3.1839 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx\)

Optimal. Leaf size=77 \[ \frac{2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac{\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac{c^2 d^2}{4 e^3 (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.12187, antiderivative size = 77, 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 \[ \frac{2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac{\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac{c^2 d^2}{4 e^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 31.0433, size = 70, normalized size = 0.91 \[ - \frac{c^{2} d^{2}}{4 e^{3} \left (d + e x\right )^{4}} - \frac{2 c d \left (a e^{2} - c d^{2}\right )}{5 e^{3} \left (d + e x\right )^{5}} - \frac{\left (a e^{2} - c d^{2}\right )^{2}}{6 e^{3} \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**9,x)

[Out]

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

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Mathematica [A]  time = 0.0443151, size = 61, normalized size = 0.79 \[ -\frac{10 a^2 e^4+4 a c d e^2 (d+6 e x)+c^2 d^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^9,x]

[Out]

-(10*a^2*e^4 + 4*a*c*d*e^2*(d + 6*e*x) + c^2*d^2*(d^2 + 6*d*e*x + 15*e^2*x^2))/(
60*e^3*(d + e*x)^6)

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Maple [A]  time = 0.008, size = 83, normalized size = 1.1 \[ -{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{6\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{2\,cd \left ( a{e}^{2}-c{d}^{2} \right ) }{5\,{e}^{3} \left ( ex+d \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^9,x)

[Out]

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

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Maxima [A]  time = 0.734167, size = 176, normalized size = 2.29 \[ -\frac{15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \,{\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d)^9,x, algorithm="maxima")

[Out]

-1/60*(15*c^2*d^2*e^2*x^2 + c^2*d^4 + 4*a*c*d^2*e^2 + 10*a^2*e^4 + 6*(c^2*d^3*e
+ 4*a*c*d*e^3)*x)/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^7*x^4 + 20*d^3*e^6*x^3 + 15*
d^4*e^5*x^2 + 6*d^5*e^4*x + d^6*e^3)

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Fricas [A]  time = 0.205128, size = 176, normalized size = 2.29 \[ -\frac{15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \,{\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d)^9,x, algorithm="fricas")

[Out]

-1/60*(15*c^2*d^2*e^2*x^2 + c^2*d^4 + 4*a*c*d^2*e^2 + 10*a^2*e^4 + 6*(c^2*d^3*e
+ 4*a*c*d*e^3)*x)/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^7*x^4 + 20*d^3*e^6*x^3 + 15*
d^4*e^5*x^2 + 6*d^5*e^4*x + d^6*e^3)

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Sympy [A]  time = 8.20987, size = 138, normalized size = 1.79 \[ - \frac{10 a^{2} e^{4} + 4 a c d^{2} e^{2} + c^{2} d^{4} + 15 c^{2} d^{2} e^{2} x^{2} + x \left (24 a c d e^{3} + 6 c^{2} d^{3} e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**9,x)

[Out]

-(10*a**2*e**4 + 4*a*c*d**2*e**2 + c**2*d**4 + 15*c**2*d**2*e**2*x**2 + x*(24*a*
c*d*e**3 + 6*c**2*d**3*e))/(60*d**6*e**3 + 360*d**5*e**4*x + 900*d**4*e**5*x**2
+ 1200*d**3*e**6*x**3 + 900*d**2*e**7*x**4 + 360*d*e**8*x**5 + 60*e**9*x**6)

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GIAC/XCAS [A]  time = 0.209669, size = 189, normalized size = 2.45 \[ -\frac{{\left (15 \, c^{2} d^{2} x^{4} e^{4} + 36 \, c^{2} d^{3} x^{3} e^{3} + 28 \, c^{2} d^{4} x^{2} e^{2} + 8 \, c^{2} d^{5} x e + c^{2} d^{6} + 24 \, a c d x^{3} e^{5} + 52 \, a c d^{2} x^{2} e^{4} + 32 \, a c d^{3} x e^{3} + 4 \, a c d^{4} e^{2} + 10 \, a^{2} x^{2} e^{6} + 20 \, a^{2} d x e^{5} + 10 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d)^9,x, algorithm="giac")

[Out]

-1/60*(15*c^2*d^2*x^4*e^4 + 36*c^2*d^3*x^3*e^3 + 28*c^2*d^4*x^2*e^2 + 8*c^2*d^5*
x*e + c^2*d^6 + 24*a*c*d*x^3*e^5 + 52*a*c*d^2*x^2*e^4 + 32*a*c*d^3*x*e^3 + 4*a*c
*d^4*e^2 + 10*a^2*x^2*e^6 + 20*a^2*d*x*e^5 + 10*a^2*d^2*e^4)*e^(-3)/(x*e + d)^8