\(\int (a+b x)^5 (a c+b c x)^2 \, dx\) [1017]

Optimal result

Integrand size = 18, antiderivative size = 17 \[ \int (a+b x)^5 (a c+b c x)^2 \, dx=\frac {c^2 (a+b x)^8}{8 b} \]

[Out]

1/8*c^2*(b*x+a)^8/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} \[ \int (a+b x)^5 (a c+b c x)^2 \, dx=\frac {c^2 (a+b x)^8}{8 b} \]

[In]

Int[(a + b*x)^5*(a*c + b*c*x)^2,x]

[Out]

(c^2*(a + b*x)^8)/(8*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^2 \int (a+b x)^7 \, dx \\ & = \frac {c^2 (a+b x)^8}{8 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int (a+b x)^5 (a c+b c x)^2 \, dx=\frac {c^2 (a+b x)^8}{8 b} \]

[In]

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

[Out]

(c^2*(a + b*x)^8)/(8*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(15)=30\).

Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.71

method	result	size
gosper	\(\frac {x \left (b^{7} x^{7}+8 a \,b^{6} x^{6}+28 a^{2} b^{5} x^{5}+56 a^{3} b^{4} x^{4}+70 a^{4} b^{3} x^{3}+56 a^{5} b^{2} x^{2}+28 a^{6} b x +8 a^{7}\right ) c^{2}}{8}\)	\(80\)
default	\(\frac {1}{8} b^{7} c^{2} x^{8}+a \,b^{6} c^{2} x^{7}+\frac {7}{2} a^{2} b^{5} c^{2} x^{6}+7 a^{3} b^{4} c^{2} x^{5}+\frac {35}{4} a^{4} b^{3} c^{2} x^{4}+7 a^{5} b^{2} c^{2} x^{3}+\frac {7}{2} a^{6} c^{2} b \,x^{2}+a^{7} c^{2} x\)	\(100\)
norman	\(\frac {1}{8} b^{7} c^{2} x^{8}+a \,b^{6} c^{2} x^{7}+\frac {7}{2} a^{2} b^{5} c^{2} x^{6}+7 a^{3} b^{4} c^{2} x^{5}+\frac {35}{4} a^{4} b^{3} c^{2} x^{4}+7 a^{5} b^{2} c^{2} x^{3}+\frac {7}{2} a^{6} c^{2} b \,x^{2}+a^{7} c^{2} x\)	\(100\)
parallelrisch	\(\frac {1}{8} b^{7} c^{2} x^{8}+a \,b^{6} c^{2} x^{7}+\frac {7}{2} a^{2} b^{5} c^{2} x^{6}+7 a^{3} b^{4} c^{2} x^{5}+\frac {35}{4} a^{4} b^{3} c^{2} x^{4}+7 a^{5} b^{2} c^{2} x^{3}+\frac {7}{2} a^{6} c^{2} b \,x^{2}+a^{7} c^{2} x\)	\(100\)
risch	\(\frac {b^{7} c^{2} x^{8}}{8}+a \,b^{6} c^{2} x^{7}+\frac {7 a^{2} b^{5} c^{2} x^{6}}{2}+7 a^{3} b^{4} c^{2} x^{5}+\frac {35 a^{4} b^{3} c^{2} x^{4}}{4}+7 a^{5} b^{2} c^{2} x^{3}+\frac {7 a^{6} c^{2} b \,x^{2}}{2}+a^{7} c^{2} x +\frac {c^{2} a^{8}}{8 b}\)	\(111\)

[In]

int((b*x+a)^5*(b*c*x+a*c)^2,x,method=_RETURNVERBOSE)

[Out]

1/8*x*(b^7*x^7+8*a*b^6*x^6+28*a^2*b^5*x^5+56*a^3*b^4*x^4+70*a^4*b^3*x^3+56*a^5*b^2*x^2+28*a^6*b*x+8*a^7)*c^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 5.82 \[ \int (a+b x)^5 (a c+b c x)^2 \, dx=\frac {1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \]

[In]

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

[Out]

1/8*b^7*c^2*x^8 + a*b^6*c^2*x^7 + 7/2*a^2*b^5*c^2*x^6 + 7*a^3*b^4*c^2*x^5 + 35/4*a^4*b^3*c^2*x^4 + 7*a^5*b^2*c
^2*x^3 + 7/2*a^6*b*c^2*x^2 + a^7*c^2*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (12) = 24\).

Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 6.47 \[ \int (a+b x)^5 (a c+b c x)^2 \, dx=a^{7} c^{2} x + \frac {7 a^{6} b c^{2} x^{2}}{2} + 7 a^{5} b^{2} c^{2} x^{3} + \frac {35 a^{4} b^{3} c^{2} x^{4}}{4} + 7 a^{3} b^{4} c^{2} x^{5} + \frac {7 a^{2} b^{5} c^{2} x^{6}}{2} + a b^{6} c^{2} x^{7} + \frac {b^{7} c^{2} x^{8}}{8} \]

[In]

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

[Out]

a**7*c**2*x + 7*a**6*b*c**2*x**2/2 + 7*a**5*b**2*c**2*x**3 + 35*a**4*b**3*c**2*x**4/4 + 7*a**3*b**4*c**2*x**5
+ 7*a**2*b**5*c**2*x**6/2 + a*b**6*c**2*x**7 + b**7*c**2*x**8/8

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 5.82 \[ \int (a+b x)^5 (a c+b c x)^2 \, dx=\frac {1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \]

[In]

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

[Out]

1/8*b^7*c^2*x^8 + a*b^6*c^2*x^7 + 7/2*a^2*b^5*c^2*x^6 + 7*a^3*b^4*c^2*x^5 + 35/4*a^4*b^3*c^2*x^4 + 7*a^5*b^2*c
^2*x^3 + 7/2*a^6*b*c^2*x^2 + a^7*c^2*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 5.82 \[ \int (a+b x)^5 (a c+b c x)^2 \, dx=\frac {1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \]

[In]

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

[Out]

1/8*b^7*c^2*x^8 + a*b^6*c^2*x^7 + 7/2*a^2*b^5*c^2*x^6 + 7*a^3*b^4*c^2*x^5 + 35/4*a^4*b^3*c^2*x^4 + 7*a^5*b^2*c
^2*x^3 + 7/2*a^6*b*c^2*x^2 + a^7*c^2*x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 5.82 \[ \int (a+b x)^5 (a c+b c x)^2 \, dx=a^7\,c^2\,x+\frac {7\,a^6\,b\,c^2\,x^2}{2}+7\,a^5\,b^2\,c^2\,x^3+\frac {35\,a^4\,b^3\,c^2\,x^4}{4}+7\,a^3\,b^4\,c^2\,x^5+\frac {7\,a^2\,b^5\,c^2\,x^6}{2}+a\,b^6\,c^2\,x^7+\frac {b^7\,c^2\,x^8}{8} \]

[In]

int((a*c + b*c*x)^2*(a + b*x)^5,x)

[Out]

a^7*c^2*x + (b^7*c^2*x^8)/8 + (7*a^6*b*c^2*x^2)/2 + a*b^6*c^2*x^7 + 7*a^5*b^2*c^2*x^3 + (35*a^4*b^3*c^2*x^4)/4
+ 7*a^3*b^4*c^2*x^5 + (7*a^2*b^5*c^2*x^6)/2