3.10.40 \(\int \frac {(\frac {b c}{d}+b x)^3}{(c+d x)^3} \, dx\)

Optimal. Leaf size=8 \[ \frac {b^3 x}{d^3} \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 8} \begin {gather*} \frac {b^3 x}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b*c)/d + b*x)^3/(c + d*x)^3,x]

[Out]

(b^3*x)/d^3

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 {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx &=\frac {b^3 \int 1 \, dx}{d^3}\\ &=\frac {b^3 x}{d^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {b^3 x}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b*c)/d + b*x)^3/(c + d*x)^3,x]

[Out]

(b^3*x)/d^3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((b*c)/d + b*x)^3/(c + d*x)^3,x]

[Out]

IntegrateAlgebraic[((b*c)/d + b*x)^3/(c + d*x)^3, x]

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fricas [A]  time = 1.19, size = 8, normalized size = 1.00 \begin {gather*} \frac {b^{3} x}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

b^3*x/d^3

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giac [A]  time = 1.05, size = 8, normalized size = 1.00 \begin {gather*} \frac {b^{3} x}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^3/(d*x+c)^3,x, algorithm="giac")

[Out]

b^3*x/d^3

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maple [A]  time = 0.00, size = 9, normalized size = 1.12 \begin {gather*} \frac {b^{3} x}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*c/d+b*x)^3/(d*x+c)^3,x)

[Out]

b^3/d^3*x

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maxima [A]  time = 1.33, size = 8, normalized size = 1.00 \begin {gather*} \frac {b^{3} x}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

b^3*x/d^3

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mupad [B]  time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} \frac {b^3\,x}{d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + (b*c)/d)^3/(c + d*x)^3,x)

[Out]

(b^3*x)/d^3

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sympy [A]  time = 0.10, size = 7, normalized size = 0.88 \begin {gather*} \frac {b^{3} x}{d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)**3/(d*x+c)**3,x)

[Out]

b**3*x/d**3

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