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

Optimal. Leaf size=94 \[ \frac{2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}-\frac{2 d^3 \log (x) (c d-2 b e)}{b^3}-\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac{d^4}{b^2 x}+\frac{e^4 x}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.233204, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}-\frac{2 d^3 \log (x) (c d-2 b e)}{b^3}-\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac{d^4}{b^2 x}+\frac{e^4 x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{4} \int \frac{1}{c^{2}}\, dx - \frac{d^{4}}{b^{2} x} - \frac{\left (b e - c d\right )^{4}}{b^{2} c^{3} \left (b + c x\right )} + \frac{2 d^{3} \left (2 b e - c d\right ) \log{\left (x \right )}}{b^{3}} - \frac{2 \left (b e - c d\right )^{3} \left (b e + c d\right ) \log{\left (b + c x \right )}}{b^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**4*Integral(c**(-2), x) - d**4/(b**2*x) - (b*e - c*d)**4/(b**2*c**3*(b + c*x))
+ 2*d**3*(2*b*e - c*d)*log(x)/b**3 - 2*(b*e - c*d)**3*(b*e + c*d)*log(b + c*x)/
(b**3*c**3)

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Mathematica [A]  time = 0.158126, size = 95, normalized size = 1.01 \[ \frac{2 (c d-b e)^3 (b e+c d) \log (b+c x)}{b^3 c^3}+\frac{2 d^3 \log (x) (2 b e-c d)}{b^3}-\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac{d^4}{b^2 x}+\frac{e^4 x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 188, normalized size = 2. \[{\frac{{e}^{4}x}{{c}^{2}}}-{\frac{{d}^{4}}{{b}^{2}x}}+4\,{\frac{{d}^{3}\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{{d}^{4}\ln \left ( x \right ) c}{{b}^{3}}}-2\,{\frac{b\ln \left ( cx+b \right ){e}^{4}}{{c}^{3}}}+4\,{\frac{\ln \left ( cx+b \right ) d{e}^{3}}{{c}^{2}}}-4\,{\frac{\ln \left ( cx+b \right ){d}^{3}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ){d}^{4}}{{b}^{3}}}-{\frac{{e}^{4}{b}^{2}}{{c}^{3} \left ( cx+b \right ) }}+4\,{\frac{bd{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-6\,{\frac{{d}^{2}{e}^{2}}{c \left ( cx+b \right ) }}+4\,{\frac{{d}^{3}e}{b \left ( cx+b \right ) }}-{\frac{{d}^{4}c}{{b}^{2} \left ( cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+b*x)^2,x)

[Out]

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

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Maxima [A]  time = 0.704239, size = 220, normalized size = 2.34 \[ \frac{e^{4} x}{c^{2}} - \frac{b c^{3} d^{4} +{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} x}{b^{2} c^{4} x^{2} + b^{3} c^{3} x} - \frac{2 \,{\left (c d^{4} - 2 \, b d^{3} e\right )} \log \left (x\right )}{b^{3}} + \frac{2 \,{\left (c^{4} d^{4} - 2 \, b c^{3} d^{3} e + 2 \, b^{3} c d e^{3} - b^{4} e^{4}\right )} \log \left (c x + b\right )}{b^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^4*x/c^2 - (b*c^3*d^4 + (2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 4*b^3*
c*d*e^3 + b^4*e^4)*x)/(b^2*c^4*x^2 + b^3*c^3*x) - 2*(c*d^4 - 2*b*d^3*e)*log(x)/b
^3 + 2*(c^4*d^4 - 2*b*c^3*d^3*e + 2*b^3*c*d*e^3 - b^4*e^4)*log(c*x + b)/(b^3*c^3
)

