3.1846 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=89 \[ \frac{1}{2} \left (a-\frac{c d^2}{e^2}\right ) (a e+c d x)^2-\frac{\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac{c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac{(a e+c d x)^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.124476, antiderivative size = 89, 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 \[ \frac{1}{2} \left (a-\frac{c d^2}{e^2}\right ) (a e+c d x)^2-\frac{\left (c d^2-a e^2\right )^3 \log (d+e x)}{e^4}+\frac{c d x \left (c d^2-a e^2\right )^2}{e^3}+\frac{(a e+c d x)^3}{3 e} \]

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)^4,x]

[Out]

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

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

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)**4,x)

[Out]

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

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Mathematica [A]  time = 0.0678054, size = 85, normalized size = 0.96 \[ \frac{c d e x \left (18 a^2 e^4+9 a c d e^2 (e x-2 d)+c^2 d^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 \left (c d^2-a e^2\right )^3 \log (d+e x)}{6 e^4} \]

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)^4,x]

[Out]

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

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Maple [A]  time = 0.004, size = 138, normalized size = 1.6 \[{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,e}}+{\frac{3\,{c}^{2}{d}^{2}{x}^{2}a}{2}}-{\frac{{c}^{3}{d}^{4}{x}^{2}}{2\,{e}^{2}}}+3\,cde{a}^{2}x-3\,{\frac{{c}^{2}{d}^{3}ax}{e}}+{\frac{{c}^{3}{d}^{5}x}{{e}^{3}}}+{e}^{2}\ln \left ( ex+d \right ){a}^{3}-3\,\ln \left ( ex+d \right ){a}^{2}c{d}^{2}+3\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{4}a}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{6}}{{e}^{4}}} \]

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)^4,x)

[Out]

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

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Maxima [A]  time = 0.731477, size = 177, normalized size = 1.99 \[ \frac{2 \, c^{3} d^{3} e^{2} x^{3} - 3 \,{\left (c^{3} d^{4} e - 3 \, a c^{2} d^{2} e^{3}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} - 3 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x}{6 \, e^{3}} - \frac{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (e x + d\right )}{e^{4}} \]

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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219164, size = 176, normalized size = 1.98 \[ \frac{2 \, c^{3} d^{3} e^{3} x^{3} - 3 \,{\left (c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \,{\left (c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 6 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \]

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)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.08411, size = 100, normalized size = 1.12 \[ \frac{c^{3} d^{3} x^{3}}{3 e} + \frac{x^{2} \left (3 a c^{2} d^{2} e^{2} - c^{3} d^{4}\right )}{2 e^{2}} + \frac{x \left (3 a^{2} c d e^{4} - 3 a c^{2} d^{3} e^{2} + c^{3} d^{5}\right )}{e^{3}} + \frac{\left (a e^{2} - c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{4}} \]

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)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214838, size = 173, normalized size = 1.94 \[ -{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, c^{3} d^{3} x^{3} e^{11} - 3 \, c^{3} d^{4} x^{2} e^{10} + 6 \, c^{3} d^{5} x e^{9} + 9 \, a c^{2} d^{2} x^{2} e^{12} - 18 \, a c^{2} d^{3} x e^{11} + 18 \, a^{2} c d x e^{13}\right )} e^{\left (-12\right )} \]

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)^4,x, algorithm="giac")

[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^(-4)*ln(abs(x*e + d))
+ 1/6*(2*c^3*d^3*x^3*e^11 - 3*c^3*d^4*x^2*e^10 + 6*c^3*d^5*x*e^9 + 9*a*c^2*d^2*
x^2*e^12 - 18*a*c^2*d^3*x*e^11 + 18*a^2*c*d*x*e^13)*e^(-12)