3.1837 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.122875, 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 \[ \frac{2 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^3}-\frac{\left (c d^2-a e^2\right )^2}{4 e^3 (d+e x)^4}-\frac{c^2 d^2}{2 e^3 (d+e x)^2} \]

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)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 30.5278, size = 70, normalized size = 0.91 \[ - \frac{c^{2} d^{2}}{2 e^{3} \left (d + e x\right )^{2}} - \frac{2 c d \left (a e^{2} - c d^{2}\right )}{3 e^{3} \left (d + e x\right )^{3}} - \frac{\left (a e^{2} - c d^{2}\right )^{2}}{4 e^{3} \left (d + e x\right )^{4}} \]

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)**7,x)

[Out]

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

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Mathematica [A]  time = 0.0439362, size = 61, normalized size = 0.79 \[ -\frac{3 a^2 e^4+2 a c d e^2 (d+4 e x)+c^2 d^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]

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)^7,x]

[Out]

-(3*a^2*e^4 + 2*a*c*d*e^2*(d + 4*e*x) + c^2*d^2*(d^2 + 4*d*e*x + 6*e^2*x^2))/(12
*e^3*(d + e*x)^4)

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

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)^7,x)

[Out]

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

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Maxima [A]  time = 0.729302, size = 146, normalized size = 1.9 \[ -\frac{6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

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)^7,x, algorithm="maxima")

[Out]

-1/12*(6*c^2*d^2*e^2*x^2 + c^2*d^4 + 2*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(c^2*d^3*e +
2*a*c*d*e^3)*x)/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Fricas [A]  time = 0.207927, size = 146, normalized size = 1.9 \[ -\frac{6 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]

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)^7,x, algorithm="fricas")

[Out]

-1/12*(6*c^2*d^2*e^2*x^2 + c^2*d^4 + 2*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(c^2*d^3*e +
2*a*c*d*e^3)*x)/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4*d^3*e^4*x + d^4*e^3)

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Sympy [A]  time = 4.29022, size = 114, normalized size = 1.48 \[ - \frac{3 a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (8 a c d e^{3} + 4 c^{2} d^{3} e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]

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)**7,x)

[Out]

-(3*a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4 + 6*c**2*d**2*e**2*x**2 + x*(8*a*c*d
*e**3 + 4*c**2*d**3*e))/(12*d**4*e**3 + 48*d**3*e**4*x + 72*d**2*e**5*x**2 + 48*
d*e**6*x**3 + 12*e**7*x**4)

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GIAC/XCAS [A]  time = 0.210078, size = 189, normalized size = 2.45 \[ -\frac{{\left (6 \, c^{2} d^{2} x^{4} e^{4} + 16 \, c^{2} d^{3} x^{3} e^{3} + 15 \, c^{2} d^{4} x^{2} e^{2} + 6 \, c^{2} d^{5} x e + c^{2} d^{6} + 8 \, a c d x^{3} e^{5} + 18 \, a c d^{2} x^{2} e^{4} + 12 \, a c d^{3} x e^{3} + 2 \, a c d^{4} e^{2} + 3 \, a^{2} x^{2} e^{6} + 6 \, a^{2} d x e^{5} + 3 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{12 \,{\left (x e + d\right )}^{6}} \]

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)^7,x, algorithm="giac")

[Out]

-1/12*(6*c^2*d^2*x^4*e^4 + 16*c^2*d^3*x^3*e^3 + 15*c^2*d^4*x^2*e^2 + 6*c^2*d^5*x
*e + c^2*d^6 + 8*a*c*d*x^3*e^5 + 18*a*c*d^2*x^2*e^4 + 12*a*c*d^3*x*e^3 + 2*a*c*d
^4*e^2 + 3*a^2*x^2*e^6 + 6*a^2*d*x*e^5 + 3*a^2*d^2*e^4)*e^(-3)/(x*e + d)^6