3.1894 \(\int \frac{d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=226 \[ -\frac{6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac{3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac{c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}-\frac{10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac{10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}-\frac{4 c d e^3}{(d+e x) \left (c d^2-a e^2\right )^5}-\frac{e^3}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4} \]

[Out]

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

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Rubi [A]  time = 0.506449, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac{3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac{c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}-\frac{10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac{10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}-\frac{4 c d e^3}{(d+e x) \left (c d^2-a e^2\right )^5}-\frac{e^3}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.371945, size = 206, normalized size = 0.91 \[ \frac{-60 c^2 d^2 e^3 \log (a e+c d x)+\frac{36 c^2 d^2 e^2 \left (a e^2-c d^2\right )}{a e+c d x}+\frac{9 c^2 d^2 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac{2 c^2 d^2 \left (a e^2-c d^2\right )^3}{(a e+c d x)^3}+\frac{24 c d e^3 \left (a e^2-c d^2\right )}{d+e x}-\frac{3 e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}+60 c^2 d^2 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 221, normalized size = 1. \[ -{\frac{{e}^{3}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{e}^{3}{c}^{2}{d}^{2}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6}}}+4\,{\frac{d{e}^{3}c}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( ex+d \right ) }}+{\frac{{c}^{2}{d}^{2}}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) ^{3}}}-10\,{\frac{{e}^{3}{c}^{2}{d}^{2}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6}}}+6\,{\frac{{c}^{2}{d}^{2}{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( cdx+ae \right ) }}+{\frac{3\,e{c}^{2}{d}^{2}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.781835, size = 1291, normalized size = 5.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.271382, size = 1677, normalized size = 7.42 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.2972, size = 1363, normalized size = 6.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*c**2*d**2*e**3*log(x + (-10*a**7*c**2*d**2*e**17/(a*e**2 - c*d**2)**6 + 70*a*
*6*c**3*d**4*e**15/(a*e**2 - c*d**2)**6 - 210*a**5*c**4*d**6*e**13/(a*e**2 - c*d
**2)**6 + 350*a**4*c**5*d**8*e**11/(a*e**2 - c*d**2)**6 - 350*a**3*c**6*d**10*e*
*9/(a*e**2 - c*d**2)**6 + 210*a**2*c**7*d**12*e**7/(a*e**2 - c*d**2)**6 - 70*a*c
**8*d**14*e**5/(a*e**2 - c*d**2)**6 + 10*a*c**2*d**2*e**5 + 10*c**9*d**16*e**3/(
a*e**2 - c*d**2)**6 + 10*c**3*d**4*e**3)/(20*c**3*d**3*e**4))/(a*e**2 - c*d**2)*
*6 - 10*c**2*d**2*e**3*log(x + (10*a**7*c**2*d**2*e**17/(a*e**2 - c*d**2)**6 - 7
0*a**6*c**3*d**4*e**15/(a*e**2 - c*d**2)**6 + 210*a**5*c**4*d**6*e**13/(a*e**2 -
c*d**2)**6 - 350*a**4*c**5*d**8*e**11/(a*e**2 - c*d**2)**6 + 350*a**3*c**6*d**1
0*e**9/(a*e**2 - c*d**2)**6 - 210*a**2*c**7*d**12*e**7/(a*e**2 - c*d**2)**6 + 70
*a*c**8*d**14*e**5/(a*e**2 - c*d**2)**6 + 10*a*c**2*d**2*e**5 - 10*c**9*d**16*e*
*3/(a*e**2 - c*d**2)**6 + 10*c**3*d**4*e**3)/(20*c**3*d**3*e**4))/(a*e**2 - c*d*
*2)**6 + (-3*a**4*e**8 + 27*a**3*c*d**2*e**6 + 47*a**2*c**2*d**4*e**4 - 13*a*c**
3*d**6*e**2 + 2*c**4*d**8 + 60*c**4*d**4*e**4*x**4 + x**3*(150*a*c**3*d**3*e**5
+ 90*c**4*d**5*e**3) + x**2*(110*a**2*c**2*d**2*e**6 + 230*a*c**3*d**4*e**4 + 20
*c**4*d**6*e**2) + x*(15*a**3*c*d*e**7 + 175*a**2*c**2*d**3*e**5 + 55*a*c**3*d**
5*e**3 - 5*c**4*d**7*e))/(6*a**8*d**2*e**13 - 30*a**7*c*d**4*e**11 + 60*a**6*c**
2*d**6*e**9 - 60*a**5*c**3*d**8*e**7 + 30*a**4*c**4*d**10*e**5 - 6*a**3*c**5*d**
12*e**3 + x**5*(6*a**5*c**3*d**3*e**12 - 30*a**4*c**4*d**5*e**10 + 60*a**3*c**5*
d**7*e**8 - 60*a**2*c**6*d**9*e**6 + 30*a*c**7*d**11*e**4 - 6*c**8*d**13*e**2) +
x**4*(18*a**6*c**2*d**2*e**13 - 78*a**5*c**3*d**4*e**11 + 120*a**4*c**4*d**6*e*
*9 - 60*a**3*c**5*d**8*e**7 - 30*a**2*c**6*d**10*e**5 + 42*a*c**7*d**12*e**3 - 1
2*c**8*d**14*e) + x**3*(18*a**7*c*d*e**14 - 54*a**6*c**2*d**3*e**12 + 6*a**5*c**
3*d**5*e**10 + 150*a**4*c**4*d**7*e**8 - 210*a**3*c**5*d**9*e**6 + 102*a**2*c**6
*d**11*e**4 - 6*a*c**7*d**13*e**2 - 6*c**8*d**15) + x**2*(6*a**8*e**15 + 6*a**7*
c*d**2*e**13 - 102*a**6*c**2*d**4*e**11 + 210*a**5*c**3*d**6*e**9 - 150*a**4*c**
4*d**8*e**7 - 6*a**3*c**5*d**10*e**5 + 54*a**2*c**6*d**12*e**3 - 18*a*c**7*d**14
*e) + x*(12*a**8*d*e**14 - 42*a**7*c*d**3*e**12 + 30*a**6*c**2*d**5*e**10 + 60*a
**5*c**3*d**7*e**8 - 120*a**4*c**4*d**9*e**6 + 78*a**3*c**5*d**11*e**4 - 18*a**2
*c**6*d**13*e**2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232494, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done