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

Optimal. Leaf size=86 \[ -\frac{c^2 \log (x) (c d-b e)}{b^4}+\frac{c^2 (c d-b e) \log (b+c x)}{b^4}+\frac{c d-b e}{2 b^2 x^2}-\frac{c (c d-b e)}{b^3 x}-\frac{d}{3 b x^3} \]

[Out]

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

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Rubi [A]  time = 0.0646612, antiderivative size = 86, 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} \[ -\frac{c^2 \log (x) (c d-b e)}{b^4}+\frac{c^2 (c d-b e) \log (b+c x)}{b^4}+\frac{c d-b e}{2 b^2 x^2}-\frac{c (c d-b e)}{b^3 x}-\frac{d}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^3*(b*x + c*x^2)),x]

[Out]

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

Mathematica [A]  time = 0.0495791, size = 81, normalized size = 0.94 \[ \frac{\frac{b \left (b^2 (-(2 d+3 e x))+3 b c x (d+2 e x)-6 c^2 d x^2\right )}{x^3}+6 c^2 \log (x) (b e-c d)+6 c^2 (c d-b e) \log (b+c x)}{6 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(x^3*(b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 101, normalized size = 1.2 \begin{align*} -{\frac{d}{3\,b{x}^{3}}}-{\frac{e}{2\,b{x}^{2}}}+{\frac{cd}{2\,{b}^{2}{x}^{2}}}+{\frac{{c}^{2}\ln \left ( x \right ) e}{{b}^{3}}}-{\frac{{c}^{3}\ln \left ( x \right ) d}{{b}^{4}}}+{\frac{ce}{{b}^{2}x}}-{\frac{{c}^{2}d}{{b}^{3}x}}-{\frac{{c}^{2}\ln \left ( cx+b \right ) e}{{b}^{3}}}+{\frac{{c}^{3}\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^3/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 1.2166, size = 120, normalized size = 1.4 \begin{align*} \frac{{\left (c^{3} d - b c^{2} e\right )} \log \left (c x + b\right )}{b^{4}} - \frac{{\left (c^{3} d - b c^{2} e\right )} \log \left (x\right )}{b^{4}} - \frac{2 \, b^{2} d + 6 \,{\left (c^{2} d - b c e\right )} x^{2} - 3 \,{\left (b c d - b^{2} e\right )} x}{6 \, b^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3*d - b*c^2*e)*log(c*x + b)/b^4 - (c^3*d - b*c^2*e)*log(x)/b^4 - 1/6*(2*b^2*d + 6*(c^2*d - b*c*e)*x^2 - 3*(
b*c*d - b^2*e)*x)/(b^3*x^3)

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Fricas [A]  time = 1.80298, size = 201, normalized size = 2.34 \begin{align*} \frac{6 \,{\left (c^{3} d - b c^{2} e\right )} x^{3} \log \left (c x + b\right ) - 6 \,{\left (c^{3} d - b c^{2} e\right )} x^{3} \log \left (x\right ) - 2 \, b^{3} d - 6 \,{\left (b c^{2} d - b^{2} c e\right )} x^{2} + 3 \,{\left (b^{2} c d - b^{3} e\right )} x}{6 \, b^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(c^3*d - b*c^2*e)*x^3*log(c*x + b) - 6*(c^3*d - b*c^2*e)*x^3*log(x) - 2*b^3*d - 6*(b*c^2*d - b^2*c*e)*x
^2 + 3*(b^2*c*d - b^3*e)*x)/(b^4*x^3)

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Sympy [B]  time = 0.986909, size = 165, normalized size = 1.92 \begin{align*} \frac{- 2 b^{2} d + x^{2} \left (6 b c e - 6 c^{2} d\right ) + x \left (- 3 b^{2} e + 3 b c d\right )}{6 b^{3} x^{3}} + \frac{c^{2} \left (b e - c d\right ) \log{\left (x + \frac{b^{2} c^{2} e - b c^{3} d - b c^{2} \left (b e - c d\right )}{2 b c^{3} e - 2 c^{4} d} \right )}}{b^{4}} - \frac{c^{2} \left (b e - c d\right ) \log{\left (x + \frac{b^{2} c^{2} e - b c^{3} d + b c^{2} \left (b e - c d\right )}{2 b c^{3} e - 2 c^{4} d} \right )}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x**3/(c*x**2+b*x),x)

[Out]

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

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Giac [A]  time = 1.19776, size = 139, normalized size = 1.62 \begin{align*} -\frac{{\left (c^{3} d - b c^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{4}} + \frac{{\left (c^{4} d - b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{4} c} - \frac{2 \, b^{3} d + 6 \,{\left (b c^{2} d - b^{2} c e\right )} x^{2} - 3 \,{\left (b^{2} c d - b^{3} e\right )} x}{6 \, b^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*d - b*c^2*e)*log(abs(x))/b^4 + (c^4*d - b*c^3*e)*log(abs(c*x + b))/(b^4*c) - 1/6*(2*b^3*d + 6*(b*c^2*d -
b^2*c*e)*x^2 - 3*(b^2*c*d - b^3*e)*x)/(b^4*x^3)