3.1887 \(\int \frac{(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=146 \[ -\frac{6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}-\frac{2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}+\frac{4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x}{c^4 d^4} \]

[Out]

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

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Rubi [A]  time = 0.320527, antiderivative size = 146, 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 \[ -\frac{6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}-\frac{2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}+\frac{4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}+\frac{e^4 x}{c^4 d^4} \]

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)^4,x]

[Out]

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

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

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)**4,x)

[Out]

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

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Mathematica [A]  time = 0.143944, size = 194, normalized size = 1.33 \[ \frac{-13 a^4 e^8+a^3 c d e^6 (22 d-27 e x)-3 a^2 c^2 d^2 e^4 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a c^3 d^3 e^2 \left (-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right )-12 e^3 \left (a e^2-c d^2\right ) (a e+c d x)^3 \log (a e+c d x)-c^4 \left (d^8+6 d^7 e x+18 d^6 e^2 x^2-3 d^4 e^4 x^4\right )}{3 c^5 d^5 (a e+c d x)^3} \]

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)^4,x]

[Out]

(-13*a^4*e^8 + a^3*c*d*e^6*(22*d - 27*e*x) - 3*a^2*c^2*d^2*e^4*(2*d^2 - 18*d*e*x
+ 3*e^2*x^2) + a*c^3*d^3*e^2*(-2*d^3 - 18*d^2*e*x + 36*d*e^2*x^2 + 9*e^3*x^3) -
c^4*(d^8 + 6*d^7*e*x + 18*d^6*e^2*x^2 - 3*d^4*e^4*x^4) - 12*e^3*(-(c*d^2) + a*e
^2)*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x)^3)

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Maple [B]  time = 0.015, size = 318, normalized size = 2.2 \[{\frac{{e}^{4}x}{{c}^{4}{d}^{4}}}-6\,{\frac{{e}^{6}{a}^{2}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) }}+12\,{\frac{{e}^{4}a}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) }}-6\,{\frac{{e}^{2}}{{c}^{3}d \left ( cdx+ae \right ) }}+2\,{\frac{{a}^{3}{e}^{7}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) ^{2}}}-6\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) ^{2}}}+6\,{\frac{a{e}^{3}}{{c}^{3}d \left ( cdx+ae \right ) ^{2}}}-2\,{\frac{de}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}-4\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ) a}{{c}^{5}{d}^{5}}}+4\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{{c}^{4}{d}^{3}}}-{\frac{{a}^{4}{e}^{8}}{3\,{c}^{5}{d}^{5} \left ( cdx+ae \right ) ^{3}}}+{\frac{4\,{a}^{3}{e}^{6}}{3\,{c}^{4}{d}^{3} \left ( cdx+ae \right ) ^{3}}}-2\,{\frac{{a}^{2}{e}^{4}}{{c}^{3}d \left ( cdx+ae \right ) ^{3}}}+{\frac{4\,ad{e}^{2}}{3\,{c}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{{d}^{3}}{3\,c \left ( cdx+ae \right ) ^{3}}} \]

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)^4,x)

[Out]

e^4*x/c^4/d^4-6/c^5/d^5*e^6/(c*d*x+a*e)*a^2+12/c^4/d^3*e^4/(c*d*x+a*e)*a-6/c^3/d
*e^2/(c*d*x+a*e)+2/d^5*e^7/c^5/(c*d*x+a*e)^2*a^3-6/d^3*e^5/c^4/(c*d*x+a*e)^2*a^2
+6/d*e^3/c^3/(c*d*x+a*e)^2*a-2*d*e/c^2/(c*d*x+a*e)^2-4/c^5/d^5*e^5*ln(c*d*x+a*e)
*a+4/c^4/d^3*e^3*ln(c*d*x+a*e)-1/3/d^5/c^5/(c*d*x+a*e)^3*a^4*e^8+4/3/d^3/c^4/(c*
d*x+a*e)^3*a^3*e^6-2/d/c^3/(c*d*x+a*e)^3*a^2*e^4+4/3*d/c^2/(c*d*x+a*e)^3*a*e^2-1
/3*d^3/c/(c*d*x+a*e)^3

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Maxima [A]  time = 0.746651, size = 328, normalized size = 2.25 \[ -\frac{c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \,{\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \,{\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \,{\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} e x^{2} + 3 \, a^{2} c^{6} d^{6} e^{2} x + a^{3} c^{5} d^{5} e^{3}\right )}} + \frac{e^{4} x}{c^{4} d^{4}} + \frac{4 \,{\left (c d^{2} e^{3} - a e^{5}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \]

