\(\int \frac {A+B x}{x^3 (a+b x)} \, dx\) [181]

Optimal result

Integrand size = 16, antiderivative size = 62 \[ \int \frac {A+B x}{x^3 (a+b x)} \, dx=-\frac {A}{2 a x^2}+\frac {A b-a B}{a^2 x}+\frac {b (A b-a B) \log (x)}{a^3}-\frac {b (A b-a B) \log (a+b x)}{a^3} \]

[Out]

-1/2*A/a/x^2+(A*b-B*a)/a^2/x+b*(A*b-B*a)*ln(x)/a^3-b*(A*b-B*a)*ln(b*x+a)/a^3

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {A+B x}{x^3 (a+b x)} \, dx=\frac {b \log (x) (A b-a B)}{a^3}-\frac {b (A b-a B) \log (a+b x)}{a^3}+\frac {A b-a B}{a^2 x}-\frac {A}{2 a x^2} \]

[In]

Int[(A + B*x)/(x^3*(a + b*x)),x]

[Out]

-1/2*A/(a*x^2) + (A*b - a*B)/(a^2*x) + (b*(A*b - a*B)*Log[x])/a^3 - (b*(A*b - a*B)*Log[a + b*x])/a^3

Rule 78

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*} \text {integral}& = \int \left (\frac {A}{a x^3}+\frac {-A b+a B}{a^2 x^2}-\frac {b (-A b+a B)}{a^3 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)}\right ) \, dx \\ & = -\frac {A}{2 a x^2}+\frac {A b-a B}{a^2 x}+\frac {b (A b-a B) \log (x)}{a^3}-\frac {b (A b-a B) \log (a+b x)}{a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{x^3 (a+b x)} \, dx=\frac {-\frac {a (a A-2 A b x+2 a B x)}{x^2}+2 b (A b-a B) \log (x)+2 b (-A b+a B) \log (a+b x)}{2 a^3} \]

[In]

Integrate[(A + B*x)/(x^3*(a + b*x)),x]

[Out]

(-((a*(a*A - 2*A*b*x + 2*a*B*x))/x^2) + 2*b*(A*b - a*B)*Log[x] + 2*b*(-(A*b) + a*B)*Log[a + b*x])/(2*a^3)

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98

method	result	size
norman	\(\frac {\frac {\left (A b -B a \right ) x}{a^{2}}-\frac {A}{2 a}}{x^{2}}+\frac {b \left (A b -B a \right ) \ln \left (x \right )}{a^{3}}-\frac {b \left (A b -B a \right ) \ln \left (b x +a \right )}{a^{3}}\)	\(61\)
default	\(-\frac {A}{2 a \,x^{2}}-\frac {-A b +B a}{x \,a^{2}}+\frac {b \left (A b -B a \right ) \ln \left (x \right )}{a^{3}}-\frac {b \left (A b -B a \right ) \ln \left (b x +a \right )}{a^{3}}\)	\(62\)
risch	\(\frac {\frac {\left (A b -B a \right ) x}{a^{2}}-\frac {A}{2 a}}{x^{2}}-\frac {b^{2} \ln \left (b x +a \right ) A}{a^{3}}+\frac {b \ln \left (b x +a \right ) B}{a^{2}}+\frac {b^{2} \ln \left (-x \right ) A}{a^{3}}-\frac {b \ln \left (-x \right ) B}{a^{2}}\)	\(76\)
parallelrisch	\(\frac {2 A \ln \left (x \right ) x^{2} b^{2}-2 A \ln \left (b x +a \right ) x^{2} b^{2}-2 B \ln \left (x \right ) x^{2} a b +2 B \ln \left (b x +a \right ) x^{2} a b +2 a A b x -2 a^{2} B x -a^{2} A}{2 a^{3} x^{2}}\)	\(79\)

[In]

int((B*x+A)/x^3/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

((A*b-B*a)/a^2*x-1/2*A/a)/x^2+b*(A*b-B*a)*ln(x)/a^3-b*(A*b-B*a)*ln(b*x+a)/a^3

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x}{x^3 (a+b x)} \, dx=\frac {2 \, {\left (B a b - A b^{2}\right )} x^{2} \log \left (b x + a\right ) - 2 \, {\left (B a b - A b^{2}\right )} x^{2} \log \left (x\right ) - A a^{2} - 2 \, {\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \]

[In]

integrate((B*x+A)/x^3/(b*x+a),x, algorithm="fricas")

[Out]

1/2*(2*(B*a*b - A*b^2)*x^2*log(b*x + a) - 2*(B*a*b - A*b^2)*x^2*log(x) - A*a^2 - 2*(B*a^2 - A*a*b)*x)/(a^3*x^2
)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (53) = 106\).

Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.11 \[ \int \frac {A+B x}{x^3 (a+b x)} \, dx=\frac {- A a + x \left (2 A b - 2 B a\right )}{2 a^{2} x^{2}} - \frac {b \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{2} + B a^{2} b - a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} + \frac {b \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{2} + B a^{2} b + a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} \]

[In]

integrate((B*x+A)/x**3/(b*x+a),x)

[Out]

(-A*a + x*(2*A*b - 2*B*a))/(2*a**2*x**2) - b*(-A*b + B*a)*log(x + (-A*a*b**2 + B*a**2*b - a*b*(-A*b + B*a))/(-
2*A*b**3 + 2*B*a*b**2))/a**3 + b*(-A*b + B*a)*log(x + (-A*a*b**2 + B*a**2*b + a*b*(-A*b + B*a))/(-2*A*b**3 + 2
*B*a*b**2))/a**3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x}{x^3 (a+b x)} \, dx=\frac {{\left (B a b - A b^{2}\right )} \log \left (b x + a\right )}{a^{3}} - \frac {{\left (B a b - A b^{2}\right )} \log \left (x\right )}{a^{3}} - \frac {A a + 2 \, {\left (B a - A b\right )} x}{2 \, a^{2} x^{2}} \]

[In]

integrate((B*x+A)/x^3/(b*x+a),x, algorithm="maxima")

[Out]

(B*a*b - A*b^2)*log(b*x + a)/a^3 - (B*a*b - A*b^2)*log(x)/a^3 - 1/2*(A*a + 2*(B*a - A*b)*x)/(a^2*x^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x}{x^3 (a+b x)} \, dx=-\frac {{\left (B a b - A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac {A a^{2} + 2 \, {\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \]

[In]

integrate((B*x+A)/x^3/(b*x+a),x, algorithm="giac")

[Out]

-(B*a*b - A*b^2)*log(abs(x))/a^3 + (B*a*b^2 - A*b^3)*log(abs(b*x + a))/(a^3*b) - 1/2*(A*a^2 + 2*(B*a^2 - A*a*b
)*x)/(a^3*x^2)

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x}{x^3 (a+b x)} \, dx=-\frac {\frac {A}{2\,a}-\frac {x\,\left (A\,b-B\,a\right )}{a^2}}{x^2}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (A\,b^2-B\,a\,b\right )}\right )\,\left (A\,b-B\,a\right )}{a^3} \]

[In]

int((A + B*x)/(x^3*(a + b*x)),x)

[Out]

- (A/(2*a) - (x*(A*b - B*a))/a^2)/x^2 - (2*b*atanh((b*(A*b - B*a)*(a + 2*b*x))/(a*(A*b^2 - B*a*b)))*(A*b - B*a
))/a^3