3.1024 \(\int \frac{(a+b x)^5}{(a c+b c x)^6} \, dx\)

Optimal. Leaf size=13 \[ \frac{\log (a+b x)}{b c^6} \]

[Out]

Log[a + b*x]/(b*c^6)

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Rubi [A]  time = 0.0037264, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {21, 31} \[ \frac{\log (a+b x)}{b c^6} \]

Antiderivative was successfully verified.

[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*} \int \frac{(a+b x)^5}{(a c+b c x)^6} \, dx &=\frac{\int \frac{1}{a+b x} \, dx}{c^6}\\ &=\frac{\log (a+b x)}{b c^6}\\ \end{align*}

Mathematica [A]  time = 0.0018879, size = 13, normalized size = 1. \[ \frac{\log (a+b x)}{b c^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*x]/(b*c^6)

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Maple [A]  time = 0.001, size = 14, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{b{c}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02807, size = 18, normalized size = 1.38 \begin{align*} \frac{\log \left (b x + a\right )}{b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + a)/(b*c^6)

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Fricas [A]  time = 1.52776, size = 30, normalized size = 2.31 \begin{align*} \frac{\log \left (b x + a\right )}{b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + a)/(b*c^6)

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Sympy [A]  time = 0.107435, size = 17, normalized size = 1.31 \begin{align*} \frac{\log{\left (a c^{6} + b c^{6} x \right )}}{b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.05024, size = 19, normalized size = 1.46 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(b*x + a))/(b*c^6)