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

Optimal. Leaf size=62 \[ \frac{c d-b e}{b^2 x}+\frac{c \log (x) (c d-b e)}{b^3}-\frac{c (c d-b e) \log (b+c x)}{b^3}-\frac{d}{2 b x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0483203, antiderivative size = 62, 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} \[ \frac{c d-b e}{b^2 x}+\frac{c \log (x) (c d-b e)}{b^3}-\frac{c (c d-b e) \log (b+c x)}{b^3}-\frac{d}{2 b x^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^2*(b*x + c*x^2)),x]

[Out]

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

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

Mathematica [A]  time = 0.0318751, size = 58, normalized size = 0.94 \[ \frac{-\frac{b (b d+2 b e x-2 c d x)}{x^2}+2 c \log (x) (c d-b e)+2 c (b e-c d) \log (b+c x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 75, normalized size = 1.2 \begin{align*} -{\frac{d}{2\,b{x}^{2}}}-{\frac{e}{bx}}+{\frac{cd}{{b}^{2}x}}-{\frac{c\ln \left ( x \right ) e}{{b}^{2}}}+{\frac{{c}^{2}\ln \left ( x \right ) d}{{b}^{3}}}+{\frac{c\ln \left ( cx+b \right ) e}{{b}^{2}}}-{\frac{{c}^{2}\ln \left ( cx+b \right ) d}{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.14581, size = 85, normalized size = 1.37 \begin{align*} -\frac{{\left (c^{2} d - b c e\right )} \log \left (c x + b\right )}{b^{3}} + \frac{{\left (c^{2} d - b c e\right )} \log \left (x\right )}{b^{3}} - \frac{b d - 2 \,{\left (c d - b e\right )} x}{2 \, b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.78322, size = 154, normalized size = 2.48 \begin{align*} -\frac{2 \,{\left (c^{2} d - b c e\right )} x^{2} \log \left (c x + b\right ) - 2 \,{\left (c^{2} d - b c e\right )} x^{2} \log \left (x\right ) + b^{2} d - 2 \,{\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.793768, size = 131, normalized size = 2.11 \begin{align*} - \frac{b d + x \left (2 b e - 2 c d\right )}{2 b^{2} x^{2}} - \frac{c \left (b e - c d\right ) \log{\left (x + \frac{b^{2} c e - b c^{2} d - b c \left (b e - c d\right )}{2 b c^{2} e - 2 c^{3} d} \right )}}{b^{3}} + \frac{c \left (b e - c d\right ) \log{\left (x + \frac{b^{2} c e - b c^{2} d + b c \left (b e - c d\right )}{2 b c^{2} e - 2 c^{3} d} \right )}}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.24864, size = 105, normalized size = 1.69 \begin{align*} \frac{{\left (c^{2} d - b c e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{{\left (c^{3} d - b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} - \frac{b^{2} d - 2 \,{\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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