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

Optimal. Leaf size=42 \[ -\frac{d \log (b+c x)}{b^2}+\frac{d \log (x)}{b^2}+\frac{c d-b e}{b c (b+c x)} \]

[Out]

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

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Rubi [A]  time = 0.0308703, antiderivative size = 42, 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} \[ -\frac{d \log (b+c x)}{b^2}+\frac{d \log (x)}{b^2}+\frac{c d-b e}{b c (b+c x)} \]

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.026959, size = 38, normalized size = 0.9 \[ \frac{\frac{b (c d-b e)}{c (b+c x)}-d \log (b+c x)+d \log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 46, normalized size = 1.1 \begin{align*}{\frac{d\ln \left ( x \right ) }{{b}^{2}}}-{\frac{e}{c \left ( cx+b \right ) }}+{\frac{d}{b \left ( cx+b \right ) }}-{\frac{d\ln \left ( cx+b \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12003, size = 58, normalized size = 1.38 \begin{align*} \frac{c d - b e}{b c^{2} x + b^{2} c} - \frac{d \log \left (c x + b\right )}{b^{2}} + \frac{d \log \left (x\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9282, size = 131, normalized size = 3.12 \begin{align*} \frac{b c d - b^{2} e -{\left (c^{2} d x + b c d\right )} \log \left (c x + b\right ) +{\left (c^{2} d x + b c d\right )} \log \left (x\right )}{b^{2} c^{2} x + b^{3} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.580045, size = 32, normalized size = 0.76 \begin{align*} - \frac{b e - c d}{b^{2} c + b c^{2} x} + \frac{d \left (\log{\left (x \right )} - \log{\left (\frac{b}{c} + x \right )}\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15894, size = 65, normalized size = 1.55 \begin{align*} -\frac{d \log \left ({\left | c x + b \right |}\right )}{b^{2}} + \frac{d \log \left ({\left | x \right |}\right )}{b^{2}} + \frac{b c d - b^{2} e}{{\left (c x + b\right )} b^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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