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)^4,x, algorithm="maxima")

[Out]

-1/3*(c^4*d^8 + 2*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 22*a^3*c*d^2*e^6 + 13*a^4*
e^8 + 18*(c^4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 6*(c^4*d^7*e +
3*a*c^3*d^5*e^3 - 9*a^2*c^2*d^3*e^5 + 5*a^3*c*d*e^7)*x)/(c^8*d^8*x^3 + 3*a*c^7*d
^7*e*x^2 + 3*a^2*c^6*d^6*e^2*x + a^3*c^5*d^5*e^3) + e^4*x/(c^4*d^4) + 4*(c*d^2*e
^3 - a*e^5)*log(c*d*x + a*e)/(c^5*d^5)

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Fricas [A]  time = 0.206267, size = 468, normalized size = 3.21 \[ \frac{3 \, c^{4} d^{4} e^{4} x^{4} + 9 \, a c^{3} d^{3} e^{5} x^{3} - c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 22 \, a^{3} c d^{2} e^{6} - 13 \, a^{4} e^{8} - 9 \,{\left (2 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 3 \,{\left (2 \, c^{4} d^{7} e + 6 \, a c^{3} d^{5} e^{3} - 18 \, a^{2} c^{2} d^{3} e^{5} + 9 \, a^{3} c d e^{7}\right )} x + 12 \,{\left (a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \,{\left (a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{3 \,{\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} e x^{2} + 3 \, a^{2} c^{6} d^{6} e^{2} x + a^{3} c^{5} d^{5} e^{3}\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

1/3*(3*c^4*d^4*e^4*x^4 + 9*a*c^3*d^3*e^5*x^3 - c^4*d^8 - 2*a*c^3*d^6*e^2 - 6*a^2
*c^2*d^4*e^4 + 22*a^3*c*d^2*e^6 - 13*a^4*e^8 - 9*(2*c^4*d^6*e^2 - 4*a*c^3*d^4*e^
4 + a^2*c^2*d^2*e^6)*x^2 - 3*(2*c^4*d^7*e + 6*a*c^3*d^5*e^3 - 18*a^2*c^2*d^3*e^5
+ 9*a^3*c*d*e^7)*x + 12*(a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^5*e^3 - a*c^3*d^3*e^5
)*x^3 + 3*(a*c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*x^2 + 3*(a^2*c^2*d^3*e^5 - a^3*c*d*e
^7)*x)*log(c*d*x + a*e))/(c^8*d^8*x^3 + 3*a*c^7*d^7*e*x^2 + 3*a^2*c^6*d^6*e^2*x
+ a^3*c^5*d^5*e^3)

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Sympy [A]  time = 27.7764, size = 257, normalized size = 1.76 \[ - \frac{13 a^{4} e^{8} - 22 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} + 2 a c^{3} d^{6} e^{2} + c^{4} d^{8} + x^{2} \left (18 a^{2} c^{2} d^{2} e^{6} - 36 a c^{3} d^{4} e^{4} + 18 c^{4} d^{6} e^{2}\right ) + x \left (30 a^{3} c d e^{7} - 54 a^{2} c^{2} d^{3} e^{5} + 18 a c^{3} d^{5} e^{3} + 6 c^{4} d^{7} e\right )}{3 a^{3} c^{5} d^{5} e^{3} + 9 a^{2} c^{6} d^{6} e^{2} x + 9 a c^{7} d^{7} e x^{2} + 3 c^{8} d^{8} x^{3}} + \frac{e^{4} x}{c^{4} d^{4}} - \frac{4 e^{3} \left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{5} d^{5}} \]

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)**4,x)

[Out]

-(13*a**4*e**8 - 22*a**3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 2*a*c**3*d**6*e**
2 + c**4*d**8 + x**2*(18*a**2*c**2*d**2*e**6 - 36*a*c**3*d**4*e**4 + 18*c**4*d**
6*e**2) + x*(30*a**3*c*d*e**7 - 54*a**2*c**2*d**3*e**5 + 18*a*c**3*d**5*e**3 + 6
*c**4*d**7*e))/(3*a**3*c**5*d**5*e**3 + 9*a**2*c**6*d**6*e**2*x + 9*a*c**7*d**7*
e*x**2 + 3*c**8*d**8*x**3) + e**4*x/(c**4*d**4) - 4*e**3*(a*e**2 - c*d**2)*log(a
*e + c*d*x)/(c**5*d**5)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)^4,x, algorithm="giac")

[Out]

Timed out