3.1884 \(\int \frac{1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=223 \[ \frac{4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac{c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac{10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac{10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6}+\frac{6 c^2 d^2 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac{3 c d e^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac{e^2}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3} \]

[Out]

-(c^3*d^3)/(2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2) + (4*c^3*d^3*e)/((c*d^2 - a*e^2
)^5*(a*e + c*d*x)) + e^2/(3*(c*d^2 - a*e^2)^3*(d + e*x)^3) + (3*c*d*e^2)/(2*(c*d
^2 - a*e^2)^4*(d + e*x)^2) + (6*c^2*d^2*e^2)/((c*d^2 - a*e^2)^5*(d + e*x)) + (10
*c^3*d^3*e^2*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^6 - (10*c^3*d^3*e^2*Log[d + e*x])
/(c*d^2 - a*e^2)^6

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Rubi [A]  time = 0.527031, antiderivative size = 223, 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 \[ \frac{4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac{c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac{10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac{10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6}+\frac{6 c^2 d^2 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac{3 c d e^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac{e^2}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-(c^3*d^3)/(2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2) + (4*c^3*d^3*e)/((c*d^2 - a*e^2
)^5*(a*e + c*d*x)) + e^2/(3*(c*d^2 - a*e^2)^3*(d + e*x)^3) + (3*c*d*e^2)/(2*(c*d
^2 - a*e^2)^4*(d + e*x)^2) + (6*c^2*d^2*e^2)/((c*d^2 - a*e^2)^5*(d + e*x)) + (10
*c^3*d^3*e^2*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^6 - (10*c^3*d^3*e^2*Log[d + e*x])
/(c*d^2 - a*e^2)^6

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.465351, size = 201, normalized size = 0.9 \[ \frac{60 c^3 d^3 e^2 \log (a e+c d x)+\frac{24 c^3 d^3 e \left (c d^2-a e^2\right )}{a e+c d x}-\frac{3 c^3 d^3 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac{36 c^2 d^2 e^2 \left (c d^2-a e^2\right )}{d+e x}+\frac{9 c d \left (c d^2 e-a e^3\right )^2}{(d+e x)^2}-\frac{2 e^2 \left (a e^2-c d^2\right )^3}{(d+e x)^3}-60 c^3 d^3 e^2 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6} \]

Antiderivative was successfully verified.

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

[Out]

((-3*c^3*d^3*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (24*c^3*d^3*e*(c*d^2 - a*e^2))
/(a*e + c*d*x) - (2*e^2*(-(c*d^2) + a*e^2)^3)/(d + e*x)^3 + (9*c*d*(c*d^2*e - a*
e^3)^2)/(d + e*x)^2 + (36*c^2*d^2*e^2*(c*d^2 - a*e^2))/(d + e*x) + 60*c^3*d^3*e^
2*Log[a*e + c*d*x] - 60*c^3*d^3*e^2*Log[d + e*x])/(6*(c*d^2 - a*e^2)^6)

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Maple [A]  time = 0.023, size = 218, normalized size = 1. \[ -{\frac{{e}^{2}}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) ^{3}}}-10\,{\frac{{c}^{3}{d}^{3}{e}^{2}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6}}}-6\,{\frac{{e}^{2}{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( ex+d \right ) }}+{\frac{3\,cd{e}^{2}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{3}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) ^{2}}}+10\,{\frac{{c}^{3}{d}^{3}{e}^{2}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6}}}-4\,{\frac{{c}^{3}{d}^{3}e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( cdx+ae \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*e^2/(a*e^2-c*d^2)^3/(e*x+d)^3-10*e^2/(a*e^2-c*d^2)^6*c^3*d^3*ln(e*x+d)-6*e^
2/(a*e^2-c*d^2)^5*c^2*d^2/(e*x+d)+3/2*e^2/(a*e^2-c*d^2)^4*c*d/(e*x+d)^2-1/2*c^3*
d^3/(a*e^2-c*d^2)^4/(c*d*x+a*e)^2+10*e^2/(a*e^2-c*d^2)^6*c^3*d^3*ln(c*d*x+a*e)-4
*c^3*d^3/(a*e^2-c*d^2)^5*e/(c*d*x+a*e)

