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

Optimal. Leaf size=113 \[ -\frac{c^2 (c d-b e)}{b^4 (b+c x)}-\frac{c^2 \log (x) (4 c d-3 b e)}{b^5}+\frac{c^2 (4 c d-3 b e) \log (b+c x)}{b^5}+\frac{2 c d-b e}{2 b^3 x^2}-\frac{c (3 c d-2 b e)}{b^4 x}-\frac{d}{3 b^2 x^3} \]

[Out]

-d/(3*b^2*x^3) + (2*c*d - b*e)/(2*b^3*x^2) - (c*(3*c*d - 2*b*e))/(b^4*x) - (c^2*(c*d - b*e))/(b^4*(b + c*x)) -
(c^2*(4*c*d - 3*b*e)*Log[x])/b^5 + (c^2*(4*c*d - 3*b*e)*Log[b + c*x])/b^5

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Rubi [A]  time = 0.103721, antiderivative size = 113, 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} \[ -\frac{c^2 (c d-b e)}{b^4 (b+c x)}-\frac{c^2 \log (x) (4 c d-3 b e)}{b^5}+\frac{c^2 (4 c d-3 b e) \log (b+c x)}{b^5}+\frac{2 c d-b e}{2 b^3 x^2}-\frac{c (3 c d-2 b e)}{b^4 x}-\frac{d}{3 b^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(3*b^2*x^3) + (2*c*d - b*e)/(2*b^3*x^2) - (c*(3*c*d - 2*b*e))/(b^4*x) - (c^2*(c*d - b*e))/(b^4*(b + c*x)) -
(c^2*(4*c*d - 3*b*e)*Log[x])/b^5 + (c^2*(4*c*d - 3*b*e)*Log[b + c*x])/b^5

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 )^2} \, dx &=\int \left (\frac{d}{b^2 x^4}+\frac{-2 c d+b e}{b^3 x^3}-\frac{c (-3 c d+2 b e)}{b^4 x^2}+\frac{c^2 (-4 c d+3 b e)}{b^5 x}-\frac{c^3 (-c d+b e)}{b^4 (b+c x)^2}-\frac{c^3 (-4 c d+3 b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac{d}{3 b^2 x^3}+\frac{2 c d-b e}{2 b^3 x^2}-\frac{c (3 c d-2 b e)}{b^4 x}-\frac{c^2 (c d-b e)}{b^4 (b+c x)}-\frac{c^2 (4 c d-3 b e) \log (x)}{b^5}+\frac{c^2 (4 c d-3 b e) \log (b+c x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0911887, size = 106, normalized size = 0.94 \[ \frac{-\frac{3 b^2 (b e-2 c d)}{x^2}-\frac{2 b^3 d}{x^3}+\frac{6 b c^2 (b e-c d)}{b+c x}+6 c^2 \log (x) (3 b e-4 c d)+6 c^2 (4 c d-3 b e) \log (b+c x)+\frac{6 b c (2 b e-3 c d)}{x}}{6 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*b^3*d)/x^3 - (3*b^2*(-2*c*d + b*e))/x^2 + (6*b*c*(-3*c*d + 2*b*e))/x + (6*b*c^2*(-(c*d) + b*e))/(b + c*x)
+ 6*c^2*(-4*c*d + 3*b*e)*Log[x] + 6*c^2*(4*c*d - 3*b*e)*Log[b + c*x])/(6*b^5)

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Maple [A]  time = 0.012, size = 134, normalized size = 1.2 \begin{align*} -{\frac{d}{3\,{b}^{2}{x}^{3}}}-{\frac{e}{2\,{b}^{2}{x}^{2}}}+{\frac{cd}{{b}^{3}{x}^{2}}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) e}{{b}^{4}}}-4\,{\frac{{c}^{3}\ln \left ( x \right ) d}{{b}^{5}}}+2\,{\frac{ec}{{b}^{3}x}}-3\,{\frac{d{c}^{2}}{{b}^{4}x}}+{\frac{e{c}^{2}}{{b}^{3} \left ( cx+b \right ) }}-{\frac{d{c}^{3}}{{b}^{4} \left ( cx+b \right ) }}-3\,{\frac{{c}^{2}\ln \left ( cx+b \right ) e}{{b}^{4}}}+4\,{\frac{{c}^{3}\ln \left ( cx+b \right ) d}{{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12709, size = 174, normalized size = 1.54 \begin{align*} -\frac{2 \, b^{3} d + 6 \,{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} x^{3} + 3 \,{\left (4 \, b c^{2} d - 3 \, b^{2} c e\right )} x^{2} -{\left (4 \, b^{2} c d - 3 \, b^{3} e\right )} x}{6 \,{\left (b^{4} c x^{4} + b^{5} x^{3}\right )}} + \frac{{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{5}} - \frac{{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (x\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^3*d + 6*(4*c^3*d - 3*b*c^2*e)*x^3 + 3*(4*b*c^2*d - 3*b^2*c*e)*x^2 - (4*b^2*c*d - 3*b^3*e)*x)/(b^4*c*
x^4 + b^5*x^3) + (4*c^3*d - 3*b*c^2*e)*log(c*x + b)/b^5 - (4*c^3*d - 3*b*c^2*e)*log(x)/b^5

