3.1893 \(\int \frac{(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=173 \[ -\frac{3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac{c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac{c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac{e^3}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac{4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac{4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]

[Out]

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

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Rubi [A]  time = 0.351951, antiderivative size = 173, 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 \[ -\frac{3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac{c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac{c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac{e^3}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac{4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac{4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 85.92, size = 160, normalized size = 0.92 \[ - \frac{4 c d e^{3} \log{\left (d + e x \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} + \frac{4 c d e^{3} \log{\left (a e + c d x \right )}}{\left (a e^{2} - c d^{2}\right )^{5}} - \frac{3 c d e^{2}}{\left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{4}} - \frac{c d e}{\left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )^{3}} - \frac{c d}{3 \left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{e^{3}}{\left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.274297, size = 157, normalized size = 0.91 \[ \frac{\frac{9 c d e^2 \left (c d^2-a e^2\right )}{a e+c d x}-\frac{3 c d e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac{c d \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}+\frac{3 c d^2 e^3-3 a e^5}{d+e x}+12 c d e^3 \log (a e+c d x)-12 c d e^3 \log (d+e x)}{3 \left (a e^2-c d^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

((c*d*(c*d^2 - a*e^2)^3)/(a*e + c*d*x)^3 - (3*c*d*e*(c*d^2 - a*e^2)^2)/(a*e + c*
d*x)^2 + (9*c*d*e^2*(c*d^2 - a*e^2))/(a*e + c*d*x) + (3*c*d^2*e^3 - 3*a*e^5)/(d
+ e*x) + 12*c*d*e^3*Log[a*e + c*d*x] - 12*c*d*e^3*Log[d + e*x])/(3*(-(c*d^2) + a
*e^2)^5)

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Maple [A]  time = 0.021, size = 173, normalized size = 1. \[ -{\frac{{e}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) }}-4\,{\frac{d{e}^{3}c\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}}-{\frac{cd}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdx+ae \right ) ^{3}}}+4\,{\frac{d{e}^{3}c\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}}-3\,{\frac{{e}^{2}cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) }}-{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(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/(e*x+d)-4*e^3/(a*e^2-c*d^2)^5*c*d*ln(e*x+d)-1/3*c*d/(a*e^2-
c*d^2)^2/(c*d*x+a*e)^3+4*e^3/(a*e^2-c*d^2)^5*c*d*ln(c*d*x+a*e)-3*c*d/(a*e^2-c*d^
2)^4*e^2/(c*d*x+a*e)-c*d/(a*e^2-c*d^2)^3*e/(c*d*x+a*e)^2