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Maxima [A]  time = 0.767065, size = 1278, normalized size = 5.73 \[ \text{result too large to display} \]

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)^3*(e*x + d)),x, algorithm="maxima")

[Out]

10*c^3*d^3*e^2*log(c*d*x + a*e)/(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^
4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12) - 10*
c^3*d^3*e^2*log(e*x + d)/(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*
a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12) + 1/6*(60*c^
4*d^4*e^4*x^4 - 3*c^4*d^8 + 27*a*c^3*d^6*e^2 + 47*a^2*c^2*d^4*e^4 - 13*a^3*c*d^2
*e^6 + 2*a^4*e^8 + 30*(5*c^4*d^5*e^3 + 3*a*c^3*d^3*e^5)*x^3 + 10*(11*c^4*d^6*e^2
+ 23*a*c^3*d^4*e^4 + 2*a^2*c^2*d^2*e^6)*x^2 + 5*(3*c^4*d^7*e + 35*a*c^3*d^5*e^3
+ 11*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)/(a^2*c^5*d^13*e^2 - 5*a^3*c^4*d^11*e^4 +
10*a^4*c^3*d^9*e^6 - 10*a^5*c^2*d^7*e^8 + 5*a^6*c*d^5*e^10 - a^7*d^3*e^12 + (c^
7*d^12*e^3 - 5*a*c^6*d^10*e^5 + 10*a^2*c^5*d^8*e^7 - 10*a^3*c^4*d^6*e^9 + 5*a^4*
c^3*d^4*e^11 - a^5*c^2*d^2*e^13)*x^5 + (3*c^7*d^13*e^2 - 13*a*c^6*d^11*e^4 + 20*
a^2*c^5*d^9*e^6 - 10*a^3*c^4*d^7*e^8 - 5*a^4*c^3*d^5*e^10 + 7*a^5*c^2*d^3*e^12 -
2*a^6*c*d*e^14)*x^4 + (3*c^7*d^14*e - 9*a*c^6*d^12*e^3 + a^2*c^5*d^10*e^5 + 25*
a^3*c^4*d^8*e^7 - 35*a^4*c^3*d^6*e^9 + 17*a^5*c^2*d^4*e^11 - a^6*c*d^2*e^13 - a^
7*e^15)*x^3 + (c^7*d^15 + a*c^6*d^13*e^2 - 17*a^2*c^5*d^11*e^4 + 35*a^3*c^4*d^9*
e^6 - 25*a^4*c^3*d^7*e^8 - a^5*c^2*d^5*e^10 + 9*a^6*c*d^3*e^12 - 3*a^7*d*e^14)*x
^2 + (2*a*c^6*d^14*e - 7*a^2*c^5*d^12*e^3 + 5*a^3*c^4*d^10*e^5 + 10*a^4*c^3*d^8*
e^7 - 20*a^5*c^2*d^6*e^9 + 13*a^6*c*d^4*e^11 - 3*a^7*d^2*e^13)*x)

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Fricas [A]  time = 0.250336, size = 1650, normalized size = 7.4 \[ \text{result too large to display} \]

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)^3*(e*x + d)),x, algorithm="fricas")

[Out]