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Fricas [A]  time = 0.242489, size = 347, normalized size = 3.69 \[ \frac{b^{3} c^{2} e^{4} x^{3} + b^{4} c e^{4} x^{2} - b^{2} c^{3} d^{4} -{\left (2 \, b c^{4} d^{4} - 4 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} + b^{5} e^{4}\right )} x + 2 \,{\left ({\left (c^{5} d^{4} - 2 \, b c^{4} d^{3} e + 2 \, b^{3} c^{2} d e^{3} - b^{4} c e^{4}\right )} x^{2} +{\left (b c^{4} d^{4} - 2 \, b^{2} c^{3} d^{3} e + 2 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (c^{5} d^{4} - 2 \, b c^{4} d^{3} e\right )} x^{2} +{\left (b c^{4} d^{4} - 2 \, b^{2} c^{3} d^{3} e\right )} x\right )} \log \left (x\right )}{b^{3} c^{4} x^{2} + b^{4} c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.567, size = 306, normalized size = 3.26 \[ - \frac{b c^{3} d^{4} + x \left (b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{b^{3} c^{3} x + b^{2} c^{4} x^{2}} + \frac{e^{4} x}{c^{2}} + \frac{2 d^{3} \left (2 b e - c d\right ) \log{\left (x + \frac{4 b^{2} c^{2} d^{3} e - 2 b c^{3} d^{4} - 2 b c^{2} d^{3} \left (2 b e - c d\right )}{2 b^{4} e^{4} - 4 b^{3} c d e^{3} + 8 b c^{3} d^{3} e - 4 c^{4} d^{4}} \right )}}{b^{3}} - \frac{2 \left (b e - c d\right )^{3} \left (b e + c d\right ) \log{\left (x + \frac{4 b^{2} c^{2} d^{3} e - 2 b c^{3} d^{4} + \frac{2 b \left (b e - c d\right )^{3} \left (b e + c d\right )}{c}}{2 b^{4} e^{4} - 4 b^{3} c d e^{3} + 8 b c^{3} d^{3} e - 4 c^{4} d^{4}} \right )}}{b^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*c**3*d**4 + x*(b**4*e**4 - 4*b**3*c*d*e**3 + 6*b**2*c**2*d**2*e**2 - 4*b*c**
3*d**3*e + 2*c**4*d**4))/(b**3*c**3*x + b**2*c**4*x**2) + e**4*x/c**2 + 2*d**3*(
2*b*e - c*d)*log(x + (4*b**2*c**2*d**3*e - 2*b*c**3*d**4 - 2*b*c**2*d**3*(2*b*e
- c*d))/(2*b**4*e**4 - 4*b**3*c*d*e**3 + 8*b*c**3*d**3*e - 4*c**4*d**4))/b**3 -
2*(b*e - c*d)**3*(b*e + c*d)*log(x + (4*b**2*c**2*d**3*e - 2*b*c**3*d**4 + 2*b*(
b*e - c*d)**3*(b*e + c*d)/c)/(2*b**4*e**4 - 4*b**3*c*d*e**3 + 8*b*c**3*d**3*e -
4*c**4*d**4))/(b**3*c**3)

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GIAC/XCAS [A]  time = 0.207053, size = 216, normalized size = 2.3 \[ \frac{x e^{4}}{c^{2}} - \frac{2 \,{\left (c d^{4} - 2 \, b d^{3} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \,{\left (c^{4} d^{4} - 2 \, b c^{3} d^{3} e + 2 \, b^{3} c d e^{3} - b^{4} e^{4}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c^{3}} - \frac{b c^{2} d^{4} + \frac{{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} x}{c}}{{\left (c x + b\right )} b^{2} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^4/c^2 - 2*(c*d^4 - 2*b*d^3*e)*ln(abs(x))/b^3 + 2*(c^4*d^4 - 2*b*c^3*d^3*e +
2*b^3*c*d*e^3 - b^4*e^4)*ln(abs(c*x + b))/(b^3*c^3) - (b*c^2*d^4 + (2*c^4*d^4 -
4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 4*b^3*c*d*e^3 + b^4*e^4)*x/c)/((c*x + b)*b^2
*c^2*x)