\(\int \frac {(\frac {b c}{d}+b x)^3}{(c+d x)^3} \, dx\) [1009]

Optimal result

Integrand size = 20, antiderivative size = 8 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx=\frac {b^3 x}{d^3} \]

[Out]

b^3*x/d^3

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 8} \[ \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx=\frac {b^3 x}{d^3} \]

[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*} \text {integral}& = \frac {b^3 \int 1 \, dx}{d^3} \\ & = \frac {b^3 x}{d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx=\frac {b^3 x}{d^3} \]

[In]

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

[Out]

(b^3*x)/d^3

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {b^{3} x}{d^{3}}\)	\(9\)
risch	\(\frac {b^{3} x}{d^{3}}\)	\(9\)
norman	\(\frac {b^{3} d \,x^{3}-\frac {2 c^{3} b^{3}}{d^{2}}-\frac {3 c^{2} b^{3} x}{d}}{d^{2} \left (d x +c \right )^{2}}\)	\(44\)

[In]

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

[Out]

b^3*x/d^3

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx=\frac {b^{3} x}{d^{3}} \]

[In]

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

[Out]

b^3*x/d^3

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx=\frac {b^{3} x}{d^{3}} \]

[In]

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

[Out]

b**3*x/d**3

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx=\frac {b^{3} x}{d^{3}} \]

[In]

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

[Out]

b^3*x/d^3

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx=\frac {b^{3} x}{d^{3}} \]

[In]

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

[Out]

b^3*x/d^3

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {b c}{d}+b x\right )^3}{(c+d x)^3} \, dx=\frac {b^3\,x}{d^3} \]

[In]

int((b*x + (b*c)/d)^3/(c + d*x)^3,x)

[Out]

(b^3*x)/d^3