3.1878 \(\int \frac{(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=85 \[ \frac{e^2 \log (a e+c d x)}{c^3 d^3}-\frac{2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.157638, antiderivative size = 85, 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 \[ \frac{e^2 \log (a e+c d x)}{c^3 d^3}-\frac{2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 37.4435, size = 78, normalized size = 0.92 \[ \frac{e^{2} \log{\left (a e + c d x \right )}}{c^{3} d^{3}} + \frac{2 e \left (a e^{2} - c d^{2}\right )}{c^{3} d^{3} \left (a e + c d x\right )} - \frac{\left (a e^{2} - c d^{2}\right )^{2}}{2 c^{3} d^{3} \left (a e + c d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0638833, size = 65, normalized size = 0.76 \[ \frac{2 e^2 \log (a e+c d x)-\frac{\left (c d^2-a e^2\right ) \left (3 a e^2+c d (d+4 e x)\right )}{(a e+c d x)^2}}{2 c^3 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.01, size = 123, normalized size = 1.5 \[ 2\,{\frac{a{e}^{3}}{{c}^{3}{d}^{3} \left ( cdx+ae \right ) }}-2\,{\frac{e}{{c}^{2}d \left ( cdx+ae \right ) }}-{\frac{{a}^{2}{e}^{4}}{2\,{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{2}}}+{\frac{a{e}^{2}}{{c}^{2}d \left ( cdx+ae \right ) ^{2}}}-{\frac{d}{2\,c \left ( cdx+ae \right ) ^{2}}}+{\frac{{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.731289, size = 142, normalized size = 1.67 \[ -\frac{c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x}{2 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}} + \frac{e^{2} \log \left (c d x + a e\right )}{c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.209026, size = 170, normalized size = 2. \[ -\frac{c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 4 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x - 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{2 \,{\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.03932, size = 109, normalized size = 1.28 \[ \frac{3 a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4} + x \left (4 a c d e^{3} - 4 c^{2} d^{3} e\right )}{2 a^{2} c^{3} d^{3} e^{2} + 4 a c^{4} d^{4} e x + 2 c^{5} d^{5} x^{2}} + \frac{e^{2} \log{\left (a e + c d x \right )}}{c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 7.23976, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Done