\(\int (a+b x)^5 (a c+b c x)^3 \, dx\) [1016]

Optimal result

Integrand size = 18, antiderivative size = 17 \[ \int (a+b x)^5 (a c+b c x)^3 \, dx=\frac {c^3 (a+b x)^9}{9 b} \]

[Out]

1/9*c^3*(b*x+a)^9/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, 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} \[ \int (a+b x)^5 (a c+b c x)^3 \, dx=\frac {c^3 (a+b x)^9}{9 b} \]

[In]

Int[(a + b*x)^5*(a*c + b*c*x)^3,x]

[Out]

(c^3*(a + b*x)^9)/(9*b)

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}& = c^3 \int (a+b x)^8 \, dx \\ & = \frac {c^3 (a+b x)^9}{9 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int (a+b x)^5 (a c+b c x)^3 \, dx=\frac {c^3 (a+b x)^9}{9 b} \]

[In]

Integrate[(a + b*x)^5*(a*c + b*c*x)^3,x]

[Out]

(c^3*(a + b*x)^9)/(9*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(15)=30\).

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 5.35

method	result	size
gosper	\(\frac {x \left (b^{8} x^{8}+9 a \,x^{7} b^{7}+36 a^{2} x^{6} b^{6}+84 a^{3} x^{5} b^{5}+126 a^{4} x^{4} b^{4}+126 a^{5} b^{3} x^{3}+84 a^{6} x^{2} b^{2}+36 a^{7} x b +9 a^{8}\right ) c^{3}}{9}\)	\(91\)
default	\(\frac {1}{9} b^{8} c^{3} x^{9}+a \,b^{7} c^{3} x^{8}+4 a^{2} b^{6} c^{3} x^{7}+\frac {28}{3} a^{3} b^{5} c^{3} x^{6}+14 a^{4} b^{4} c^{3} x^{5}+14 a^{5} b^{3} c^{3} x^{4}+\frac {28}{3} a^{6} c^{3} b^{2} x^{3}+4 a^{7} c^{3} b \,x^{2}+a^{8} c^{3} x\)	\(114\)
norman	\(\frac {1}{9} b^{8} c^{3} x^{9}+a \,b^{7} c^{3} x^{8}+4 a^{2} b^{6} c^{3} x^{7}+\frac {28}{3} a^{3} b^{5} c^{3} x^{6}+14 a^{4} b^{4} c^{3} x^{5}+14 a^{5} b^{3} c^{3} x^{4}+\frac {28}{3} a^{6} c^{3} b^{2} x^{3}+4 a^{7} c^{3} b \,x^{2}+a^{8} c^{3} x\)	\(114\)
parallelrisch	\(\frac {1}{9} b^{8} c^{3} x^{9}+a \,b^{7} c^{3} x^{8}+4 a^{2} b^{6} c^{3} x^{7}+\frac {28}{3} a^{3} b^{5} c^{3} x^{6}+14 a^{4} b^{4} c^{3} x^{5}+14 a^{5} b^{3} c^{3} x^{4}+\frac {28}{3} a^{6} c^{3} b^{2} x^{3}+4 a^{7} c^{3} b \,x^{2}+a^{8} c^{3} x\)	\(114\)
risch	\(\frac {b^{8} c^{3} x^{9}}{9}+a \,b^{7} c^{3} x^{8}+4 a^{2} b^{6} c^{3} x^{7}+\frac {28 a^{3} b^{5} c^{3} x^{6}}{3}+14 a^{4} b^{4} c^{3} x^{5}+14 a^{5} b^{3} c^{3} x^{4}+\frac {28 a^{6} c^{3} b^{2} x^{3}}{3}+4 a^{7} c^{3} b \,x^{2}+a^{8} c^{3} x +\frac {c^{3} a^{9}}{9 b}\)	\(125\)

[In]

int((b*x+a)^5*(b*c*x+a*c)^3,x,method=_RETURNVERBOSE)

[Out]

1/9*x*(b^8*x^8+9*a*b^7*x^7+36*a^2*b^6*x^6+84*a^3*b^5*x^5+126*a^4*b^4*x^4+126*a^5*b^3*x^3+84*a^6*b^2*x^2+36*a^7
*b*x+9*a^8)*c^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 6.65 \[ \int (a+b x)^5 (a c+b c x)^3 \, dx=\frac {1}{9} \, b^{8} c^{3} x^{9} + a b^{7} c^{3} x^{8} + 4 \, a^{2} b^{6} c^{3} x^{7} + \frac {28}{3} \, a^{3} b^{5} c^{3} x^{6} + 14 \, a^{4} b^{4} c^{3} x^{5} + 14 \, a^{5} b^{3} c^{3} x^{4} + \frac {28}{3} \, a^{6} b^{2} c^{3} x^{3} + 4 \, a^{7} b c^{3} x^{2} + a^{8} c^{3} x \]

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

1/9*b^8*c^3*x^9 + a*b^7*c^3*x^8 + 4*a^2*b^6*c^3*x^7 + 28/3*a^3*b^5*c^3*x^6 + 14*a^4*b^4*c^3*x^5 + 14*a^5*b^3*c
^3*x^4 + 28/3*a^6*b^2*c^3*x^3 + 4*a^7*b*c^3*x^2 + a^8*c^3*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (12) = 24\).

Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 7.29 \[ \int (a+b x)^5 (a c+b c x)^3 \, dx=a^{8} c^{3} x + 4 a^{7} b c^{3} x^{2} + \frac {28 a^{6} b^{2} c^{3} x^{3}}{3} + 14 a^{5} b^{3} c^{3} x^{4} + 14 a^{4} b^{4} c^{3} x^{5} + \frac {28 a^{3} b^{5} c^{3} x^{6}}{3} + 4 a^{2} b^{6} c^{3} x^{7} + a b^{7} c^{3} x^{8} + \frac {b^{8} c^{3} x^{9}}{9} \]

[In]

integrate((b*x+a)**5*(b*c*x+a*c)**3,x)

[Out]

a**8*c**3*x + 4*a**7*b*c**3*x**2 + 28*a**6*b**2*c**3*x**3/3 + 14*a**5*b**3*c**3*x**4 + 14*a**4*b**4*c**3*x**5
+ 28*a**3*b**5*c**3*x**6/3 + 4*a**2*b**6*c**3*x**7 + a*b**7*c**3*x**8 + b**8*c**3*x**9/9

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 6.65 \[ \int (a+b x)^5 (a c+b c x)^3 \, dx=\frac {1}{9} \, b^{8} c^{3} x^{9} + a b^{7} c^{3} x^{8} + 4 \, a^{2} b^{6} c^{3} x^{7} + \frac {28}{3} \, a^{3} b^{5} c^{3} x^{6} + 14 \, a^{4} b^{4} c^{3} x^{5} + 14 \, a^{5} b^{3} c^{3} x^{4} + \frac {28}{3} \, a^{6} b^{2} c^{3} x^{3} + 4 \, a^{7} b c^{3} x^{2} + a^{8} c^{3} x \]

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

1/9*b^8*c^3*x^9 + a*b^7*c^3*x^8 + 4*a^2*b^6*c^3*x^7 + 28/3*a^3*b^5*c^3*x^6 + 14*a^4*b^4*c^3*x^5 + 14*a^5*b^3*c
^3*x^4 + 28/3*a^6*b^2*c^3*x^3 + 4*a^7*b*c^3*x^2 + a^8*c^3*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 6.65 \[ \int (a+b x)^5 (a c+b c x)^3 \, dx=\frac {1}{9} \, b^{8} c^{3} x^{9} + a b^{7} c^{3} x^{8} + 4 \, a^{2} b^{6} c^{3} x^{7} + \frac {28}{3} \, a^{3} b^{5} c^{3} x^{6} + 14 \, a^{4} b^{4} c^{3} x^{5} + 14 \, a^{5} b^{3} c^{3} x^{4} + \frac {28}{3} \, a^{6} b^{2} c^{3} x^{3} + 4 \, a^{7} b c^{3} x^{2} + a^{8} c^{3} x \]

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^3,x, algorithm="giac")

[Out]

1/9*b^8*c^3*x^9 + a*b^7*c^3*x^8 + 4*a^2*b^6*c^3*x^7 + 28/3*a^3*b^5*c^3*x^6 + 14*a^4*b^4*c^3*x^5 + 14*a^5*b^3*c
^3*x^4 + 28/3*a^6*b^2*c^3*x^3 + 4*a^7*b*c^3*x^2 + a^8*c^3*x

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 6.65 \[ \int (a+b x)^5 (a c+b c x)^3 \, dx=a^8\,c^3\,x+4\,a^7\,b\,c^3\,x^2+\frac {28\,a^6\,b^2\,c^3\,x^3}{3}+14\,a^5\,b^3\,c^3\,x^4+14\,a^4\,b^4\,c^3\,x^5+\frac {28\,a^3\,b^5\,c^3\,x^6}{3}+4\,a^2\,b^6\,c^3\,x^7+a\,b^7\,c^3\,x^8+\frac {b^8\,c^3\,x^9}{9} \]

[In]

int((a*c + b*c*x)^3*(a + b*x)^5,x)

[Out]

a^8*c^3*x + (b^8*c^3*x^9)/9 + 4*a^7*b*c^3*x^2 + a*b^7*c^3*x^8 + (28*a^6*b^2*c^3*x^3)/3 + 14*a^5*b^3*c^3*x^4 +
14*a^4*b^4*c^3*x^5 + (28*a^3*b^5*c^3*x^6)/3 + 4*a^2*b^6*c^3*x^7