3.1019 \(\int \frac{(a+b x)^5}{a c+b c x} \, dx\)

Optimal. Leaf size=17 \[ \frac{(a+b x)^5}{5 b c} \]

[Out]

(a + b*x)^5/(5*b*c)

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Rubi [A]  time = 0.0035367, 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} \[ \frac{(a+b x)^5}{5 b c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x),x]

[Out]

(a + b*x)^5/(5*b*c)

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

Mathematica [A]  time = 0.0014878, size = 17, normalized size = 1. \[ \frac{(a+b x)^5}{5 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^5/(5*b*c)

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{5}}{5\,bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b*x+a)^5/b/c

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Maxima [B]  time = 1.01414, size = 65, normalized size = 3.82 \begin{align*} \frac{b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)/c

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Fricas [B]  time = 1.41319, size = 99, normalized size = 5.82 \begin{align*} \frac{b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)/c

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Sympy [B]  time = 0.089336, size = 51, normalized size = 3. \begin{align*} \frac{a^{4} x}{c} + \frac{2 a^{3} b x^{2}}{c} + \frac{2 a^{2} b^{2} x^{3}}{c} + \frac{a b^{3} x^{4}}{c} + \frac{b^{4} x^{5}}{5 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*x/c + 2*a**3*b*x**2/c + 2*a**2*b**2*x**3/c + a*b**3*x**4/c + b**4*x**5/(5*c)

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Giac [B]  time = 1.08084, size = 65, normalized size = 3.82 \begin{align*} \frac{b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x}{5 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)/c