\(\int (a+b x)^5 (a c+b c x)^n \, dx\) [1015]

Optimal result

Integrand size = 18, antiderivative size = 24 \[ \int (a+b x)^5 (a c+b c x)^n \, dx=\frac {(a c+b c x)^{6+n}}{b c^6 (6+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32} \[ \int (a+b x)^5 (a c+b c x)^n \, dx=\frac {(a c+b c x)^{n+6}}{b c^6 (n+6)} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int (a+b x)^5 (a c+b c x)^n \, dx=\frac {(a+b x)^6 (c (a+b x))^n}{b (6+n)} \]

[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))

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12

method	result	size
gosper	\(\frac {\left (b x +a \right )^{6} \left (b c x +a c \right )^{n}}{b \left (6+n \right )}\)	\(27\)
risch	\(\frac {\left (b^{6} x^{6}+6 a \,x^{5} b^{5}+15 a^{2} x^{4} b^{4}+20 a^{3} x^{3} b^{3}+15 a^{4} x^{2} b^{2}+6 a^{5} x b +a^{6}\right ) \left (c \left (b x +a \right )\right )^{n}}{b \left (6+n \right )}\)	\(79\)
parallelrisch	\(\frac {x^{6} \left (c \left (b x +a \right )\right )^{n} a \,b^{6}+6 x^{5} \left (c \left (b x +a \right )\right )^{n} a^{2} b^{5}+15 x^{4} \left (c \left (b x +a \right )\right )^{n} a^{3} b^{4}+20 x^{3} \left (c \left (b x +a \right )\right )^{n} a^{4} b^{3}+15 x^{2} \left (c \left (b x +a \right )\right )^{n} a^{5} b^{2}+6 x \left (c \left (b x +a \right )\right )^{n} a^{6} b +\left (c \left (b x +a \right )\right )^{n} a^{7}}{\left (6+n \right ) b a}\)	\(140\)
norman	\(\frac {a^{6} {\mathrm e}^{n \ln \left (b c x +a c \right )}}{b \left (6+n \right )}+\frac {b^{5} x^{6} {\mathrm e}^{n \ln \left (b c x +a c \right )}}{6+n}+\frac {6 a^{5} x \,{\mathrm e}^{n \ln \left (b c x +a c \right )}}{6+n}+\frac {6 a \,b^{4} x^{5} {\mathrm e}^{n \ln \left (b c x +a c \right )}}{6+n}+\frac {15 a^{2} b^{3} x^{4} {\mathrm e}^{n \ln \left (b c x +a c \right )}}{6+n}+\frac {20 a^{3} b^{2} x^{3} {\mathrm e}^{n \ln \left (b c x +a c \right )}}{6+n}+\frac {15 a^{4} b \,x^{2} {\mathrm e}^{n \ln \left (b c x +a c \right )}}{6+n}\)	\(181\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).

Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int (a+b x)^5 (a c+b c x)^n \, dx=\frac {{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} {\left (b c x + a c\right )}^{n}}{b n + 6 \, b} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (19) = 38\).

Time = 1.48 (sec) , antiderivative size = 212, normalized size of antiderivative = 8.83 \[ \int (a+b x)^5 (a c+b c x)^n \, dx=\begin {cases} \frac {x}{a c^{6}} & \text {for}\: b = 0 \wedge n = -6 \\a^{5} x \left (a c\right )^{n} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + x \right )}}{b c^{6}} & \text {for}\: n = -6 \\\frac {a^{6} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {6 a^{5} b x \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {15 a^{4} b^{2} x^{2} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {20 a^{3} b^{3} x^{3} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {15 a^{2} b^{4} x^{4} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {6 a b^{5} x^{5} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {b^{6} x^{6} \left (a c + b c x\right )^{n}}{b n + 6 b} & \text {otherwise} \end {cases} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (24) = 48\).

