### 3.271 $$\int \frac{d+e x}{(b x+c x^2)^2} \, dx$$

Optimal. Leaf size=65 $-\frac{c d-b e}{b^2 (b+c x)}-\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{d}{b^2 x}$

[Out]

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

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Rubi [A]  time = 0.0503731, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {631} $-\frac{c d-b e}{b^2 (b+c x)}-\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{d}{b^2 x}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(b*x + c*x^2)^2,x]

[Out]

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

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0396616, size = 56, normalized size = 0.86 $\frac{\frac{b (b e-c d)}{b+c x}+\log (x) (b e-2 c d)+(2 c d-b e) \log (b+c x)-\frac{b d}{x}}{b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.054, size = 78, normalized size = 1.2 \begin{align*} -{\frac{d}{{b}^{2}x}}+{\frac{\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{\ln \left ( x \right ) cd}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) e}{{b}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) cd}{{b}^{3}}}+{\frac{e}{b \left ( cx+b \right ) }}-{\frac{cd}{{b}^{2} \left ( cx+b \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13112, size = 93, normalized size = 1.43 \begin{align*} -\frac{b d +{\left (2 \, c d - b e\right )} x}{b^{2} c x^{2} + b^{3} x} + \frac{{\left (2 \, c d - b e\right )} \log \left (c x + b\right )}{b^{3}} - \frac{{\left (2 \, c d - b e\right )} \log \left (x\right )}{b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.68922, size = 227, normalized size = 3.49 \begin{align*} -\frac{b^{2} d +{\left (2 \, b c d - b^{2} e\right )} x -{\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (c x + b\right ) +{\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.45253, size = 128, normalized size = 1.97 \begin{align*} \frac{- b d + x \left (b e - 2 c d\right )}{b^{3} x + b^{2} c x^{2}} + \frac{\left (b e - 2 c d\right ) \log{\left (x + \frac{b^{2} e - 2 b c d - b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} - \frac{\left (b e - 2 c d\right ) \log{\left (x + \frac{b^{2} e - 2 b c d + b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2503, size = 104, normalized size = 1.6 \begin{align*} -\frac{{\left (2 \, c d - b e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c d x - b x e + b d}{{\left (c x^{2} + b x\right )} b^{2}} + \frac{{\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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