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

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

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Rubi [A]  time = 0.10, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.08, size = 106, normalized size = 0.94 \begin {gather*} \frac {-\frac {2 b^3 d}{x^3}-\frac {3 b^2 (b e-2 c d)}{x^2}+\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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{x^2 \left (b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 180, normalized size = 1.59 \begin {gather*} -\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 \relax (x)}{6 \, {\left (b^{5} c x^{4} + b^{6} x^{3}\right )}} \end {gather*}

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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giac [A]  time = 0.15, size = 139, normalized size = 1.23 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.12, size = 134, normalized size = 1.19 \begin {gather*} \frac {c^{2} e}{\left (c x +b \right ) b^{3}}-\frac {c^{3} d}{\left (c x +b \right ) b^{4}}+\frac {3 c^{2} e \ln \relax (x )}{b^{4}}-\frac {3 c^{2} e \ln \left (c x +b \right )}{b^{4}}-\frac {4 c^{3} d \ln \relax (x )}{b^{5}}+\frac {4 c^{3} d \ln \left (c x +b \right )}{b^{5}}+\frac {2 c e}{b^{3} x}-\frac {3 c^{2} d}{b^{4} x}-\frac {e}{2 b^{2} x^{2}}+\frac {c d}{b^{3} x^{2}}-\frac {d}{3 b^{2} x^{3}} \end {gather*}

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]

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

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maxima [A]  time = 0.97, size = 129, normalized size = 1.14 \begin {gather*} -\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 \relax (x)}{b^{5}} \end {gather*}

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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mupad [B]  time = 1.08, size = 132, normalized size = 1.17 \begin {gather*} \frac {2\,c^2\,\mathrm {atanh}\left (\frac {c^2\,\left (3\,b\,e-4\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (4\,c^3\,d-3\,b\,c^2\,e\right )}\right )\,\left (3\,b\,e-4\,c\,d\right )}{b^5}-\frac {\frac {d}{3\,b}+\frac {x\,\left (3\,b\,e-4\,c\,d\right )}{6\,b^2}-\frac {c\,x^2\,\left (3\,b\,e-4\,c\,d\right )}{2\,b^3}-\frac {c^2\,x^3\,\left (3\,b\,e-4\,c\,d\right )}{b^4}}{c\,x^4+b\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.66, size = 219, normalized size = 1.94 \begin {gather*} \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 {gather*}

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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