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Maxima [A]  time = 0.763985, size = 886, normalized size = 5.12 \[ -\frac{4 \, c d e^{3} \log \left (c d x + a e\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} + \frac{4 \, c d e^{3} \log \left (e x + d\right )}{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}} - \frac{12 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} - 5 \, a c^{2} d^{4} e^{2} + 13 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} + 5 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} - 11 \, a^{2} c d e^{5}\right )} x}{3 \,{\left (a^{3} c^{4} d^{9} e^{3} - 4 \, a^{4} c^{3} d^{7} e^{5} + 6 \, a^{5} c^{2} d^{5} e^{7} - 4 \, a^{6} c d^{3} e^{9} + a^{7} d e^{11} +{\left (c^{7} d^{11} e - 4 \, a c^{6} d^{9} e^{3} + 6 \, a^{2} c^{5} d^{7} e^{5} - 4 \, a^{3} c^{4} d^{5} e^{7} + a^{4} c^{3} d^{3} e^{9}\right )} x^{4} +{\left (c^{7} d^{12} - a c^{6} d^{10} e^{2} - 6 \, a^{2} c^{5} d^{8} e^{4} + 14 \, a^{3} c^{4} d^{6} e^{6} - 11 \, a^{4} c^{3} d^{4} e^{8} + 3 \, a^{5} c^{2} d^{2} e^{10}\right )} x^{3} + 3 \,{\left (a c^{6} d^{11} e - 3 \, a^{2} c^{5} d^{9} e^{3} + 2 \, a^{3} c^{4} d^{7} e^{5} + 2 \, a^{4} c^{3} d^{5} e^{7} - 3 \, a^{5} c^{2} d^{3} e^{9} + a^{6} c d e^{11}\right )} x^{2} +{\left (3 \, a^{2} c^{5} d^{10} e^{2} - 11 \, a^{3} c^{4} d^{8} e^{4} + 14 \, a^{4} c^{3} d^{6} e^{6} - 6 \, a^{5} c^{2} d^{4} e^{8} - a^{6} c d^{2} e^{10} + a^{7} e^{12}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*c*d*e^3*log(c*d*x + a*e)/(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 1
0*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10) + 4*c*d*e^3*log(e*x + d)/(c^5*d^
10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8
- a^5*e^10) - 1/3*(12*c^3*d^3*e^3*x^3 + c^3*d^6 - 5*a*c^2*d^4*e^2 + 13*a^2*c*d^
2*e^4 + 3*a^3*e^6 + 6*(c^3*d^4*e^2 + 5*a*c^2*d^2*e^4)*x^2 - 2*(c^3*d^5*e - 8*a*c
^2*d^3*e^3 - 11*a^2*c*d*e^5)*x)/(a^3*c^4*d^9*e^3 - 4*a^4*c^3*d^7*e^5 + 6*a^5*c^2
*d^5*e^7 - 4*a^6*c*d^3*e^9 + a^7*d*e^11 + (c^7*d^11*e - 4*a*c^6*d^9*e^3 + 6*a^2*
c^5*d^7*e^5 - 4*a^3*c^4*d^5*e^7 + a^4*c^3*d^3*e^9)*x^4 + (c^7*d^12 - a*c^6*d^10*
e^2 - 6*a^2*c^5*d^8*e^4 + 14*a^3*c^4*d^6*e^6 - 11*a^4*c^3*d^4*e^8 + 3*a^5*c^2*d^
2*e^10)*x^3 + 3*(a*c^6*d^11*e - 3*a^2*c^5*d^9*e^3 + 2*a^3*c^4*d^7*e^5 + 2*a^4*c^
3*d^5*e^7 - 3*a^5*c^2*d^3*e^9 + a^6*c*d*e^11)*x^2 + (3*a^2*c^5*d^10*e^2 - 11*a^3
*c^4*d^8*e^4 + 14*a^4*c^3*d^6*e^6 - 6*a^5*c^2*d^4*e^8 - a^6*c*d^2*e^10 + a^7*e^1
2)*x)

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Fricas [A]  time = 0.245457, size = 1130, normalized size = 6.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(c^4*d^8 - 6*a*c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 10*a^3*c*d^2*e^6 - 3*a^4*
e^8 + 12*(c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + 6*(c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 -
5*a^2*c^2*d^2*e^6)*x^2 - 2*(c^4*d^7*e - 9*a*c^3*d^5*e^3 - 3*a^2*c^2*d^3*e^5 + 11
*a^3*c*d*e^7)*x + 12*(c^4*d^4*e^4*x^4 + a^3*c*d^2*e^6 + (c^4*d^5*e^3 + 3*a*c^3*d
^3*e^5)*x^3 + 3*(a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + (3*a^2*c^2*d^3*e^5 + a^3
*c*d*e^7)*x)*log(c*d*x + a*e) - 12*(c^4*d^4*e^4*x^4 + a^3*c*d^2*e^6 + (c^4*d^5*e
^3 + 3*a*c^3*d^3*e^5)*x^3 + 3*(a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + (3*a^2*c^2
*d^3*e^5 + a^3*c*d*e^7)*x)*log(e*x + d))/(a^3*c^5*d^11*e^3 - 5*a^4*c^4*d^9*e^5 +
10*a^5*c^3*d^7*e^7 - 10*a^6*c^2*d^5*e^9 + 5*a^7*c*d^3*e^11 - a^8*d*e^13 + (c^8*
d^13*e - 5*a*c^7*d^11*e^3 + 10*a^2*c^6*d^9*e^5 - 10*a^3*c^5*d^7*e^7 + 5*a^4*c^4*
d^5*e^9 - a^5*c^3*d^3*e^11)*x^4 + (c^8*d^14 - 2*a*c^7*d^12*e^2 - 5*a^2*c^6*d^10*
e^4 + 20*a^3*c^5*d^8*e^6 - 25*a^4*c^4*d^6*e^8 + 14*a^5*c^3*d^4*e^10 - 3*a^6*c^2*
d^2*e^12)*x^3 + 3*(a*c^7*d^13*e - 4*a^2*c^6*d^11*e^3 + 5*a^3*c^5*d^9*e^5 - 5*a^5
*c^3*d^5*e^9 + 4*a^6*c^2*d^3*e^11 - a^7*c*d*e^13)*x^2 + (3*a^2*c^6*d^12*e^2 - 14
*a^3*c^5*d^10*e^4 + 25*a^4*c^4*d^8*e^6 - 20*a^5*c^3*d^6*e^8 + 5*a^6*c^2*d^4*e^10
+ 2*a^7*c*d^2*e^12 - a^8*e^14)*x)

