3.1864 \(\int \frac{(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=184 \[ \frac{e^5 (a e+c d x)^4}{4 c^6 d^6}-\frac{\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac{5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}+\frac{5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac{5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac{10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5} \]

[Out]

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

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Rubi [A]  time = 0.43705, antiderivative size = 184, 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 \[ \frac{e^5 (a e+c d x)^4}{4 c^6 d^6}-\frac{\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac{5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}+\frac{5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac{5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac{10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 77.3356, size = 173, normalized size = 0.94 \[ - \frac{10 e^{2} x \left (a e^{2} - c d^{2}\right )^{3}}{c^{5} d^{5}} + \frac{e^{5} \left (a e + c d x\right )^{4}}{4 c^{6} d^{6}} - \frac{5 e^{4} \left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )}{3 c^{6} d^{6}} + \frac{5 e^{3} \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}}{c^{6} d^{6}} + \frac{5 e \left (a e^{2} - c d^{2}\right )^{4} \log{\left (a e + c d x \right )}}{c^{6} d^{6}} + \frac{\left (a e^{2} - c d^{2}\right )^{5}}{c^{6} d^{6} \left (a e + c d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.166755, size = 263, normalized size = 1.43 \[ \frac{12 a^5 e^{10}-12 a^4 c d e^8 (5 d+4 e x)+30 a^3 c^2 d^2 e^6 \left (4 d^2+6 d e x-e^2 x^2\right )-10 a^2 c^3 d^3 e^4 \left (12 d^3+24 d^2 e x-12 d e^2 x^2-e^3 x^3\right )+5 a c^4 d^4 e^2 \left (12 d^4+24 d^3 e x-36 d^2 e^2 x^2-8 d e^3 x^3-e^4 x^4\right )+60 e \left (c d^2-a e^2\right )^4 (a e+c d x) \log (a e+c d x)+c^5 d^5 \left (-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right )}{12 c^6 d^6 (a e+c d x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(12*a^5*e^10 - 12*a^4*c*d*e^8*(5*d + 4*e*x) + 30*a^3*c^2*d^2*e^6*(4*d^2 + 6*d*e*
x - e^2*x^2) - 10*a^2*c^3*d^3*e^4*(12*d^3 + 24*d^2*e*x - 12*d*e^2*x^2 - e^3*x^3)
+ 5*a*c^4*d^4*e^2*(12*d^4 + 24*d^3*e*x - 36*d^2*e^2*x^2 - 8*d*e^3*x^3 - e^4*x^4
) + c^5*d^5*(-12*d^5 + 120*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 20*d*e^4*x^4 + 3*e^5*x
^5) + 60*e*(c*d^2 - a*e^2)^4*(a*e + c*d*x)*Log[a*e + c*d*x])/(12*c^6*d^6*(a*e +
c*d*x))

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Maple [B]  time = 0.014, size = 378, normalized size = 2.1 \[{\frac{{e}^{5}{x}^{4}}{4\,{c}^{2}{d}^{2}}}-{\frac{2\,{e}^{6}{x}^{3}a}{3\,{c}^{3}{d}^{3}}}+{\frac{5\,{e}^{4}{x}^{3}}{3\,{c}^{2}d}}+{\frac{3\,{e}^{7}{x}^{2}{a}^{2}}{2\,{c}^{4}{d}^{4}}}-5\,{\frac{{e}^{5}{x}^{2}a}{{c}^{3}{d}^{2}}}+5\,{\frac{{e}^{3}{x}^{2}}{{c}^{2}}}-4\,{\frac{{a}^{3}{e}^{8}x}{{c}^{5}{d}^{5}}}+15\,{\frac{{a}^{2}{e}^{6}x}{{c}^{4}{d}^{3}}}-20\,{\frac{a{e}^{4}x}{{c}^{3}d}}+10\,{\frac{d{e}^{2}x}{{c}^{2}}}+{\frac{{a}^{5}{e}^{10}}{{c}^{6}{d}^{6} \left ( cdx+ae \right ) }}-5\,{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{4} \left ( cdx+ae \right ) }}+10\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}{d}^{2} \left ( cdx+ae \right ) }}-10\,{\frac{{a}^{2}{e}^{4}}{{c}^{3} \left ( cdx+ae \right ) }}+5\,{\frac{a{d}^{2}{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) }}-{\frac{{d}^{4}}{c \left ( cdx+ae \right ) }}+5\,{\frac{{e}^{9}\ln \left ( cdx+ae \right ){a}^{4}}{{c}^{6}{d}^{6}}}-20\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{5}{d}^{4}}}+30\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{4}{d}^{2}}}-20\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}}}+5\,{\frac{{d}^{2}e\ln \left ( cdx+ae \right ) }{{c}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^7/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.734513, size = 405, normalized size = 2.2 \[ -\frac{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}}{c^{7} d^{7} x + a c^{6} d^{6} e} + \frac{3 \, c^{3} d^{3} e^{5} x^{4} + 4 \,{\left (5 \, c^{3} d^{4} e^{4} - 2 \, a c^{2} d^{2} e^{6}\right )} x^{3} + 6 \,{\left (10 \, c^{3} d^{5} e^{3} - 10 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{2} + 12 \,{\left (10 \, c^{3} d^{6} e^{2} - 20 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} - 4 \, a^{3} e^{8}\right )} x}{12 \, c^{5} d^{5}} + \frac{5 \,{\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.205255, size = 563, normalized size = 3.06 \[ \frac{3 \, c^{5} d^{5} e^{5} x^{5} - 12 \, c^{5} d^{10} + 60 \, a c^{4} d^{8} e^{2} - 120 \, a^{2} c^{3} d^{6} e^{4} + 120 \, a^{3} c^{2} d^{4} e^{6} - 60 \, a^{4} c d^{2} e^{8} + 12 \, a^{5} e^{10} + 5 \,{\left (4 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 10 \,{\left (6 \, c^{5} d^{7} e^{3} - 4 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 30 \,{\left (4 \, c^{5} d^{8} e^{2} - 6 \, a c^{4} d^{6} e^{4} + 4 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 12 \,{\left (10 \, a c^{4} d^{7} e^{3} - 20 \, a^{2} c^{3} d^{5} e^{5} + 15 \, a^{3} c^{2} d^{3} e^{7} - 4 \, a^{4} c d e^{9}\right )} x + 60 \,{\left (a c^{4} d^{8} e^{2} - 4 \, a^{2} c^{3} d^{6} e^{4} + 6 \, a^{3} c^{2} d^{4} e^{6} - 4 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} +{\left (c^{5} d^{9} e - 4 \, a c^{4} d^{7} e^{3} + 6 \, a^{2} c^{3} d^{5} e^{5} - 4 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{12 \,{\left (c^{7} d^{7} x + a c^{6} d^{6} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")