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Fricas [A]  time = 1.88719, size = 378, normalized size = 3.35 \begin{align*} -\frac{2 \, b^{4} d + 6 \,{\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 3 \,{\left (4 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} -{\left (4 \, b^{3} c d - 3 \, b^{4} e\right )} x - 6 \,{\left ({\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} +{\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3}\right )} \log \left (c x + b\right ) + 6 \,{\left ({\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} +{\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (b^{5} c x^{4} + b^{6} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^4*d + 6*(4*b*c^3*d - 3*b^2*c^2*e)*x^3 + 3*(4*b^2*c^2*d - 3*b^3*c*e)*x^2 - (4*b^3*c*d - 3*b^4*e)*x -
6*((4*c^4*d - 3*b*c^3*e)*x^4 + (4*b*c^3*d - 3*b^2*c^2*e)*x^3)*log(c*x + b) + 6*((4*c^4*d - 3*b*c^3*e)*x^4 + (4
*b*c^3*d - 3*b^2*c^2*e)*x^3)*log(x))/(b^5*c*x^4 + b^6*x^3)

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Sympy [B]  time = 1.47464, size = 219, normalized size = 1.94 \begin{align*} \frac{- 2 b^{3} d + x^{3} \left (18 b c^{2} e - 24 c^{3} d\right ) + x^{2} \left (9 b^{2} c e - 12 b c^{2} d\right ) + x \left (- 3 b^{3} e + 4 b^{2} c d\right )}{6 b^{5} x^{3} + 6 b^{4} c x^{4}} + \frac{c^{2} \left (3 b e - 4 c d\right ) \log{\left (x + \frac{3 b^{2} c^{2} e - 4 b c^{3} d - b c^{2} \left (3 b e - 4 c d\right )}{6 b c^{3} e - 8 c^{4} d} \right )}}{b^{5}} - \frac{c^{2} \left (3 b e - 4 c d\right ) \log{\left (x + \frac{3 b^{2} c^{2} e - 4 b c^{3} d + b c^{2} \left (3 b e - 4 c d\right )}{6 b c^{3} e - 8 c^{4} d} \right )}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*b**3*d + x**3*(18*b*c**2*e - 24*c**3*d) + x**2*(9*b**2*c*e - 12*b*c**2*d) + x*(-3*b**3*e + 4*b**2*c*d))/(6
*b**5*x**3 + 6*b**4*c*x**4) + c**2*(3*b*e - 4*c*d)*log(x + (3*b**2*c**2*e - 4*b*c**3*d - b*c**2*(3*b*e - 4*c*d
))/(6*b*c**3*e - 8*c**4*d))/b**5 - c**2*(3*b*e - 4*c*d)*log(x + (3*b**2*c**2*e - 4*b*c**3*d + b*c**2*(3*b*e -
4*c*d))/(6*b*c**3*e - 8*c**4*d))/b**5

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Giac [A]  time = 1.20579, size = 188, normalized size = 1.66 \begin{align*} -\frac{{\left (4 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{{\left (4 \, c^{4} d - 3 \, b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac{2 \, b^{4} d + 6 \,{\left (4 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 3 \,{\left (4 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} -{\left (4 \, b^{3} c d - 3 \, b^{4} e\right )} x}{6 \,{\left (c x + b\right )} b^{5} x^{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)^2,x, algorithm="giac")

[Out]

-(4*c^3*d - 3*b*c^2*e)*log(abs(x))/b^5 + (4*c^4*d - 3*b*c^3*e)*log(abs(c*x + b))/(b^5*c) - 1/6*(2*b^4*d + 6*(4
*b*c^3*d - 3*b^2*c^2*e)*x^3 + 3*(4*b^2*c^2*d - 3*b^3*c*e)*x^2 - (4*b^3*c*d - 3*b^4*e)*x)/((c*x + b)*b^5*x^3)