3.1022 \(\int \frac{(a+b x)^5}{(a c+b c x)^4} \, dx\)

Optimal. Leaf size=18 \[ \frac{a x}{c^4}+\frac{b x^2}{2 c^4} \]

[Out]

(a*x)/c^4 + (b*x^2)/(2*c^4)

________________________________________________________________________________________

Rubi [A]  time = 0.004596, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {21} \[ \frac{a x}{c^4}+\frac{b x^2}{2 c^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^4,x]

[Out]

(a*x)/c^4 + (b*x^2)/(2*c^4)

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)^4} \, dx &=\frac{\int (a+b x) \, dx}{c^4}\\ &=\frac{a x}{c^4}+\frac{b x^2}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0006691, size = 16, normalized size = 0.89 \[ \frac{a x+\frac{b x^2}{2}}{c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^4,x]

[Out]

(a*x + (b*x^2)/2)/c^4

________________________________________________________________________________________

Maple [A]  time = 0., size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{4}} \left ( ax+{\frac{b{x}^{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^4,x)

[Out]

1/c^4*(a*x+1/2*b*x^2)

________________________________________________________________________________________

Maxima [A]  time = 1.02607, size = 20, normalized size = 1.11 \begin{align*} \frac{b x^{2} + 2 \, a x}{2 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

1/2*(b*x^2 + 2*a*x)/c^4

________________________________________________________________________________________

Fricas [A]  time = 1.56144, size = 34, normalized size = 1.89 \begin{align*} \frac{b x^{2} + 2 \, a x}{2 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

1/2*(b*x^2 + 2*a*x)/c^4

________________________________________________________________________________________

Sympy [A]  time = 0.144051, size = 15, normalized size = 0.83 \begin{align*} \frac{a x}{c^{4}} + \frac{b x^{2}}{2 c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**4,x)

[Out]

a*x/c**4 + b*x**2/(2*c**4)

________________________________________________________________________________________

Giac [A]  time = 1.0618, size = 20, normalized size = 1.11 \begin{align*} \frac{b x^{2} + 2 \, a x}{2 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^4,x, algorithm="giac")

[Out]

1/2*(b*x^2 + 2*a*x)/c^4