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

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 62, 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 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} \end {gather*}

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 = 0.03, size = 58, normalized size = 0.94 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________

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 )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 68, normalized size = 1.10 \begin {gather*} -\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 \relax (x) + b^{2} d - 2 \, {\left (b c d - b^{2} e\right )} x}{2 \, b^{3} x^{2}} \end {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 78, normalized size = 1.26 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 75, normalized size = 1.21 \begin {gather*} -\frac {c e \ln \relax (x )}{b^{2}}+\frac {c e \ln \left (c x +b \right )}{b^{2}}+\frac {c^{2} d \ln \relax (x )}{b^{3}}-\frac {c^{2} d \ln \left (c x +b \right )}{b^{3}}-\frac {e}{b x}+\frac {c d}{b^{2} x}-\frac {d}{2 b \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.92, size = 63, normalized size = 1.02 \begin {gather*} -\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 \relax (x)}{b^{3}} - \frac {b d - 2 \, {\left (c d - b e\right )} x}{2 \, b^{2} x^{2}} \end {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 73, normalized size = 1.18 \begin {gather*} -\frac {\frac {d}{2\,b}+\frac {x\,\left (b\,e-c\,d\right )}{b^2}}{x^2}-\frac {2\,c\,\mathrm {atanh}\left (\frac {c\,\left (b\,e-c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (c^2\,d-b\,c\,e\right )}\right )\,\left (b\,e-c\,d\right )}{b^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)),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 0.45, size = 131, normalized size = 2.11 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________