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Sympy [A]  time = 13.59, size = 1005, normalized size = 5.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*c*d*e**3*log(x + (-4*a**6*c*d*e**15/(a*e**2 - c*d**2)**5 + 24*a**5*c**2*d**3*
e**13/(a*e**2 - c*d**2)**5 - 60*a**4*c**3*d**5*e**11/(a*e**2 - c*d**2)**5 + 80*a
**3*c**4*d**7*e**9/(a*e**2 - c*d**2)**5 - 60*a**2*c**5*d**9*e**7/(a*e**2 - c*d**
2)**5 + 24*a*c**6*d**11*e**5/(a*e**2 - c*d**2)**5 + 4*a*c*d*e**5 - 4*c**7*d**13*
e**3/(a*e**2 - c*d**2)**5 + 4*c**2*d**3*e**3)/(8*c**2*d**2*e**4))/(a*e**2 - c*d*
*2)**5 + 4*c*d*e**3*log(x + (4*a**6*c*d*e**15/(a*e**2 - c*d**2)**5 - 24*a**5*c**
2*d**3*e**13/(a*e**2 - c*d**2)**5 + 60*a**4*c**3*d**5*e**11/(a*e**2 - c*d**2)**5
- 80*a**3*c**4*d**7*e**9/(a*e**2 - c*d**2)**5 + 60*a**2*c**5*d**9*e**7/(a*e**2
- c*d**2)**5 - 24*a*c**6*d**11*e**5/(a*e**2 - c*d**2)**5 + 4*a*c*d*e**5 + 4*c**7
*d**13*e**3/(a*e**2 - c*d**2)**5 + 4*c**2*d**3*e**3)/(8*c**2*d**2*e**4))/(a*e**2
- c*d**2)**5 - (3*a**3*e**6 + 13*a**2*c*d**2*e**4 - 5*a*c**2*d**4*e**2 + c**3*d
**6 + 12*c**3*d**3*e**3*x**3 + x**2*(30*a*c**2*d**2*e**4 + 6*c**3*d**4*e**2) + x
*(22*a**2*c*d*e**5 + 16*a*c**2*d**3*e**3 - 2*c**3*d**5*e))/(3*a**7*d*e**11 - 12*
a**6*c*d**3*e**9 + 18*a**5*c**2*d**5*e**7 - 12*a**4*c**3*d**7*e**5 + 3*a**3*c**4
*d**9*e**3 + x**4*(3*a**4*c**3*d**3*e**9 - 12*a**3*c**4*d**5*e**7 + 18*a**2*c**5
*d**7*e**5 - 12*a*c**6*d**9*e**3 + 3*c**7*d**11*e) + x**3*(9*a**5*c**2*d**2*e**1
0 - 33*a**4*c**3*d**4*e**8 + 42*a**3*c**4*d**6*e**6 - 18*a**2*c**5*d**8*e**4 - 3
*a*c**6*d**10*e**2 + 3*c**7*d**12) + x**2*(9*a**6*c*d*e**11 - 27*a**5*c**2*d**3*
e**9 + 18*a**4*c**3*d**5*e**7 + 18*a**3*c**4*d**7*e**5 - 27*a**2*c**5*d**9*e**3
+ 9*a*c**6*d**11*e) + x*(3*a**7*e**12 - 3*a**6*c*d**2*e**10 - 18*a**5*c**2*d**4*
e**8 + 42*a**4*c**3*d**6*e**6 - 33*a**3*c**4*d**8*e**4 + 9*a**2*c**5*d**10*e**2)
)

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GIAC/XCAS [A]  time = 0.263636, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done