3.1847 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx\)

Optimal. Leaf size=94 \[ -\frac{c^2 d^2 x \left (2 c d^2-3 a e^2\right )}{e^3}+\frac{\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac{3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}+\frac{c^3 d^3 x^2}{2 e^2} \]

[Out]

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

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Rubi [A]  time = 0.19619, antiderivative size = 94, 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 \[ -\frac{c^2 d^2 x \left (2 c d^2-3 a e^2\right )}{e^3}+\frac{\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac{3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}+\frac{c^3 d^3 x^2}{2 e^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^5,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**5,x)

[Out]

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

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Mathematica [A]  time = 0.084942, size = 129, normalized size = 1.37 \[ \frac{-2 a^3 e^6+6 a^2 c d^2 e^4+6 a c^2 d^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 c d (d+e x) \left (c d^2-a e^2\right )^2 \log (d+e x)+c^3 d^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )}{2 e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^5,x]

[Out]

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

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Maple [A]  time = 0.011, size = 156, normalized size = 1.7 \[{\frac{{c}^{3}{d}^{3}{x}^{2}}{2\,{e}^{2}}}+3\,{\frac{a{c}^{2}{d}^{2}x}{e}}-2\,{\frac{{c}^{3}{d}^{4}x}{{e}^{3}}}+3\,dc\ln \left ( ex+d \right ){a}^{2}-6\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) a}{{e}^{2}}}+3\,{\frac{{c}^{3}{d}^{5}\ln \left ( ex+d \right ) }{{e}^{4}}}-{\frac{{e}^{2}{a}^{3}}{ex+d}}+3\,{\frac{{a}^{2}c{d}^{2}}{ex+d}}-3\,{\frac{{c}^{2}{d}^{4}a}{{e}^{2} \left ( ex+d \right ) }}+{\frac{{c}^{3}{d}^{6}}{{e}^{4} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 0.733098, size = 184, normalized size = 1.96 \[ \frac{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{e^{5} x + d e^{4}} + \frac{c^{3} d^{3} e x^{2} - 2 \,{\left (2 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} x}{2 \, e^{3}} + \frac{3 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/(e*x + d)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.220731, size = 261, normalized size = 2.78 \[ \frac{c^{3} d^{3} e^{3} x^{3} + 2 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} - 3 \,{\left (c^{3} d^{4} e^{2} - 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x + 6 \,{\left (c^{3} d^{6} - 2 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x + d e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/(e*x + d)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.19058, size = 119, normalized size = 1.27 \[ \frac{c^{3} d^{3} x^{2}}{2 e^{2}} + \frac{3 c d \left (a e^{2} - c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}}{d e^{4} + e^{5} x} + \frac{x \left (3 a c^{2} d^{2} e^{2} - 2 c^{3} d^{4}\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.222785, size = 240, normalized size = 2.55 \[ \frac{1}{2} \,{\left (c^{3} d^{3} - \frac{6 \,{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} - 3 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} e^{\left (-4\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{c^{3} d^{6} e^{20}}{x e + d} - \frac{3 \, a c^{2} d^{4} e^{22}}{x e + d} + \frac{3 \, a^{2} c d^{2} e^{24}}{x e + d} - \frac{a^{3} e^{26}}{x e + d}\right )} e^{\left (-24\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3/(e*x + d)^5,x, algorithm="giac")

[Out]

1/2*(c^3*d^3 - 6*(c^3*d^4*e - a*c^2*d^2*e^3)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-4
) - 3*(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*e^(-4)*ln(abs(x*e + d)*e^(-1)/(x
*e + d)^2) + (c^3*d^6*e^20/(x*e + d) - 3*a*c^2*d^4*e^22/(x*e + d) + 3*a^2*c*d^2*
e^24/(x*e + d) - a^3*e^26/(x*e + d))*e^(-24)