3.1862 \(\int \frac{1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.240033, 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 \[ \frac{c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}+\frac{c^2 d^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{c d}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{1}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 54.9795, size = 121, normalized size = 0.87 \[ - \frac{c^{3} d^{3} \log{\left (d + e x \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{c^{3} d^{3} \log{\left (a e + c d x \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{c^{2} d^{2}}{\left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{3}} + \frac{c d}{2 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{1}{3 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.127038, size = 135, normalized size = 0.97 \[ \frac{\left (c d^2-a e^2\right ) \left (2 a^2 e^4-a c d e^2 (7 d+3 e x)+c^2 d^2 \left (11 d^2+15 d e x+6 e^2 x^2\right )\right )+6 c^3 d^3 (d+e x)^3 \log (a e+c d x)-6 c^3 d^3 (d+e x)^3 \log (d+e x)}{6 (d+e x)^3 \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

((c*d^2 - a*e^2)*(2*a^2*e^4 - a*c*d*e^2*(7*d + 3*e*x) + c^2*d^2*(11*d^2 + 15*d*e
*x + 6*e^2*x^2)) + 6*c^3*d^3*(d + e*x)^3*Log[a*e + c*d*x] - 6*c^3*d^3*(d + e*x)^
3*Log[d + e*x])/(6*(c*d^2 - a*e^2)^4*(d + e*x)^3)

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Maple [A]  time = 0.016, size = 137, normalized size = 1. \[ -{\frac{1}{ \left ( 3\,a{e}^{2}-3\,c{d}^{2} \right ) \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) }}+{\frac{cd}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}}+{\frac{{c}^{3}{d}^{3}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

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

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Maxima [A]  time = 0.743767, size = 531, normalized size = 3.82 \[ \frac{c^{3} d^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac{c^{3} d^{3} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac{6 \, c^{2} d^{2} e^{2} x^{2} + 11 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + 3 \,{\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{6 \,{\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} +{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \,{\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \,{\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219827, size = 613, normalized size = 4.41 \[ \frac{11 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \,{\left (5 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (c d x + a e\right ) - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{6 \,{\left (c^{4} d^{11} - 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} - 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} +{\left (c^{4} d^{8} e^{3} - 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} - 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \,{\left (c^{4} d^{9} e^{2} - 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} - 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \,{\left (c^{4} d^{10} e - 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} - 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^3),x, algorithm="fricas")

[Out]

1/6*(11*c^3*d^6 - 18*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 2*a^3*e^6 + 6*(c^3*d^4*e^
2 - a*c^2*d^2*e^4)*x^2 + 3*(5*c^3*d^5*e - 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x + 6*(
c^3*d^3*e^3*x^3 + 3*c^3*d^4*e^2*x^2 + 3*c^3*d^5*e*x + c^3*d^6)*log(c*d*x + a*e)
- 6*(c^3*d^3*e^3*x^3 + 3*c^3*d^4*e^2*x^2 + 3*c^3*d^5*e*x + c^3*d^6)*log(e*x + d)
)/(c^4*d^11 - 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 - 4*a^3*c*d^5*e^6 + a^4*d^3*e^
8 + (c^4*d^8*e^3 - 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 - 4*a^3*c*d^2*e^9 + a^4*e
^11)*x^3 + 3*(c^4*d^9*e^2 - 4*a*c^3*d^7*e^4 + 6*a^2*c^2*d^5*e^6 - 4*a^3*c*d^3*e^
8 + a^4*d*e^10)*x^2 + 3*(c^4*d^10*e - 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^6*e^5 - 4*a^
3*c*d^4*e^7 + a^4*d^2*e^9)*x)

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Sympy [A]  time = 7.26805, size = 672, normalized size = 4.83 \[ - \frac{c^{3} d^{3} \log{\left (x + \frac{- \frac{a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} + \frac{c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{c^{3} d^{3} \log{\left (x + \frac{\frac{a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} - \frac{c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{2 a^{2} e^{4} - 7 a c d^{2} e^{2} + 11 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} + 15 c^{2} d^{3} e\right )}{6 a^{3} d^{3} e^{6} - 18 a^{2} c d^{5} e^{4} + 18 a c^{2} d^{7} e^{2} - 6 c^{3} d^{9} + x^{3} \left (6 a^{3} e^{9} - 18 a^{2} c d^{2} e^{7} + 18 a c^{2} d^{4} e^{5} - 6 c^{3} d^{6} e^{3}\right ) + x^{2} \left (18 a^{3} d e^{8} - 54 a^{2} c d^{3} e^{6} + 54 a c^{2} d^{5} e^{4} - 18 c^{3} d^{7} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{7} - 54 a^{2} c d^{4} e^{5} + 54 a c^{2} d^{6} e^{3} - 18 c^{3} d^{8} e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError