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

Optimal. Leaf size=140 \[ -\frac{c^2 (4 c d-3 b e)}{b^5 (b+c x)}-\frac{c^2 (c d-b e)}{2 b^4 (b+c x)^2}-\frac{2 c^2 \log (x) (5 c d-3 b e)}{b^6}+\frac{2 c^2 (5 c d-3 b e) \log (b+c x)}{b^6}+\frac{3 c d-b e}{2 b^4 x^2}-\frac{3 c (2 c d-b e)}{b^5 x}-\frac{d}{3 b^3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.124081, antiderivative size = 140, 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} \[ -\frac{c^2 (4 c d-3 b e)}{b^5 (b+c x)}-\frac{c^2 (c d-b e)}{2 b^4 (b+c x)^2}-\frac{2 c^2 \log (x) (5 c d-3 b e)}{b^6}+\frac{2 c^2 (5 c d-3 b e) \log (b+c x)}{b^6}+\frac{3 c d-b e}{2 b^4 x^2}-\frac{3 c (2 c d-b e)}{b^5 x}-\frac{d}{3 b^3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x*(b*x + c*x^2)^3),x]

[Out]

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

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 \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac{d}{b^3 x^4}+\frac{-3 c d+b e}{b^4 x^3}-\frac{3 c (-2 c d+b e)}{b^5 x^2}+\frac{2 c^2 (-5 c d+3 b e)}{b^6 x}-\frac{c^3 (-c d+b e)}{b^4 (b+c x)^3}-\frac{c^3 (-4 c d+3 b e)}{b^5 (b+c x)^2}-\frac{2 c^3 (-5 c d+3 b e)}{b^6 (b+c x)}\right ) \, dx\\ &=-\frac{d}{3 b^3 x^3}+\frac{3 c d-b e}{2 b^4 x^2}-\frac{3 c (2 c d-b e)}{b^5 x}-\frac{c^2 (c d-b e)}{2 b^4 (b+c x)^2}-\frac{c^2 (4 c d-3 b e)}{b^5 (b+c x)}-\frac{2 c^2 (5 c d-3 b e) \log (x)}{b^6}+\frac{2 c^2 (5 c d-3 b e) \log (b+c x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.128242, size = 129, normalized size = 0.92 \[ \frac{\frac{b \left (2 b^2 c^2 x^2 (27 e x-10 d)+b^3 c x (5 d+12 e x)+b^4 (-(2 d+3 e x))+18 b c^3 x^3 (2 e x-5 d)-60 c^4 d x^4\right )}{x^3 (b+c x)^2}+12 c^2 \log (x) (3 b e-5 c d)+12 c^2 (5 c d-3 b e) \log (b+c x)}{6 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*(-60*c^4*d*x^4 + 18*b*c^3*x^3*(-5*d + 2*e*x) - b^4*(2*d + 3*e*x) + b^3*c*x*(5*d + 12*e*x) + 2*b^2*c^2*x^2*
(-10*d + 27*e*x)))/(x^3*(b + c*x)^2) + 12*c^2*(-5*c*d + 3*b*e)*Log[x] + 12*c^2*(5*c*d - 3*b*e)*Log[b + c*x])/(
6*b^6)

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Maple [A]  time = 0.012, size = 168, normalized size = 1.2 \begin{align*} -{\frac{d}{3\,{b}^{3}{x}^{3}}}-{\frac{e}{2\,{b}^{3}{x}^{2}}}+{\frac{3\,cd}{2\,{b}^{4}{x}^{2}}}+3\,{\frac{ec}{{b}^{4}x}}-6\,{\frac{d{c}^{2}}{{b}^{5}x}}+6\,{\frac{{c}^{2}\ln \left ( x \right ) e}{{b}^{5}}}-10\,{\frac{{c}^{3}\ln \left ( x \right ) d}{{b}^{6}}}+{\frac{e{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-{\frac{d{c}^{3}}{2\,{b}^{4} \left ( cx+b \right ) ^{2}}}-6\,{\frac{{c}^{2}\ln \left ( cx+b \right ) e}{{b}^{5}}}+10\,{\frac{{c}^{3}\ln \left ( cx+b \right ) d}{{b}^{6}}}+3\,{\frac{e{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}-4\,{\frac{d{c}^{3}}{{b}^{5} \left ( cx+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*d/b^3/x^3-1/2/b^3/x^2*e+3/2/b^4/x^2*c*d+3*c/b^4/x*e-6*c^2/b^5/x*d+6*c^2/b^5*ln(x)*e-10*c^3/b^6*ln(x)*d+1/
2*c^2/b^3/(c*x+b)^2*e-1/2*c^3/b^4/(c*x+b)^2*d-6*c^2/b^5*ln(c*x+b)*e+10*c^3/b^6*ln(c*x+b)*d+3*c^2/b^4/(c*x+b)*e
-4*c^3/b^5/(c*x+b)*d

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Maxima [A]  time = 1.10913, size = 223, normalized size = 1.59 \begin{align*} -\frac{2 \, b^{4} d + 12 \,{\left (5 \, c^{4} d - 3 \, b c^{3} e\right )} x^{4} + 18 \,{\left (5 \, b c^{3} d - 3 \, b^{2} c^{2} e\right )} x^{3} + 4 \,{\left (5 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x^{2} -{\left (5 \, b^{3} c d - 3 \, b^{4} e\right )} x}{6 \,{\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\right )}} + \frac{2 \,{\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{6}} - \frac{2 \,{\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (x\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^4*d + 12*(5*c^4*d - 3*b*c^3*e)*x^4 + 18*(5*b*c^3*d - 3*b^2*c^2*e)*x^3 + 4*(5*b^2*c^2*d - 3*b^3*c*e)*
x^2 - (5*b^3*c*d - 3*b^4*e)*x)/(b^5*c^2*x^5 + 2*b^6*c*x^4 + b^7*x^3) + 2*(5*c^3*d - 3*b*c^2*e)*log(c*x + b)/b^
6 - 2*(5*c^3*d - 3*b*c^2*e)*log(x)/b^6

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Fricas [A]  time = 2.00282, size = 551, normalized size = 3.94 \begin{align*} -\frac{2 \, b^{5} d + 12 \,{\left (5 \, b c^{4} d - 3 \, b^{2} c^{3} e\right )} x^{4} + 18 \,{\left (5 \, b^{2} c^{3} d - 3 \, b^{3} c^{2} e\right )} x^{3} + 4 \,{\left (5 \, b^{3} c^{2} d - 3 \, b^{4} c e\right )} x^{2} -{\left (5 \, b^{4} c d - 3 \, b^{5} e\right )} x - 12 \,{\left ({\left (5 \, c^{5} d - 3 \, b c^{4} e\right )} x^{5} + 2 \,{\left (5 \, b c^{4} d - 3 \, b^{2} c^{3} e\right )} x^{4} +{\left (5 \, b^{2} c^{3} d - 3 \, b^{3} c^{2} e\right )} x^{3}\right )} \log \left (c x + b\right ) + 12 \,{\left ({\left (5 \, c^{5} d - 3 \, b c^{4} e\right )} x^{5} + 2 \,{\left (5 \, b c^{4} d - 3 \, b^{2} c^{3} e\right )} x^{4} +{\left (5 \, b^{2} c^{3} d - 3 \, b^{3} c^{2} e\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (b^{6} c^{2} x^{5} + 2 \, b^{7} c x^{4} + b^{8} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*b^5*d + 12*(5*b*c^4*d - 3*b^2*c^3*e)*x^4 + 18*(5*b^2*c^3*d - 3*b^3*c^2*e)*x^3 + 4*(5*b^3*c^2*d - 3*b^4
*c*e)*x^2 - (5*b^4*c*d - 3*b^5*e)*x - 12*((5*c^5*d - 3*b*c^4*e)*x^5 + 2*(5*b*c^4*d - 3*b^2*c^3*e)*x^4 + (5*b^2
*c^3*d - 3*b^3*c^2*e)*x^3)*log(c*x + b) + 12*((5*c^5*d - 3*b*c^4*e)*x^5 + 2*(5*b*c^4*d - 3*b^2*c^3*e)*x^4 + (5
*b^2*c^3*d - 3*b^3*c^2*e)*x^3)*log(x))/(b^6*c^2*x^5 + 2*b^7*c*x^4 + b^8*x^3)

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Sympy [A]  time = 1.84975, size = 262, normalized size = 1.87 \begin{align*} \frac{- 2 b^{4} d + x^{4} \left (36 b c^{3} e - 60 c^{4} d\right ) + x^{3} \left (54 b^{2} c^{2} e - 90 b c^{3} d\right ) + x^{2} \left (12 b^{3} c e - 20 b^{2} c^{2} d\right ) + x \left (- 3 b^{4} e + 5 b^{3} c d\right )}{6 b^{7} x^{3} + 12 b^{6} c x^{4} + 6 b^{5} c^{2} x^{5}} + \frac{2 c^{2} \left (3 b e - 5 c d\right ) \log{\left (x + \frac{6 b^{2} c^{2} e - 10 b c^{3} d - 2 b c^{2} \left (3 b e - 5 c d\right )}{12 b c^{3} e - 20 c^{4} d} \right )}}{b^{6}} - \frac{2 c^{2} \left (3 b e - 5 c d\right ) \log{\left (x + \frac{6 b^{2} c^{2} e - 10 b c^{3} d + 2 b c^{2} \left (3 b e - 5 c d\right )}{12 b c^{3} e - 20 c^{4} d} \right )}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*b**4*d + x**4*(36*b*c**3*e - 60*c**4*d) + x**3*(54*b**2*c**2*e - 90*b*c**3*d) + x**2*(12*b**3*c*e - 20*b**
2*c**2*d) + x*(-3*b**4*e + 5*b**3*c*d))/(6*b**7*x**3 + 12*b**6*c*x**4 + 6*b**5*c**2*x**5) + 2*c**2*(3*b*e - 5*
c*d)*log(x + (6*b**2*c**2*e - 10*b*c**3*d - 2*b*c**2*(3*b*e - 5*c*d))/(12*b*c**3*e - 20*c**4*d))/b**6 - 2*c**2
*(3*b*e - 5*c*d)*log(x + (6*b**2*c**2*e - 10*b*c**3*d + 2*b*c**2*(3*b*e - 5*c*d))/(12*b*c**3*e - 20*c**4*d))/b
**6

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Giac [A]  time = 1.13678, size = 223, normalized size = 1.59 \begin{align*} -\frac{2 \,{\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{6}} + \frac{2 \,{\left (5 \, c^{4} d - 3 \, b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{6} c} - \frac{2 \, b^{5} d + 12 \,{\left (5 \, b c^{4} d - 3 \, b^{2} c^{3} e\right )} x^{4} + 18 \,{\left (5 \, b^{2} c^{3} d - 3 \, b^{3} c^{2} e\right )} x^{3} + 4 \,{\left (5 \, b^{3} c^{2} d - 3 \, b^{4} c e\right )} x^{2} -{\left (5 \, b^{4} c d - 3 \, b^{5} e\right )} x}{6 \,{\left (c x + b\right )}^{2} b^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(5*c^3*d - 3*b*c^2*e)*log(abs(x))/b^6 + 2*(5*c^4*d - 3*b*c^3*e)*log(abs(c*x + b))/(b^6*c) - 1/6*(2*b^5*d +
12*(5*b*c^4*d - 3*b^2*c^3*e)*x^4 + 18*(5*b^2*c^3*d - 3*b^3*c^2*e)*x^3 + 4*(5*b^3*c^2*d - 3*b^4*c*e)*x^2 - (5*b
^4*c*d - 3*b^5*e)*x)/((c*x + b)^2*b^6*x^3)