} c^{3} d^{6} e^{4} - 150 \, a^{3} c^{2} d^{4} e^{6} + 105 \, a^{4} c d^{2} e^{8} - 27 \, a^{5} e^{10} + 5 \,{\left (3 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 20 \,{\left (3 \, c^{5} d^{7} e^{3} - 3 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \,{\left (40 \, a c^{4} d^{6} e^{4} - 55 \, a^{2} c^{3} d^{4} e^{6} + 21 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 6 \,{\left (5 \, c^{5} d^{9} e - 20 \, a c^{4} d^{7} e^{3} + 20 \, a^{2} c^{3} d^{5} e^{5} - 5 \, a^{3} c^{2} d^{3} e^{7} - a^{4} c d e^{9}\right )} x + 60 \,{\left (a^{2} c^{3} d^{6} e^{4} - 3 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - a^{5} e^{10} +{\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \,{\left (a c^{4} d^{7} e^{3} - 3 \, a^{2} c^{3} d^{5} e^{5} + 3 \, a^{3} c^{2} d^{3} e^{7} - a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{6 \,{\left (c^{8} d^{8} x^{2} + 2 \, a c^{7} d^{7} e x + a^{2} c^{6} d^{6} e^{2}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/6*(2*c^5*d^5*e^5*x^5 - 3*c^5*d^10 - 15*a*c^4*d^8*e^2 + 90*a^2*c^3*d^6*e^4 - 15
0*a^3*c^2*d^4*e^6 + 105*a^4*c*d^2*e^8 - 27*a^5*e^10 + 5*(3*c^5*d^6*e^4 - a*c^4*d
^4*e^6)*x^4 + 20*(3*c^5*d^7*e^3 - 3*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^3 + 3*(40
*a*c^4*d^6*e^4 - 55*a^2*c^3*d^4*e^6 + 21*a^3*c^2*d^2*e^8)*x^2 - 6*(5*c^5*d^9*e -
20*a*c^4*d^7*e^3 + 20*a^2*c^3*d^5*e^5 - 5*a^3*c^2*d^3*e^7 - a^4*c*d*e^9)*x + 60
*(a^2*c^3*d^6*e^4 - 3*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2*e^8 - a^5*e^10 + (c^5*d^8*e^
2 - 3*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 + 2*(a*c^4*d^7*e^
3 - 3*a^2*c^3*d^5*e^5 + 3*a^3*c^2*d^3*e^7 - a^4*c*d*e^9)*x)*log(c*d*x + a*e))/(c
^8*d^8*x^2 + 2*a*c^7*d^7*e*x + a^2*c^6*d^6*e^2)

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Sympy [A]  time = 12.8974, size = 299, normalized size = 1.62 \[ - \frac{9 a^{5} e^{10} - 35 a^{4} c d^{2} e^{8} + 50 a^{3} c^{2} d^{4} e^{6} - 30 a^{2} c^{3} d^{6} e^{4} + 5 a c^{4} d^{8} e^{2} + c^{5} d^{10} + x \left (10 a^{4} c d e^{9} - 40 a^{3} c^{2} d^{3} e^{7} + 60 a^{2} c^{3} d^{5} e^{5} - 40 a c^{4} d^{7} e^{3} + 10 c^{5} d^{9} e\right )}{2 a^{2} c^{6} d^{6} e^{2} + 4 a c^{7} d^{7} e x + 2 c^{8} d^{8} x^{2}} + \frac{e^{5} x^{3}}{3 c^{3} d^{3}} - \frac{x^{2} \left (3 a e^{6} - 5 c d^{2} e^{4}\right )}{2 c^{4} d^{4}} + \frac{x \left (6 a^{2} e^{7} - 15 a c d^{2} e^{5} + 10 c^{2} d^{4} e^{3}\right )}{c^{5} d^{5}} - \frac{10 e^{2} \left (a e^{2} - c d^{2}\right )^{3} \log{\left (a e + c d x \right )}}{c^{6} d^{6}} \]

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)**3,x)

[Out]

-(9*a**5*e**10 - 35*a**4*c*d**2*e**8 + 50*a**3*c**2*d**4*e**6 - 30*a**2*c**3*d**
6*e**4 + 5*a*c**4*d**8*e**2 + c**5*d**10 + x*(10*a**4*c*d*e**9 - 40*a**3*c**2*d*
*3*e**7 + 60*a**2*c**3*d**5*e**5 - 40*a*c**4*d**7*e**3 + 10*c**5*d**9*e))/(2*a**
2*c**6*d**6*e**2 + 4*a*c**7*d**7*e*x + 2*c**8*d**8*x**2) + e**5*x**3/(3*c**3*d**
3) - x**2*(3*a*e**6 - 5*c*d**2*e**4)/(2*c**4*d**4) + x*(6*a**2*e**7 - 15*a*c*d**
2*e**5 + 10*c**2*d**4*e**3)/(c**5*d**5) - 10*e**2*(a*e**2 - c*d**2)**3*log(a*e +
c*d*x)/(c**6*d**6)

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GIAC/XCAS [A]  time = 27.3622, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

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)^3,x, algorithm="giac")

[Out]

Done