3.1020 \(\int \frac{(a+b x)^5}{(a c+b c x)^2} \, dx\)

Optimal. Leaf size=17 \[ \frac{(a+b x)^4}{4 b c^2} \]

[Out]

(a + b*x)^4/(4*b*c^2)

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Rubi [A]  time = 0.0040407, antiderivative size = 17, normalized size of antiderivative = 1., 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, 32} \[ \frac{(a+b x)^4}{4 b c^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^2,x]

[Out]

(a + b*x)^4/(4*b*c^2)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^5}{(a c+b c x)^2} \, dx &=\frac{\int (a+b x)^3 \, dx}{c^2}\\ &=\frac{(a+b x)^4}{4 b c^2}\\ \end{align*}

Mathematica [A]  time = 0.0013063, size = 17, normalized size = 1. \[ \frac{(a+b x)^4}{4 b c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^2,x]

[Out]

(a + b*x)^4/(4*b*c^2)

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Maple [A]  time = 0., size = 16, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{4}}{4\,b{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^2,x)

[Out]

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

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Maxima [B]  time = 0.987722, size = 50, normalized size = 2.94 \begin{align*} \frac{b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^2,x, algorithm="maxima")

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)/c^2

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Fricas [B]  time = 1.42017, size = 77, normalized size = 4.53 \begin{align*} \frac{b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^2,x, algorithm="fricas")

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)/c^2

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Sympy [B]  time = 0.094837, size = 46, normalized size = 2.71 \begin{align*} \frac{a^{3} x}{c^{2}} + \frac{3 a^{2} b x^{2}}{2 c^{2}} + \frac{a b^{2} x^{3}}{c^{2}} + \frac{b^{3} x^{4}}{4 c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**2,x)

[Out]

a**3*x/c**2 + 3*a**2*b*x**2/(2*c**2) + a*b**2*x**3/c**2 + b**3*x**4/(4*c**2)

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Giac [A]  time = 1.07711, size = 24, normalized size = 1.41 \begin{align*} \frac{{\left (b c x + a c\right )}^{4}}{4 \, b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^2,x, algorithm="giac")

[Out]

1/4*(b*c*x + a*c)^4/(b*c^6)