3.1888 \(\int \frac{(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=122 \[ \frac{e^3 \log (a e+c d x)}{c^4 d^4}-\frac{3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3} \]

[Out]

-(c*d^2 - a*e^2)^3/(3*c^4*d^4*(a*e + c*d*x)^3) - (3*e*(c*d^2 - a*e^2)^2)/(2*c^4*
d^4*(a*e + c*d*x)^2) - (3*e^2*(c*d^2 - a*e^2))/(c^4*d^4*(a*e + c*d*x)) + (e^3*Lo
g[a*e + c*d*x])/(c^4*d^4)

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Rubi [A]  time = 0.22851, antiderivative size = 122, 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 \[ \frac{e^3 \log (a e+c d x)}{c^4 d^4}-\frac{3 e^2 \left (c d^2-a e^2\right )}{c^4 d^4 (a e+c d x)}-\frac{3 e \left (c d^2-a e^2\right )^2}{2 c^4 d^4 (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^3}{3 c^4 d^4 (a e+c d x)^3} \]

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)^4,x]

[Out]

-(c*d^2 - a*e^2)^3/(3*c^4*d^4*(a*e + c*d*x)^3) - (3*e*(c*d^2 - a*e^2)^2)/(2*c^4*
d^4*(a*e + c*d*x)^2) - (3*e^2*(c*d^2 - a*e^2))/(c^4*d^4*(a*e + c*d*x)) + (e^3*Lo
g[a*e + c*d*x])/(c^4*d^4)

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Rubi in Sympy [A]  time = 51.4672, size = 114, normalized size = 0.93 \[ \frac{e^{3} \log{\left (a e + c d x \right )}}{c^{4} d^{4}} + \frac{3 e^{2} \left (a e^{2} - c d^{2}\right )}{c^{4} d^{4} \left (a e + c d x\right )} - \frac{3 e \left (a e^{2} - c d^{2}\right )^{2}}{2 c^{4} d^{4} \left (a e + c d x\right )^{2}} + \frac{\left (a e^{2} - c d^{2}\right )^{3}}{3 c^{4} d^{4} \left (a e + c d x\right )^{3}} \]

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)**4,x)

[Out]

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

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Mathematica [A]  time = 0.0978415, size = 99, normalized size = 0.81 \[ \frac{6 e^3 \log (a e+c d x)-\frac{\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (5 d+27 e x)+c^2 d^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )}{(a e+c d x)^3}}{6 c^4 d^4} \]

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)^4,x]

[Out]

(-(((c*d^2 - a*e^2)*(11*a^2*e^4 + a*c*d*e^2*(5*d + 27*e*x) + c^2*d^2*(2*d^2 + 9*
d*e*x + 18*e^2*x^2)))/(a*e + c*d*x)^3) + 6*e^3*Log[a*e + c*d*x])/(6*c^4*d^4)

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Maple [A]  time = 0.01, size = 210, normalized size = 1.7 \[ 3\,{\frac{{e}^{4}a}{{c}^{4}{d}^{4} \left ( cdx+ae \right ) }}-3\,{\frac{{e}^{2}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) }}-{\frac{3\,{a}^{2}{e}^{5}}{2\,{c}^{4}{d}^{4} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{a{e}^{3}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{3\,e}{2\,{c}^{2} \left ( cdx+ae \right ) ^{2}}}+{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{{c}^{4}{d}^{4}}}+{\frac{{a}^{3}{e}^{6}}{3\,{c}^{4}{d}^{4} \left ( cdx+ae \right ) ^{3}}}-{\frac{{a}^{2}{e}^{4}}{{c}^{3}{d}^{2} \left ( cdx+ae \right ) ^{3}}}+{\frac{a{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{{d}^{2}}{3\,c \left ( cdx+ae \right ) ^{3}}} \]

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)^4,x)

[Out]

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

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Maxima [A]  time = 0.733848, size = 242, normalized size = 1.98 \[ -\frac{2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{6 \,{\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} + \frac{e^{3} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]

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)^4,x, algorithm="maxima")

[Out]

-1/6*(2*c^3*d^6 + 3*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 - 11*a^3*e^6 + 18*(c^3*d^4*e
^2 - a*c^2*d^2*e^4)*x^2 + 9*(c^3*d^5*e + 2*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x)/(c^
7*d^7*x^3 + 3*a*c^6*d^6*e*x^2 + 3*a^2*c^5*d^5*e^2*x + a^3*c^4*d^4*e^3) + e^3*log
(c*d*x + a*e)/(c^4*d^4)

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Fricas [A]  time = 0.206915, size = 293, normalized size = 2.4 \[ -\frac{2 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 18 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 9 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{6 \,{\left (c^{7} d^{7} x^{3} + 3 \, a c^{6} d^{6} e x^{2} + 3 \, a^{2} c^{5} d^{5} e^{2} x + a^{3} c^{4} d^{4} e^{3}\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

-1/6*(2*c^3*d^6 + 3*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 - 11*a^3*e^6 + 18*(c^3*d^4*e
^2 - a*c^2*d^2*e^4)*x^2 + 9*(c^3*d^5*e + 2*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x - 6*
(c^3*d^3*e^3*x^3 + 3*a*c^2*d^2*e^4*x^2 + 3*a^2*c*d*e^5*x + a^3*e^6)*log(c*d*x +
a*e))/(c^7*d^7*x^3 + 3*a*c^6*d^6*e*x^2 + 3*a^2*c^5*d^5*e^2*x + a^3*c^4*d^4*e^3)

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Sympy [A]  time = 8.6542, size = 189, normalized size = 1.55 \[ \frac{11 a^{3} e^{6} - 6 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} - 2 c^{3} d^{6} + x^{2} \left (18 a c^{2} d^{2} e^{4} - 18 c^{3} d^{4} e^{2}\right ) + x \left (27 a^{2} c d e^{5} - 18 a c^{2} d^{3} e^{3} - 9 c^{3} d^{5} e\right )}{6 a^{3} c^{4} d^{4} e^{3} + 18 a^{2} c^{5} d^{5} e^{2} x + 18 a c^{6} d^{6} e x^{2} + 6 c^{7} d^{7} x^{3}} + \frac{e^{3} \log{\left (a e + c d x \right )}}{c^{4} d^{4}} \]

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)**4,x)

[Out]

(11*a**3*e**6 - 6*a**2*c*d**2*e**4 - 3*a*c**2*d**4*e**2 - 2*c**3*d**6 + x**2*(18
*a*c**2*d**2*e**4 - 18*c**3*d**4*e**2) + x*(27*a**2*c*d*e**5 - 18*a*c**2*d**3*e*
*3 - 9*c**3*d**5*e))/(6*a**3*c**4*d**4*e**3 + 18*a**2*c**5*d**5*e**2*x + 18*a*c*
*6*d**6*e*x**2 + 6*c**7*d**7*x**3) + e**3*log(a*e + c*d*x)/(c**4*d**4)

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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)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="giac")

[Out]

Timed out