3.1015 \(\int (a+b x)^5 (a c+b c x)^n \, dx\)

Optimal. Leaf size=24 \[ \frac{(a c+b c x)^{n+6}}{b c^6 (n+6)} \]

[Out]

(a*c + b*c*x)^(6 + n)/(b*c^6*(6 + n))

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Rubi [A]  time = 0.0088863, antiderivative size = 24, 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} \[ \frac{(a c+b c x)^{n+6}}{b c^6 (n+6)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*(a*c + b*c*x)^n,x]

[Out]

(a*c + b*c*x)^(6 + n)/(b*c^6*(6 + n))

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 (a+b x)^5 (a c+b c x)^n \, dx &=\frac{\int (a c+b c x)^{5+n} \, dx}{c^5}\\ &=\frac{(a c+b c x)^{6+n}}{b c^6 (6+n)}\\ \end{align*}

Mathematica [A]  time = 0.0196693, size = 25, normalized size = 1.04 \[ \frac{(a+b x)^6 (c (a+b x))^n}{b (n+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^6*(c*(a + b*x))^n)/(b*(6 + n))

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Maple [A]  time = 0.002, size = 27, normalized size = 1.1 \begin{align*}{\frac{ \left ( bx+a \right ) ^{6} \left ( bcx+ac \right ) ^{n}}{b \left ( 6+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^6/b/(6+n)*(b*c*x+a*c)^n

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.6815, size = 165, normalized size = 6.88 \begin{align*} \frac{{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )}{\left (b c x + a c\right )}^{n}}{b n + 6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^6*x^6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a^6)*(b*c*x + a*c)^n/(
b*n + 6*b)

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Sympy [A]  time = 1.76593, size = 212, normalized size = 8.83 \begin{align*} \begin{cases} \frac{x}{a c^{6}} & \text{for}\: b = 0 \wedge n = -6 \\a^{5} x \left (a c\right )^{n} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + x \right )}}{b c^{6}} & \text{for}\: n = -6 \\\frac{a^{6} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac{6 a^{5} b x \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac{15 a^{4} b^{2} x^{2} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac{20 a^{3} b^{3} x^{3} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac{15 a^{2} b^{4} x^{4} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac{6 a b^{5} x^{5} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac{b^{6} x^{6} \left (a c + b c x\right )^{n}}{b n + 6 b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(a*c**6), Eq(b, 0) & Eq(n, -6)), (a**5*x*(a*c)**n, Eq(b, 0)), (log(a/b + x)/(b*c**6), Eq(n, -6)),
(a**6*(a*c + b*c*x)**n/(b*n + 6*b) + 6*a**5*b*x*(a*c + b*c*x)**n/(b*n + 6*b) + 15*a**4*b**2*x**2*(a*c + b*c*x
)**n/(b*n + 6*b) + 20*a**3*b**3*x**3*(a*c + b*c*x)**n/(b*n + 6*b) + 15*a**2*b**4*x**4*(a*c + b*c*x)**n/(b*n +
6*b) + 6*a*b**5*x**5*(a*c + b*c*x)**n/(b*n + 6*b) + b**6*x**6*(a*c + b*c*x)**n/(b*n + 6*b), True))

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Giac [B]  time = 1.07213, size = 190, normalized size = 7.92 \begin{align*} \frac{{\left (b c x + a c\right )}^{n} b^{6} x^{6} + 6 \,{\left (b c x + a c\right )}^{n} a b^{5} x^{5} + 15 \,{\left (b c x + a c\right )}^{n} a^{2} b^{4} x^{4} + 20 \,{\left (b c x + a c\right )}^{n} a^{3} b^{3} x^{3} + 15 \,{\left (b c x + a c\right )}^{n} a^{4} b^{2} x^{2} + 6 \,{\left (b c x + a c\right )}^{n} a^{5} b x +{\left (b c x + a c\right )}^{n} a^{6}}{b n + 6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*c*x + a*c)^n*b^6*x^6 + 6*(b*c*x + a*c)^n*a*b^5*x^5 + 15*(b*c*x + a*c)^n*a^2*b^4*x^4 + 20*(b*c*x + a*c)^n*a
^3*b^3*x^3 + 15*(b*c*x + a*c)^n*a^4*b^2*x^2 + 6*(b*c*x + a*c)^n*a^5*b*x + (b*c*x + a*c)^n*a^6)/(b*n + 6*b)