-1/6*(3*c^5*d^10 - 30*a*c^4*d^8*e^2 - 20*a^2*c^3*d^6*e^4 + 60*a^3*c^2*d^4*e^6 -
15*a^4*c*d^2*e^8 + 2*a^5*e^10 - 60*(c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 - 30*(5*c^5
*d^7*e^3 - 2*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*x^3 - 10*(11*c^5*d^8*e^2 + 12*a*
c^4*d^6*e^4 - 21*a^2*c^3*d^4*e^6 - 2*a^3*c^2*d^2*e^8)*x^2 - 5*(3*c^5*d^9*e + 32*
a*c^4*d^7*e^3 - 24*a^2*c^3*d^5*e^5 - 12*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)*x - 60*(c
^5*d^5*e^5*x^5 + a^2*c^3*d^6*e^4 + (3*c^5*d^6*e^4 + 2*a*c^4*d^4*e^6)*x^4 + (3*c^
5*d^7*e^3 + 6*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^3 + (c^5*d^8*e^2 + 6*a*c^4*d^6*
e^4 + 3*a^2*c^3*d^4*e^6)*x^2 + (2*a*c^4*d^7*e^3 + 3*a^2*c^3*d^5*e^5)*x)*log(c*d*
x + a*e) + 60*(c^5*d^5*e^5*x^5 + a^2*c^3*d^6*e^4 + (3*c^5*d^6*e^4 + 2*a*c^4*d^4*
e^6)*x^4 + (3*c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^3 + (c^5*d^8*e^
2 + 6*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6)*x^2 + (2*a*c^4*d^7*e^3 + 3*a^2*c^3*d^5*
e^5)*x)*log(e*x + d))/(a^2*c^6*d^15*e^2 - 6*a^3*c^5*d^13*e^4 + 15*a^4*c^4*d^11*e
^6 - 20*a^5*c^3*d^9*e^8 + 15*a^6*c^2*d^7*e^10 - 6*a^7*c*d^5*e^12 + a^8*d^3*e^14
+ (c^8*d^14*e^3 - 6*a*c^7*d^12*e^5 + 15*a^2*c^6*d^10*e^7 - 20*a^3*c^5*d^8*e^9 +
15*a^4*c^4*d^6*e^11 - 6*a^5*c^3*d^4*e^13 + a^6*c^2*d^2*e^15)*x^5 + (3*c^8*d^15*e
^2 - 16*a*c^7*d^13*e^4 + 33*a^2*c^6*d^11*e^6 - 30*a^3*c^5*d^9*e^8 + 5*a^4*c^4*d^
7*e^10 + 12*a^5*c^3*d^5*e^12 - 9*a^6*c^2*d^3*e^14 + 2*a^7*c*d*e^16)*x^4 + (3*c^8
*d^16*e - 12*a*c^7*d^14*e^3 + 10*a^2*c^6*d^12*e^5 + 24*a^3*c^5*d^10*e^7 - 60*a^4
*c^4*d^8*e^9 + 52*a^5*c^3*d^6*e^11 - 18*a^6*c^2*d^4*e^13 + a^8*e^17)*x^3 + (c^8*
d^17 - 18*a^2*c^6*d^13*e^4 + 52*a^3*c^5*d^11*e^6 - 60*a^4*c^4*d^9*e^8 + 24*a^5*c
^3*d^7*e^10 + 10*a^6*c^2*d^5*e^12 - 12*a^7*c*d^3*e^14 + 3*a^8*d*e^16)*x^2 + (2*a
*c^7*d^16*e - 9*a^2*c^6*d^14*e^3 + 12*a^3*c^5*d^12*e^5 + 5*a^4*c^4*d^10*e^7 - 30
*a^5*c^3*d^8*e^9 + 33*a^6*c^2*d^6*e^11 - 16*a^7*c*d^4*e^13 + 3*a^8*d^2*e^15)*x)

