3.64 \(\int \frac{x (d+e x)}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=88 \[ -\frac{\log (x) (3 c d-b e)}{b^4}+\frac{(3 c d-b e) \log (b+c x)}{b^4}-\frac{2 c d-b e}{b^3 (b+c x)}-\frac{d}{b^3 x}-\frac{c d-b e}{2 b^2 (b+c x)^2} \]

[Out]

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

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Rubi [A]  time = 0.166646, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\log (x) (3 c d-b e)}{b^4}+\frac{(3 c d-b e) \log (b+c x)}{b^4}-\frac{2 c d-b e}{b^3 (b+c x)}-\frac{d}{b^3 x}-\frac{c d-b e}{2 b^2 (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.2197, size = 75, normalized size = 0.85 \[ \frac{b e - c d}{2 b^{2} \left (b + c x\right )^{2}} - \frac{d}{b^{3} x} + \frac{b e - 2 c d}{b^{3} \left (b + c x\right )} + \frac{\left (b e - 3 c d\right ) \log{\left (x \right )}}{b^{4}} - \frac{\left (b e - 3 c d\right ) \log{\left (b + c x \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0910758, size = 81, normalized size = 0.92 \[ \frac{\frac{b^2 (b e-c d)}{(b+c x)^2}+\frac{2 b (b e-2 c d)}{b+c x}+2 \log (x) (b e-3 c d)+2 (3 c d-b e) \log (b+c x)-\frac{2 b d}{x}}{2 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 105, normalized size = 1.2 \[ -{\frac{d}{{b}^{3}x}}+{\frac{e\ln \left ( x \right ) }{{b}^{3}}}-3\,{\frac{\ln \left ( x \right ) cd}{{b}^{4}}}-{\frac{\ln \left ( cx+b \right ) e}{{b}^{3}}}+3\,{\frac{\ln \left ( cx+b \right ) cd}{{b}^{4}}}+{\frac{e}{{b}^{2} \left ( cx+b \right ) }}-2\,{\frac{cd}{{b}^{3} \left ( cx+b \right ) }}+{\frac{e}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{cd}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x+d)/(c*x^2+b*x)^3,x)

[Out]

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

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Maxima [A]  time = 0.709238, size = 140, normalized size = 1.59 \[ -\frac{2 \, b^{2} d + 2 \,{\left (3 \, c^{2} d - b c e\right )} x^{2} + 3 \,{\left (3 \, b c d - b^{2} e\right )} x}{2 \,{\left (b^{3} c^{2} x^{3} + 2 \, b^{4} c x^{2} + b^{5} x\right )}} + \frac{{\left (3 \, c d - b e\right )} \log \left (c x + b\right )}{b^{4}} - \frac{{\left (3 \, c d - b e\right )} \log \left (x\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.282644, size = 263, normalized size = 2.99 \[ -\frac{2 \, b^{3} d + 2 \,{\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + 3 \,{\left (3 \, b^{2} c d - b^{3} e\right )} x - 2 \,{\left ({\left (3 \, c^{3} d - b c^{2} e\right )} x^{3} + 2 \,{\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} +{\left (3 \, b^{2} c d - b^{3} e\right )} x\right )} \log \left (c x + b\right ) + 2 \,{\left ({\left (3 \, c^{3} d - b c^{2} e\right )} x^{3} + 2 \,{\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} +{\left (3 \, b^{2} c d - b^{3} e\right )} x\right )} \log \left (x\right )}{2 \,{\left (b^{4} c^{2} x^{3} + 2 \, b^{5} c x^{2} + b^{6} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.06443, size = 168, normalized size = 1.91 \[ \frac{- 2 b^{2} d + x^{2} \left (2 b c e - 6 c^{2} d\right ) + x \left (3 b^{2} e - 9 b c d\right )}{2 b^{5} x + 4 b^{4} c x^{2} + 2 b^{3} c^{2} x^{3}} + \frac{\left (b e - 3 c d\right ) \log{\left (x + \frac{b^{2} e - 3 b c d - b \left (b e - 3 c d\right )}{2 b c e - 6 c^{2} d} \right )}}{b^{4}} - \frac{\left (b e - 3 c d\right ) \log{\left (x + \frac{b^{2} e - 3 b c d + b \left (b e - 3 c d\right )}{2 b c e - 6 c^{2} d} \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*b**2*d + x**2*(2*b*c*e - 6*c**2*d) + x*(3*b**2*e - 9*b*c*d))/(2*b**5*x + 4*b
**4*c*x**2 + 2*b**3*c**2*x**3) + (b*e - 3*c*d)*log(x + (b**2*e - 3*b*c*d - b*(b*
e - 3*c*d))/(2*b*c*e - 6*c**2*d))/b**4 - (b*e - 3*c*d)*log(x + (b**2*e - 3*b*c*d
+ b*(b*e - 3*c*d))/(2*b*c*e - 6*c**2*d))/b**4

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GIAC/XCAS [A]  time = 0.272641, size = 144, normalized size = 1.64 \[ -\frac{{\left (3 \, c d - b e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{4}} + \frac{{\left (3 \, c^{2} d - b c e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac{2 \, b^{3} d + 2 \,{\left (3 \, b c^{2} d - b^{2} c e\right )} x^{2} + 3 \,{\left (3 \, b^{2} c d - b^{3} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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