\(\int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx\) [1024]

Optimal result

Integrand size = 18, antiderivative size = 13 \[ \int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx=\frac {\log (a+b x)}{b c^6} \]

[Out]

ln(b*x+a)/b/c^6

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 31} \[ \int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx=\frac {\log (a+b x)}{b c^6} \]

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^6,x]

[Out]

Log[a + b*x]/(b*c^6)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{a+b x} \, dx}{c^6} \\ & = \frac {\log (a+b x)}{b c^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx=\frac {\log (a+b x)}{b c^6} \]

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^6,x]

[Out]

Log[a + b*x]/(b*c^6)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
default	\(\frac {\ln \left (b x +a \right )}{b \,c^{6}}\)	\(14\)
norman	\(\frac {\ln \left (b x +a \right )}{b \,c^{6}}\)	\(14\)
risch	\(\frac {\ln \left (b x +a \right )}{b \,c^{6}}\)	\(14\)
parallelrisch	\(\frac {\ln \left (b x +a \right )}{b \,c^{6}}\)	\(14\)

[In]

int((b*x+a)^5/(b*c*x+a*c)^6,x,method=_RETURNVERBOSE)

[Out]

ln(b*x+a)/b/c^6

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx=\frac {\log \left (b x + a\right )}{b c^{6}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^6,x, algorithm="fricas")

[Out]

log(b*x + a)/(b*c^6)

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx=\frac {\log {\left (a c^{6} + b c^{6} x \right )}}{b c^{6}} \]

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**6,x)

[Out]

log(a*c**6 + b*c**6*x)/(b*c**6)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx=\frac {\log \left (b x + a\right )}{b c^{6}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^6,x, algorithm="maxima")

[Out]

log(b*x + a)/(b*c^6)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{b c^{6}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^6,x, algorithm="giac")

[Out]

log(abs(b*x + a))/(b*c^6)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^6} \, dx=\frac {\ln \left (a+b\,x\right )}{b\,c^6} \]

[In]

int((a + b*x)^5/(a*c + b*c*x)^6,x)

[Out]

log(a + b*x)/(b*c^6)