3.1001 \(\int \frac{(a+b x)^3}{(\frac{a d}{b}+d x)^3} \, dx\)

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

[Out]

(b^3*x)/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.001698, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 8} \[ \frac{b^3 x}{d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/((a*d)/b + d*x)^3,x]

[Out]

(b^3*x)/d^3

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

Mathematica [A]  time = 0.0003215, size = 8, normalized size = 1. \[ \frac{b^3 x}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*x)/d^3

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 9, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}x}{{d}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x/d^3

________________________________________________________________________________________

Maxima [A]  time = 1.00235, size = 11, normalized size = 1.38 \begin{align*} \frac{b^{3} x}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x/d^3

________________________________________________________________________________________

Fricas [A]  time = 1.50043, size = 15, normalized size = 1.88 \begin{align*} \frac{b^{3} x}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x/d^3

________________________________________________________________________________________

Sympy [A]  time = 0.086447, size = 7, normalized size = 0.88 \begin{align*} \frac{b^{3} x}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*x/d**3

________________________________________________________________________________________

Giac [A]  time = 1.06505, size = 11, normalized size = 1.38 \begin{align*} \frac{b^{3} x}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x/d^3