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Sympy [A]  time = 20.9409, size = 1353, normalized size = 6.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*c**3*d**3*e**2*log(x + (-10*a**7*c**3*d**3*e**16/(a*e**2 - c*d**2)**6 + 70*a
**6*c**4*d**5*e**14/(a*e**2 - c*d**2)**6 - 210*a**5*c**5*d**7*e**12/(a*e**2 - c*
d**2)**6 + 350*a**4*c**6*d**9*e**10/(a*e**2 - c*d**2)**6 - 350*a**3*c**7*d**11*e
**8/(a*e**2 - c*d**2)**6 + 210*a**2*c**8*d**13*e**6/(a*e**2 - c*d**2)**6 - 70*a*
c**9*d**15*e**4/(a*e**2 - c*d**2)**6 + 10*a*c**3*d**3*e**4 + 10*c**10*d**17*e**2
/(a*e**2 - c*d**2)**6 + 10*c**4*d**5*e**2)/(20*c**4*d**4*e**3))/(a*e**2 - c*d**2
)**6 + 10*c**3*d**3*e**2*log(x + (10*a**7*c**3*d**3*e**16/(a*e**2 - c*d**2)**6 -
70*a**6*c**4*d**5*e**14/(a*e**2 - c*d**2)**6 + 210*a**5*c**5*d**7*e**12/(a*e**2
- c*d**2)**6 - 350*a**4*c**6*d**9*e**10/(a*e**2 - c*d**2)**6 + 350*a**3*c**7*d*
*11*e**8/(a*e**2 - c*d**2)**6 - 210*a**2*c**8*d**13*e**6/(a*e**2 - c*d**2)**6 +
70*a*c**9*d**15*e**4/(a*e**2 - c*d**2)**6 + 10*a*c**3*d**3*e**4 - 10*c**10*d**17
*e**2/(a*e**2 - c*d**2)**6 + 10*c**4*d**5*e**2)/(20*c**4*d**4*e**3))/(a*e**2 - c
*d**2)**6 - (2*a**4*e**8 - 13*a**3*c*d**2*e**6 + 47*a**2*c**2*d**4*e**4 + 27*a*c
**3*d**6*e**2 - 3*c**4*d**8 + 60*c**4*d**4*e**4*x**4 + x**3*(90*a*c**3*d**3*e**5
+ 150*c**4*d**5*e**3) + x**2*(20*a**2*c**2*d**2*e**6 + 230*a*c**3*d**4*e**4 + 1
10*c**4*d**6*e**2) + x*(-5*a**3*c*d*e**7 + 55*a**2*c**2*d**3*e**5 + 175*a*c**3*d
**5*e**3 + 15*c**4*d**7*e))/(6*a**7*d**3*e**12 - 30*a**6*c*d**5*e**10 + 60*a**5*
c**2*d**7*e**8 - 60*a**4*c**3*d**9*e**6 + 30*a**3*c**4*d**11*e**4 - 6*a**2*c**5*
d**13*e**2 + x**5*(6*a**5*c**2*d**2*e**13 - 30*a**4*c**3*d**4*e**11 + 60*a**3*c*
*4*d**6*e**9 - 60*a**2*c**5*d**8*e**7 + 30*a*c**6*d**10*e**5 - 6*c**7*d**12*e**3
) + x**4*(12*a**6*c*d*e**14 - 42*a**5*c**2*d**3*e**12 + 30*a**4*c**3*d**5*e**10
+ 60*a**3*c**4*d**7*e**8 - 120*a**2*c**5*d**9*e**6 + 78*a*c**6*d**11*e**4 - 18*c
**7*d**13*e**2) + x**3*(6*a**7*e**15 + 6*a**6*c*d**2*e**13 - 102*a**5*c**2*d**4*
e**11 + 210*a**4*c**3*d**6*e**9 - 150*a**3*c**4*d**8*e**7 - 6*a**2*c**5*d**10*e*
*5 + 54*a*c**6*d**12*e**3 - 18*c**7*d**14*e) + x**2*(18*a**7*d*e**14 - 54*a**6*c
*d**3*e**12 + 6*a**5*c**2*d**5*e**10 + 150*a**4*c**3*d**7*e**8 - 210*a**3*c**4*d
**9*e**6 + 102*a**2*c**5*d**11*e**4 - 6*a*c**6*d**13*e**2 - 6*c**7*d**15) + x*(1
8*a**7*d**2*e**13 - 78*a**6*c*d**4*e**11 + 120*a**5*c**2*d**6*e**9 - 60*a**4*c**
3*d**8*e**7 - 30*a**3*c**4*d**10*e**5 + 42*a**2*c**5*d**12*e**3 - 12*a*c**6*d**1
4*e))

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GIAC/XCAS [A]  time = 8.17915, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^3*(e*x + d)),x, algorithm="giac")

[Out]

Done