\(\int \frac {1}{x^3 (a+b x)} \, dx\) [48]

Optimal result

Integrand size = 11, antiderivative size = 37 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}-\frac {b^2 \log \left (\frac {a+b x}{x}\right )}{a^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^3 (a+b x)} \, dx=\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \]

[In]

Int[1/(x^3*(a + b*x)),x]

[Out]

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx \\ & = -\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3} \]

[In]

Integrate[1/(x^3*(a + b*x)),x]

[Out]

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

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {1}{2 a \,x^{2}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}+\frac {b}{a^{2} x}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}\)	\(41\)
norman	\(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{x^{2}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}\)	\(41\)
risch	\(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{x^{2}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}+\frac {b^{2} \ln \left (-x \right )}{a^{3}}\)	\(43\)
parallelrisch	\(\frac {2 b^{2} \ln \left (x \right ) x^{2}-2 \ln \left (b x +a \right ) x^{2} b^{2}+2 b a x -a^{2}}{2 a^{3} x^{2}}\)	\(44\)

[In]

int(1/x^3/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \]

[In]

integrate(1/x^3/(b*x+a),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 (a+b x)} \, dx=\frac {- a + 2 b x}{2 a^{2} x^{2}} + \frac {b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \]

[In]

integrate(1/x**3/(b*x+a),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {b^{2} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} \log \left (x\right )}{a^{3}} + \frac {2 \, b x - a}{2 \, a^{2} x^{2}} \]

[In]

integrate(1/x^3/(b*x+a),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \]

[In]

integrate(1/x^3/(b*x+a),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {\frac {a^2}{2}-a\,b\,x}{a^3\,x^2}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3} \]

[In]

int(1/(x^3*(a + b*x)),x)

[Out]

- (a^2/2 - a*b*x)/(a^3*x^2) - (2*b^2*atanh((2*b*x)/a + 1))/a^3