3.1023 \(\int \frac{(a+b x)^5}{(a c+b c x)^5} \, dx\)

Optimal. Leaf size=5 \[ \frac{x}{c^5} \]

[Out]

x/c^5

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Rubi [A]  time = 0.0013297, antiderivative size = 5, 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, 8} \[ \frac{x}{c^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^5,x]

[Out]

x/c^5

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0003026, size = 5, normalized size = 1. \[ \frac{x}{c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/c^5

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Maple [A]  time = 0.002, size = 6, normalized size = 1.2 \begin{align*}{\frac{x}{{c}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^5

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Maxima [A]  time = 1.03508, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^5

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Fricas [A]  time = 1.45602, size = 9, normalized size = 1.8 \begin{align*} \frac{x}{c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^5

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Sympy [A]  time = 0.098591, size = 3, normalized size = 0.6 \begin{align*} \frac{x}{c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c**5

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Giac [B]  time = 1.07033, size = 20, normalized size = 4. \begin{align*} \frac{b c x + a c}{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)^5,x, algorithm="giac")

[Out]

(b*c*x + a*c)/(b*c^6)