3.1877 \(\int \frac{(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}+\frac{3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4}+\frac{e^3 x}{c^3 d^3} \]

[Out]

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

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Rubi [A]  time = 0.226887, antiderivative size = 111, 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 \[ -\frac{3 e \left (c d^2-a e^2\right )^2}{c^4 d^4 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^3}{2 c^4 d^4 (a e+c d x)^2}+\frac{3 e^2 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^4 d^4}+\frac{e^3 x}{c^3 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.106576, size = 139, normalized size = 1.25 \[ \frac{-5 a^3 e^6+a^2 c d e^4 (9 d-4 e x)+a c^2 d^2 e^2 \left (-3 d^2+12 d e x+4 e^2 x^2\right )-6 e^2 \left (a e^2-c d^2\right ) (a e+c d x)^2 \log (a e+c d x)-c^3 \left (d^6+6 d^5 e x-2 d^3 e^3 x^3\right )}{2 c^4 d^4 (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(-5*a^3*e^6 + a^2*c*d*e^4*(9*d - 4*e*x) + a*c^2*d^2*e^2*(-3*d^2 + 12*d*e*x + 4*e
^2*x^2) - c^3*(d^6 + 6*d^5*e*x - 2*d^3*e^3*x^3) - 6*e^2*(-(c*d^2) + a*e^2)*(a*e
+ c*d*x)^2*Log[a*e + c*d*x])/(2*c^4*d^4*(a*e + c*d*x)^2)

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Maple [A]  time = 0.01, size = 201, normalized size = 1.8 \[{\frac{{e}^{3}x}{{c}^{3}{d}^{3}}}-3\,{\frac{{a}^{2}{e}^{5}}{{c}^{4}{d}^{4} \left ( cdx+ae \right ) }}+6\,{\frac{a{e}^{3}}{{d}^{2}{c}^{3} \left ( cdx+ae \right ) }}-3\,{\frac{e}{{c}^{2} \left ( cdx+ae \right ) }}+{\frac{{a}^{3}{e}^{6}}{2\,{c}^{4}{d}^{4} \left ( cdx+ae \right ) ^{2}}}-{\frac{3\,{a}^{2}{e}^{4}}{2\,{d}^{2}{c}^{3} \left ( cdx+ae \right ) ^{2}}}+{\frac{3\,a{e}^{2}}{2\,{c}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{{d}^{2}}{2\,c \left ( cdx+ae \right ) ^{2}}}-3\,{\frac{{e}^{4}\ln \left ( cdx+ae \right ) a}{{c}^{4}{d}^{4}}}+3\,{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{{d}^{2}{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^6/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.732277, size = 211, normalized size = 1.9 \[ -\frac{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} - 9 \, a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \,{\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}} + \frac{e^{3} x}{c^{3} d^{3}} + \frac{3 \,{\left (c d^{2} e^{2} - a e^{4}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.208952, size = 306, normalized size = 2.76 \[ \frac{2 \, c^{3} d^{3} e^{3} x^{3} + 4 \, a c^{2} d^{2} e^{4} x^{2} - c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} - 2 \,{\left (3 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 2 \, a^{2} c d e^{5}\right )} x + 6 \,{\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \log \left (c d x + a e\right )}{2 \,{\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.03195, size = 163, normalized size = 1.47 \[ - \frac{5 a^{3} e^{6} - 9 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6} + x \left (6 a^{2} c d e^{5} - 12 a c^{2} d^{3} e^{3} + 6 c^{3} d^{5} e\right )}{2 a^{2} c^{4} d^{4} e^{2} + 4 a c^{5} d^{5} e x + 2 c^{6} d^{6} x^{2}} + \frac{e^{3} x}{c^{3} d^{3}} - \frac{3 e^{2} \left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

-(5*a**3*e**6 - 9*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6 + x*(6*a**2*
c*d*e**5 - 12*a*c**2*d**3*e**3 + 6*c**3*d**5*e))/(2*a**2*c**4*d**4*e**2 + 4*a*c*
*5*d**5*e*x + 2*c**6*d**6*x**2) + e**3*x/(c**3*d**3) - 3*e**2*(a*e**2 - c*d**2)*
log(a*e + c*d*x)/(c**4*d**4)

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GIAC/XCAS [A]  time = 21.3928, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Done