[Out]

1/12*(3*c^5*d^5*e^5*x^5 - 12*c^5*d^10 + 60*a*c^4*d^8*e^2 - 120*a^2*c^3*d^6*e^4 +
120*a^3*c^2*d^4*e^6 - 60*a^4*c*d^2*e^8 + 12*a^5*e^10 + 5*(4*c^5*d^6*e^4 - a*c^4
*d^4*e^6)*x^4 + 10*(6*c^5*d^7*e^3 - 4*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^3 + 30*
(4*c^5*d^8*e^2 - 6*a*c^4*d^6*e^4 + 4*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^2 + 12
*(10*a*c^4*d^7*e^3 - 20*a^2*c^3*d^5*e^5 + 15*a^3*c^2*d^3*e^7 - 4*a^4*c*d*e^9)*x
+ 60*(a*c^4*d^8*e^2 - 4*a^2*c^3*d^6*e^4 + 6*a^3*c^2*d^4*e^6 - 4*a^4*c*d^2*e^8 +
a^5*e^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)*log(c*d*x + a*e))/(c^7*d^7*x + a*c^6*d^6*e)

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Sympy [A]  time = 5.33305, size = 265, normalized size = 1.44 \[ \frac{a^{5} e^{10} - 5 a^{4} c d^{2} e^{8} + 10 a^{3} c^{2} d^{4} e^{6} - 10 a^{2} c^{3} d^{6} e^{4} + 5 a c^{4} d^{8} e^{2} - c^{5} d^{10}}{a c^{6} d^{6} e + c^{7} d^{7} x} + \frac{e^{5} x^{4}}{4 c^{2} d^{2}} - \frac{x^{3} \left (2 a e^{6} - 5 c d^{2} e^{4}\right )}{3 c^{3} d^{3}} + \frac{x^{2} \left (3 a^{2} e^{7} - 10 a c d^{2} e^{5} + 10 c^{2} d^{4} e^{3}\right )}{2 c^{4} d^{4}} - \frac{x \left (4 a^{3} e^{8} - 15 a^{2} c d^{2} e^{6} + 20 a c^{2} d^{4} e^{4} - 10 c^{3} d^{6} e^{2}\right )}{c^{5} d^{5}} + \frac{5 e \left (a e^{2} - c d^{2}\right )^{4} \log{\left (a e + c d x \right )}}{c^{6} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

(a**5*e**10 - 5*a**4*c*d**2*e**8 + 10*a**3*c**2*d**4*e**6 - 10*a**2*c**3*d**6*e*
*4 + 5*a*c**4*d**8*e**2 - c**5*d**10)/(a*c**6*d**6*e + c**7*d**7*x) + e**5*x**4/
(4*c**2*d**2) - x**3*(2*a*e**6 - 5*c*d**2*e**4)/(3*c**3*d**3) + x**2*(3*a**2*e**
7 - 10*a*c*d**2*e**5 + 10*c**2*d**4*e**3)/(2*c**4*d**4) - x*(4*a**3*e**8 - 15*a*
*2*c*d**2*e**6 + 20*a*c**2*d**4*e**4 - 10*c**3*d**6*e**2)/(c**5*d**5) + 5*e*(a*e
**2 - c*d**2)**4*log(a*e + c*d*x)/(c**6*d**6)

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GIAC/XCAS [A]  time = 0.331119, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")

[Out]

Done