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

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

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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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{(a c+b c x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.22, 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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giac [B]  time = 1.11, 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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maple [A]  time = 0.00, size = 6, normalized size = 1.20 \begin {gather*} \frac {x}{c^{5}} \end {gather*}

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 = 1.40, 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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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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sympy [A]  time = 0.11, 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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