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

Optimal. Leaf size=43 \[ -\frac {\log (x) (b c-a d)}{a^2}+\frac {(b c-a d) \log (a+b x)}{a^2}-\frac {c}{a x} \]

[Out]

-c/a/x-(-a*d+b*c)*ln(x)/a^2+(-a*d+b*c)*ln(b*x+a)/a^2

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Rubi [A]  time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ -\frac {\log (x) (b c-a d)}{a^2}+\frac {(b c-a d) \log (a+b x)}{a^2}-\frac {c}{a x} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(x^2*(a + b*x)),x]

[Out]

-(c/(a*x)) - ((b*c - a*d)*Log[x])/a^2 + ((b*c - a*d)*Log[a + b*x])/a^2

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 42, normalized size = 0.98 \[ \frac {\log (x) (a d-b c)}{a^2}+\frac {(b c-a d) \log (a+b x)}{a^2}-\frac {c}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c/(a*x)) + ((-(b*c) + a*d)*Log[x])/a^2 + ((b*c - a*d)*Log[a + b*x])/a^2

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fricas [A]  time = 1.20, size = 41, normalized size = 0.95 \[ \frac {{\left (b c - a d\right )} x \log \left (b x + a\right ) - {\left (b c - a d\right )} x \log \relax (x) - a c}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*c - a*d)*x*log(b*x + a) - (b*c - a*d)*x*log(x) - a*c)/(a^2*x)

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giac [A]  time = 0.88, size = 51, normalized size = 1.19 \[ -\frac {{\left (b c - a d\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {c}{a x} + \frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | b x + a \right |}\right )}{a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*c - a*d)*log(abs(x))/a^2 - c/(a*x) + (b^2*c - a*b*d)*log(abs(b*x + a))/(a^2*b)

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maple [A]  time = 0.01, size = 51, normalized size = 1.19 \[ \frac {d \ln \relax (x )}{a}-\frac {d \ln \left (b x +a \right )}{a}-\frac {b c \ln \relax (x )}{a^{2}}+\frac {b c \ln \left (b x +a \right )}{a^{2}}-\frac {c}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 43, normalized size = 1.00 \[ \frac {{\left (b c - a d\right )} \log \left (b x + a\right )}{a^{2}} - \frac {{\left (b c - a d\right )} \log \relax (x)}{a^{2}} - \frac {c}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c - a*d)*log(b*x + a)/a^2 - (b*c - a*d)*log(x)/a^2 - c/(a*x)

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mupad [B]  time = 0.33, size = 33, normalized size = 0.77 \[ -\frac {c}{a\,x}-\frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )\,\left (a\,d-b\,c\right )}{a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(x^2*(a + b*x)),x)

[Out]

- c/(a*x) - (2*atanh((2*b*x)/a + 1)*(a*d - b*c))/a^2

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sympy [B]  time = 0.39, size = 95, normalized size = 2.21 \[ - \frac {c}{a x} + \frac {\left (a d - b c\right ) \log {\left (x + \frac {a^{2} d - a b c - a \left (a d - b c\right )}{2 a b d - 2 b^{2} c} \right )}}{a^{2}} - \frac {\left (a d - b c\right ) \log {\left (x + \frac {a^{2} d - a b c + a \left (a d - b c\right )}{2 a b d - 2 b^{2} c} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/(a*x) + (a*d - b*c)*log(x + (a**2*d - a*b*c - a*(a*d - b*c))/(2*a*b*d - 2*b**2*c))/a**2 - (a*d - b*c)*log(x
+ (a**2*d - a*b*c + a*(a*d - b*c))/(2*a*b*d - 2*b**2*c))/a**2

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