3.1892 \(\int \frac{(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=139 \[ -\frac{e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac{e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac{1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac{e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac{e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]

[Out]

-1/(3*(c*d^2 - a*e^2)*(a*e + c*d*x)^3) + e/(2*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^2)
- e^2/((c*d^2 - a*e^2)^3*(a*e + c*d*x)) - (e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2
)^4 + (e^3*Log[d + e*x])/(c*d^2 - a*e^2)^4

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Rubi [A]  time = 0.229225, antiderivative size = 139, 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 \[ -\frac{e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac{e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac{1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac{e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac{e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-1/(3*(c*d^2 - a*e^2)*(a*e + c*d*x)^3) + e/(2*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^2)
- e^2/((c*d^2 - a*e^2)^3*(a*e + c*d*x)) - (e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2
)^4 + (e^3*Log[d + e*x])/(c*d^2 - a*e^2)^4

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Rubi in Sympy [A]  time = 49.7337, size = 119, normalized size = 0.86 \[ \frac{e^{3} \log{\left (d + e x \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{e^{3} \log{\left (a e + c d x \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{e^{2}}{\left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{3}} + \frac{e}{2 \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{3 \left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

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

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Mathematica [A]  time = 0.192082, size = 117, normalized size = 0.84 \[ -\frac{\frac{\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (15 e x-7 d)+c^2 d^2 \left (2 d^2-3 d e x+6 e^2 x^2\right )\right )}{(a e+c d x)^3}+6 e^3 \log (a e+c d x)-6 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(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*(-7*d + 15*e*x) + c^2*d^2*(2*d^2 - 3*
d*e*x + 6*e^2*x^2)))/(a*e + c*d*x)^3 + 6*e^3*Log[a*e + c*d*x] - 6*e^3*Log[d + e*
x])/(6*(c*d^2 - a*e^2)^4)

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Maple [A]  time = 0.015, size = 135, normalized size = 1. \[{\frac{{e}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}}+{\frac{1}{ \left ( 3\,a{e}^{2}-3\,c{d}^{2} \right ) \left ( cdx+ae \right ) ^{3}}}+{\frac{e}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdx+ae \right ) ^{2}}}+{\frac{{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) }}-{\frac{{e}^{3}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

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

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Maxima [A]  time = 0.744466, size = 556, normalized size = 4. \[ -\frac{e^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac{e^{3} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac{6 \, c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 11 \, a^{2} e^{4} - 3 \,{\left (c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x}{6 \,{\left (a^{3} c^{3} d^{6} e^{3} - 3 \, a^{4} c^{2} d^{4} e^{5} + 3 \, a^{5} c d^{2} e^{7} - a^{6} e^{9} +{\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \,{\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + 3 \,{\left (a^{2} c^{4} d^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="maxima")

[Out]

-e^3*log(c*d*x + a*e)/(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d
^2*e^6 + a^4*e^8) + e^3*log(e*x + d)/(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*
e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8) - 1/6*(6*c^2*d^2*e^2*x^2 + 2*c^2*d^4 - 7*a*c*d^
2*e^2 + 11*a^2*e^4 - 3*(c^2*d^3*e - 5*a*c*d*e^3)*x)/(a^3*c^3*d^6*e^3 - 3*a^4*c^2
*d^4*e^5 + 3*a^5*c*d^2*e^7 - a^6*e^9 + (c^6*d^9 - 3*a*c^5*d^7*e^2 + 3*a^2*c^4*d^
5*e^4 - a^3*c^3*d^3*e^6)*x^3 + 3*(a*c^5*d^8*e - 3*a^2*c^4*d^6*e^3 + 3*a^3*c^3*d^
4*e^5 - a^4*c^2*d^2*e^7)*x^2 + 3*(a^2*c^4*d^7*e^2 - 3*a^3*c^3*d^5*e^4 + 3*a^4*c^
2*d^3*e^6 - a^5*c*d*e^8)*x)

