\(\int (a+b x) (a c-b c x)^5 \, dx\) [31]

Optimal result

Integrand size = 17, antiderivative size = 38 \[ \int (a+b x) (a c-b c x)^5 \, dx=-\frac {a c^5 (a-b x)^6}{3 b}+\frac {c^5 (a-b x)^7}{7 b} \]

[Out]

-1/3*a*c^5*(-b*x+a)^6/b+1/7*c^5*(-b*x+a)^7/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int (a+b x) (a c-b c x)^5 \, dx=\frac {c^5 (a-b x)^7}{7 b}-\frac {a c^5 (a-b x)^6}{3 b} \]

[In]

Int[(a + b*x)*(a*c - b*c*x)^5,x]

[Out]

-1/3*(a*c^5*(a - b*x)^6)/b + (c^5*(a - b*x)^7)/(7*b)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (2 a (a c-b c x)^5-\frac {(a c-b c x)^6}{c}\right ) \, dx \\ & = -\frac {a c^5 (a-b x)^6}{3 b}+\frac {c^5 (a-b x)^7}{7 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.68 \[ \int (a+b x) (a c-b c x)^5 \, dx=c^5 \left (a^6 x-2 a^5 b x^2+\frac {5}{3} a^4 b^2 x^3-a^2 b^4 x^5+\frac {2}{3} a b^5 x^6-\frac {b^6 x^7}{7}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.55

method	result	size
gosper	\(\frac {x \left (-3 b^{6} x^{6}+14 a \,x^{5} b^{5}-21 a^{2} x^{4} b^{4}+35 a^{4} x^{2} b^{2}-42 a^{5} x b +21 a^{6}\right ) c^{5}}{21}\)	\(59\)
default	\(-\frac {1}{7} b^{6} c^{5} x^{7}+\frac {2}{3} a \,b^{5} c^{5} x^{6}-a^{2} b^{4} c^{5} x^{5}+\frac {5}{3} a^{4} b^{2} c^{5} x^{3}-2 a^{5} b \,c^{5} x^{2}+a^{6} c^{5} x\)	\(73\)
norman	\(-\frac {1}{7} b^{6} c^{5} x^{7}+\frac {2}{3} a \,b^{5} c^{5} x^{6}-a^{2} b^{4} c^{5} x^{5}+\frac {5}{3} a^{4} b^{2} c^{5} x^{3}-2 a^{5} b \,c^{5} x^{2}+a^{6} c^{5} x\)	\(73\)
risch	\(-\frac {1}{7} b^{6} c^{5} x^{7}+\frac {2}{3} a \,b^{5} c^{5} x^{6}-a^{2} b^{4} c^{5} x^{5}+\frac {5}{3} a^{4} b^{2} c^{5} x^{3}-2 a^{5} b \,c^{5} x^{2}+a^{6} c^{5} x\)	\(73\)
parallelrisch	\(-\frac {1}{7} b^{6} c^{5} x^{7}+\frac {2}{3} a \,b^{5} c^{5} x^{6}-a^{2} b^{4} c^{5} x^{5}+\frac {5}{3} a^{4} b^{2} c^{5} x^{3}-2 a^{5} b \,c^{5} x^{2}+a^{6} c^{5} x\)	\(73\)

[In]

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

[Out]

1/21*x*(-3*b^6*x^6+14*a*b^5*x^5-21*a^2*b^4*x^4+35*a^4*b^2*x^2-42*a^5*b*x+21*a^6)*c^5

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.89 \[ \int (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{7} \, b^{6} c^{5} x^{7} + \frac {2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac {5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (27) = 54\).

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.05 \[ \int (a+b x) (a c-b c x)^5 \, dx=a^{6} c^{5} x - 2 a^{5} b c^{5} x^{2} + \frac {5 a^{4} b^{2} c^{5} x^{3}}{3} - a^{2} b^{4} c^{5} x^{5} + \frac {2 a b^{5} c^{5} x^{6}}{3} - \frac {b^{6} c^{5} x^{7}}{7} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.89 \[ \int (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{7} \, b^{6} c^{5} x^{7} + \frac {2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac {5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.89 \[ \int (a+b x) (a c-b c x)^5 \, dx=-\frac {1}{7} \, b^{6} c^{5} x^{7} + \frac {2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac {5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.89 \[ \int (a+b x) (a c-b c x)^5 \, dx=a^6\,c^5\,x-2\,a^5\,b\,c^5\,x^2+\frac {5\,a^4\,b^2\,c^5\,x^3}{3}-a^2\,b^4\,c^5\,x^5+\frac {2\,a\,b^5\,c^5\,x^6}{3}-\frac {b^6\,c^5\,x^7}{7} \]

[In]

int((a*c - b*c*x)^5*(a + b*x),x)

[Out]

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