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

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

[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 [A]  time = 0.0748076, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {765} \[ -\frac{2 c d-b e}{b^3 (b+c x)}-\frac{c d-b e}{2 b^2 (b+c x)^2}-\frac{\log (x) (3 c d-b e)}{b^4}+\frac{(3 c d-b e) \log (b+c x)}{b^4}-\frac{d}{b^3 x} \]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{x (d+e x)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d}{b^3 x^2}+\frac{-3 c d+b e}{b^4 x}-\frac{c (-c d+b e)}{b^2 (b+c x)^3}-\frac{c (-2 c d+b e)}{b^3 (b+c x)^2}-\frac{c (-3 c d+b e)}{b^4 (b+c x)}\right ) \, dx\\ &=-\frac{d}{b^3 x}-\frac{c d-b e}{2 b^2 (b+c x)^2}-\frac{2 c d-b e}{b^3 (b+c x)}-\frac{(3 c d-b e) \log (x)}{b^4}+\frac{(3 c d-b e) \log (b+c x)}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.0491672, 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 = 0.011, size = 105, normalized size = 1.2 \begin{align*} -{\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}}} \end{align*}

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

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Maxima [A]  time = 1.04973, size = 140, normalized size = 1.59 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)/(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 [B]  time = 1.96162, size = 400, normalized size = 4.55 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)/(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 [B]  time = 1.2191, size = 168, normalized size = 1.91 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.14254, size = 144, normalized size = 1.64 \begin{align*} -\frac{{\left (3 \, c d - b e\right )} \log \left ({\left | x \right |}\right )}{b^{4}} + \frac{{\left (3 \, c^{2} d - b c e\right )} \log \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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