Time = 0.24 (sec) , antiderivative size = 649, normalized size of antiderivative = 27.04 \[ \int (a+b x)^5 (a c+b c x)^n \, dx=\frac {5 \, {\left (b^{2} c^{n} {\left (n + 1\right )} x^{2} + a b c^{n} n x - a^{2} c^{n}\right )} {\left (b x + a\right )}^{n} a^{4}}{{\left (n^{2} + 3 \, n + 2\right )} b} + \frac {10 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} c^{n} x^{3} + {\left (n^{2} + n\right )} a b^{2} c^{n} x^{2} - 2 \, a^{2} b c^{n} n x + 2 \, a^{3} c^{n}\right )} {\left (b x + a\right )}^{n} a^{3}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b} + \frac {{\left (b c x + a c\right )}^{n + 1} a^{5}}{b c {\left (n + 1\right )}} + \frac {10 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} c^{n} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} c^{n} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} c^{n} x^{2} + 6 \, a^{3} b c^{n} n x - 6 \, a^{4} c^{n}\right )} {\left (b x + a\right )}^{n} a^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b} + \frac {5 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} c^{n} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} c^{n} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} c^{n} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} c^{n} x^{2} - 24 \, a^{4} b c^{n} n x + 24 \, a^{5} c^{n}\right )} {\left (b x + a\right )}^{n} a}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} c^{n} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} c^{n} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} c^{n} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} c^{n} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} c^{n} x^{2} + 120 \, a^{5} b c^{n} n x - 120 \, a^{6} c^{n}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b} \]

[In]

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

[Out]

5*(b^2*c^n*(n + 1)*x^2 + a*b*c^n*n*x - a^2*c^n)*(b*x + a)^n*a^4/((n^2 + 3*n + 2)*b) + 10*((n^2 + 3*n + 2)*b^3*
c^n*x^3 + (n^2 + n)*a*b^2*c^n*x^2 - 2*a^2*b*c^n*n*x + 2*a^3*c^n)*(b*x + a)^n*a^3/((n^3 + 6*n^2 + 11*n + 6)*b)
+ (b*c*x + a*c)^(n + 1)*a^5/(b*c*(n + 1)) + 10*((n^3 + 6*n^2 + 11*n + 6)*b^4*c^n*x^4 + (n^3 + 3*n^2 + 2*n)*a*b
^3*c^n*x^3 - 3*(n^2 + n)*a^2*b^2*c^n*x^2 + 6*a^3*b*c^n*n*x - 6*a^4*c^n)*(b*x + a)^n*a^2/((n^4 + 10*n^3 + 35*n^
2 + 50*n + 24)*b) + 5*((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*c^n*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*c^
n*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*c^n*x^3 + 12*(n^2 + n)*a^3*b^2*c^n*x^2 - 24*a^4*b*c^n*n*x + 24*a^5*c^n)*
(b*x + a)^n*a/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b) + ((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n
+ 120)*b^6*c^n*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a*b^5*c^n*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^
2*b^4*c^n*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^3*c^n*x^3 - 60*(n^2 + n)*a^4*b^2*c^n*x^2 + 120*a^5*b*c^n*n*x - 12
0*a^6*c^n)*(b*x + a)^n/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (24) = 48\).

Time = 0.30 (sec) , antiderivative size = 141, normalized size of antiderivative = 5.88 \[ \int (a+b x)^5 (a c+b c x)^n \, dx=\frac {{\left (b c x + a c\right )}^{n} b^{6} x^{6} + 6 \, {\left (b c x + a c\right )}^{n} a b^{5} x^{5} + 15 \, {\left (b c x + a c\right )}^{n} a^{2} b^{4} x^{4} + 20 \, {\left (b c x + a c\right )}^{n} a^{3} b^{3} x^{3} + 15 \, {\left (b c x + a c\right )}^{n} a^{4} b^{2} x^{2} + 6 \, {\left (b c x + a c\right )}^{n} a^{5} b x + {\left (b c x + a c\right )}^{n} a^{6}}{b n + 6 \, b} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.46 \[ \int (a+b x)^5 (a c+b c x)^n \, dx={\left (a\,c+b\,c\,x\right )}^n\,\left (\frac {a^6}{b\,\left (n+6\right )}+\frac {b^5\,x^6}{n+6}+\frac {6\,a^5\,x}{n+6}+\frac {15\,a^4\,b\,x^2}{n+6}+\frac {6\,a\,b^4\,x^5}{n+6}+\frac {20\,a^3\,b^2\,x^3}{n+6}+\frac {15\,a^2\,b^3\,x^4}{n+6}\right ) \]

[In]

int((a*c + b*c*x)^n*(a + b*x)^5,x)

[Out]

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