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

Optimal. Leaf size=85 \[ \frac{2 c d-b e}{b^3 x}+\frac{c (c d-b e)}{b^3 (b+c x)}+\frac{c \log (x) (3 c d-2 b e)}{b^4}-\frac{c (3 c d-2 b e) \log (b+c x)}{b^4}-\frac{d}{2 b^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0758893, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{2 c d-b e}{b^3 x}+\frac{c (c d-b e)}{b^3 (b+c x)}+\frac{c \log (x) (3 c d-2 b e)}{b^4}-\frac{c (3 c d-2 b e) \log (b+c x)}{b^4}-\frac{d}{2 b^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x*(b*x + c*x^2)^2),x]

[Out]

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

Mathematica [A]  time = 0.0688269, size = 85, normalized size = 1. \[ \frac{-\frac{b \left (b^2 (d+2 e x)+b c x (4 e x-3 d)-6 c^2 d x^2\right )}{x^2 (b+c x)}+2 c \log (x) (3 c d-2 b e)+2 c (2 b e-3 c d) \log (b+c x)}{2 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 107, normalized size = 1.3 \begin{align*} -{\frac{d}{2\,{b}^{2}{x}^{2}}}-{\frac{e}{{b}^{2}x}}+2\,{\frac{cd}{{b}^{3}x}}-2\,{\frac{c\ln \left ( x \right ) e}{{b}^{3}}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) d}{{b}^{4}}}-{\frac{ec}{{b}^{2} \left ( cx+b \right ) }}+{\frac{d{c}^{2}}{{b}^{3} \left ( cx+b \right ) }}+2\,{\frac{c\ln \left ( cx+b \right ) e}{{b}^{3}}}-3\,{\frac{{c}^{2}\ln \left ( cx+b \right ) d}{{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/x/(c*x^2+b*x)^2,x)

[Out]

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

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Maxima [A]  time = 1.11687, size = 135, normalized size = 1.59 \begin{align*} -\frac{b^{2} d - 2 \,{\left (3 \, c^{2} d - 2 \, b c e\right )} x^{2} -{\left (3 \, b c d - 2 \, b^{2} e\right )} x}{2 \,{\left (b^{3} c x^{3} + b^{4} x^{2}\right )}} - \frac{{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left (c x + b\right )}{b^{4}} + \frac{{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left (x\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.85703, size = 321, normalized size = 3.78 \begin{align*} -\frac{b^{3} d - 2 \,{\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2} -{\left (3 \, b^{2} c d - 2 \, b^{3} e\right )} x + 2 \,{\left ({\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} x^{3} +{\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} x^{3} +{\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{4} c x^{3} + b^{5} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.982621, size = 184, normalized size = 2.16 \begin{align*} - \frac{b^{2} d + x^{2} \left (4 b c e - 6 c^{2} d\right ) + x \left (2 b^{2} e - 3 b c d\right )}{2 b^{4} x^{2} + 2 b^{3} c x^{3}} - \frac{c \left (2 b e - 3 c d\right ) \log{\left (x + \frac{2 b^{2} c e - 3 b c^{2} d - b c \left (2 b e - 3 c d\right )}{4 b c^{2} e - 6 c^{3} d} \right )}}{b^{4}} + \frac{c \left (2 b e - 3 c d\right ) \log{\left (x + \frac{2 b^{2} c e - 3 b c^{2} d + b c \left (2 b e - 3 c d\right )}{4 b c^{2} e - 6 c^{3} d} \right )}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x/(c*x**2+b*x)**2,x)

[Out]

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

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Giac [A]  time = 1.1521, size = 150, normalized size = 1.76 \begin{align*} \frac{{\left (3 \, c^{2} d - 2 \, b c e\right )} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac{{\left (3 \, c^{3} d - 2 \, b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac{b^{3} d - 2 \,{\left (3 \, b c^{2} d - 2 \, b^{2} c e\right )} x^{2} -{\left (3 \, b^{2} c d - 2 \, b^{3} e\right )} x}{2 \,{\left (c x + b\right )} b^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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