3.11.23 \(\int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx\) [1023]

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

[Out]

x/c^5

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Rubi [A]
time = 0.00, antiderivative size = 5, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x}{c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^5,x]

[Out]

x/c^5

Rule 8

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

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])

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*}

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Mathematica [A]
time = 0.00, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c^5} \end {gather*}

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 = 0.15, size = 6, normalized size = 1.20

method	result	size
default	\(\frac {x}{c^{5}}\)	\(6\)
risch	\(\frac {x}{c^{5}}\)	\(6\)
norman	\(\frac {\frac {b^{4} x^{5}}{c}+\frac {a^{4} x}{c}+\frac {4 a \,b^{3} x^{4}}{c}+\frac {4 a^{3} b \,x^{2}}{c}+\frac {6 b^{2} a^{2} x^{3}}{c}}{c^{4} \left (b x +a \right )^{4}}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c^5

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Maxima [A]
time = 0.30, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c^{5}} \end {gather*}

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 = 0.45, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c^{5}} \end {gather*}

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 = 0.03, size = 3, normalized size = 0.60 \begin {gather*} \frac {x}{c^{5}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).
time = 2.82, size = 15, normalized size = 3.00 \begin {gather*} \frac {b c x + a c}{b c^{6}} \end {gather*}

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)

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Mupad [B]
time = 0.01, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{c^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(a*c + b*c*x)^5,x)

[Out]

x/c^5

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