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Fricas [A]  time = 0.22383, size = 653, normalized size = 4.7 \[ -\frac{2 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \,{\left (c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right ) - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (e x + d\right )}{6 \,{\left (a^{3} c^{4} d^{8} e^{3} - 4 \, a^{4} c^{3} d^{6} e^{5} + 6 \, a^{5} c^{2} d^{4} e^{7} - 4 \, a^{6} c d^{2} e^{9} + a^{7} e^{11} +{\left (c^{7} d^{11} - 4 \, a c^{6} d^{9} e^{2} + 6 \, a^{2} c^{5} d^{7} e^{4} - 4 \, a^{3} c^{4} d^{5} e^{6} + a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \,{\left (a c^{6} d^{10} e - 4 \, a^{2} c^{5} d^{8} e^{3} + 6 \, a^{3} c^{4} d^{6} e^{5} - 4 \, a^{4} c^{3} d^{4} e^{7} + a^{5} c^{2} d^{2} e^{9}\right )} x^{2} + 3 \,{\left (a^{2} c^{5} d^{9} e^{2} - 4 \, a^{3} c^{4} d^{7} e^{4} + 6 \, a^{4} c^{3} d^{5} e^{6} - 4 \, a^{5} c^{2} d^{3} e^{8} + a^{6} c d e^{10}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(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 - 9*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 - 11*a^3*e^6 + 6*(c^3*d^4*e
^2 - a*c^2*d^2*e^4)*x^2 - 3*(c^3*d^5*e - 6*a*c^2*d^3*e^3 + 5*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) - 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
(e*x + d))/(a^3*c^4*d^8*e^3 - 4*a^4*c^3*d^6*e^5 + 6*a^5*c^2*d^4*e^7 - 4*a^6*c*d^
2*e^9 + a^7*e^11 + (c^7*d^11 - 4*a*c^6*d^9*e^2 + 6*a^2*c^5*d^7*e^4 - 4*a^3*c^4*d
^5*e^6 + a^4*c^3*d^3*e^8)*x^3 + 3*(a*c^6*d^10*e - 4*a^2*c^5*d^8*e^3 + 6*a^3*c^4*
d^6*e^5 - 4*a^4*c^3*d^4*e^7 + a^5*c^2*d^2*e^9)*x^2 + 3*(a^2*c^5*d^9*e^2 - 4*a^3*
c^4*d^7*e^4 + 6*a^4*c^3*d^5*e^6 - 4*a^5*c^2*d^3*e^8 + a^6*c*d*e^10)*x)

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Sympy [A]  time = 8.04754, size = 668, normalized size = 4.81 \[ \frac{e^{3} \log{\left (x + \frac{- \frac{a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} + \frac{c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{e^{3} \log{\left (x + \frac{\frac{a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} - \frac{c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{11 a^{2} e^{4} - 7 a c d^{2} e^{2} + 2 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (15 a c d e^{3} - 3 c^{2} d^{3} e\right )}{6 a^{6} e^{9} - 18 a^{5} c d^{2} e^{7} + 18 a^{4} c^{2} d^{4} e^{5} - 6 a^{3} c^{3} d^{6} e^{3} + x^{3} \left (6 a^{3} c^{3} d^{3} e^{6} - 18 a^{2} c^{4} d^{5} e^{4} + 18 a c^{5} d^{7} e^{2} - 6 c^{6} d^{9}\right ) + x^{2} \left (18 a^{4} c^{2} d^{2} e^{7} - 54 a^{3} c^{3} d^{4} e^{5} + 54 a^{2} c^{4} d^{6} e^{3} - 18 a c^{5} d^{8} e\right ) + x \left (18 a^{5} c d e^{8} - 54 a^{4} c^{2} d^{3} e^{6} + 54 a^{3} c^{3} d^{5} e^{4} - 18 a^{2} c^{4} d^{7} e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

e**3*log(x + (-a**5*e**13/(a*e**2 - c*d**2)**4 + 5*a**4*c*d**2*e**11/(a*e**2 - c
*d**2)**4 - 10*a**3*c**2*d**4*e**9/(a*e**2 - c*d**2)**4 + 10*a**2*c**3*d**6*e**7
/(a*e**2 - c*d**2)**4 - 5*a*c**4*d**8*e**5/(a*e**2 - c*d**2)**4 + a*e**5 + c**5*
d**10*e**3/(a*e**2 - c*d**2)**4 + c*d**2*e**3)/(2*c*d*e**4))/(a*e**2 - c*d**2)**
4 - e**3*log(x + (a**5*e**13/(a*e**2 - c*d**2)**4 - 5*a**4*c*d**2*e**11/(a*e**2
- c*d**2)**4 + 10*a**3*c**2*d**4*e**9/(a*e**2 - c*d**2)**4 - 10*a**2*c**3*d**6*e
**7/(a*e**2 - c*d**2)**4 + 5*a*c**4*d**8*e**5/(a*e**2 - c*d**2)**4 + a*e**5 - c*
*5*d**10*e**3/(a*e**2 - c*d**2)**4 + c*d**2*e**3)/(2*c*d*e**4))/(a*e**2 - c*d**2
)**4 + (11*a**2*e**4 - 7*a*c*d**2*e**2 + 2*c**2*d**4 + 6*c**2*d**2*e**2*x**2 + x
*(15*a*c*d*e**3 - 3*c**2*d**3*e))/(6*a**6*e**9 - 18*a**5*c*d**2*e**7 + 18*a**4*c
**2*d**4*e**5 - 6*a**3*c**3*d**6*e**3 + x**3*(6*a**3*c**3*d**3*e**6 - 18*a**2*c*
*4*d**5*e**4 + 18*a*c**5*d**7*e**2 - 6*c**6*d**9) + x**2*(18*a**4*c**2*d**2*e**7
- 54*a**3*c**3*d**4*e**5 + 54*a**2*c**4*d**6*e**3 - 18*a*c**5*d**8*e) + x*(18*a
**5*c*d*e**8 - 54*a**4*c**2*d**3*e**6 + 54*a**3*c**3*d**5*e**4 - 18*a**2*c**4*d*
*7*e**2))

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GIAC/XCAS [A]  time = 0.238654, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="giac")

[Out]

Done