\(\int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx\) [12]

Optimal result

Integrand size = 20, antiderivative size = 55 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx=-\frac {a^4 c^3}{7 x^7}+\frac {a^3 b c^3}{3 x^6}-\frac {a b^3 c^3}{2 x^4}+\frac {b^4 c^3}{3 x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx=-\frac {a^4 c^3}{7 x^7}+\frac {a^3 b c^3}{3 x^6}-\frac {a b^3 c^3}{2 x^4}+\frac {b^4 c^3}{3 x^3} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^8,x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^4 c^3}{x^8}-\frac {2 a^3 b c^3}{x^7}+\frac {2 a b^3 c^3}{x^5}-\frac {b^4 c^3}{x^4}\right ) \, dx \\ & = -\frac {a^4 c^3}{7 x^7}+\frac {a^3 b c^3}{3 x^6}-\frac {a b^3 c^3}{2 x^4}+\frac {b^4 c^3}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx=c^3 \left (-\frac {a^4}{7 x^7}+\frac {a^3 b}{3 x^6}-\frac {a b^3}{2 x^4}+\frac {b^4}{3 x^3}\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^8,x]

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71

method	result	size
gosper	\(-\frac {c^{3} \left (-14 b^{4} x^{4}+21 a \,b^{3} x^{3}-14 a^{3} b x +6 a^{4}\right )}{42 x^{7}}\)	\(39\)
default	\(c^{3} \left (\frac {a^{3} b}{3 x^{6}}-\frac {a^{4}}{7 x^{7}}+\frac {b^{4}}{3 x^{3}}-\frac {a \,b^{3}}{2 x^{4}}\right )\)	\(40\)
norman	\(\frac {-\frac {1}{7} a^{4} c^{3}+\frac {1}{3} b^{4} c^{3} x^{4}-\frac {1}{2} a \,b^{3} c^{3} x^{3}+\frac {1}{3} a^{3} b \,c^{3} x}{x^{7}}\)	\(47\)
risch	\(\frac {-\frac {1}{7} a^{4} c^{3}+\frac {1}{3} b^{4} c^{3} x^{4}-\frac {1}{2} a \,b^{3} c^{3} x^{3}+\frac {1}{3} a^{3} b \,c^{3} x}{x^{7}}\)	\(47\)
parallelrisch	\(\frac {14 b^{4} c^{3} x^{4}-21 a \,b^{3} c^{3} x^{3}+14 a^{3} b \,c^{3} x -6 a^{4} c^{3}}{42 x^{7}}\)	\(48\)

[In]

int((b*x+a)*(-b*c*x+a*c)^3/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/42*c^3*(-14*b^4*x^4+21*a*b^3*x^3-14*a^3*b*x+6*a^4)/x^7

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx=\frac {14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^8,x, algorithm="fricas")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx=- \frac {6 a^{4} c^{3} - 14 a^{3} b c^{3} x + 21 a b^{3} c^{3} x^{3} - 14 b^{4} c^{3} x^{4}}{42 x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**8,x)

[Out]

-(6*a**4*c**3 - 14*a**3*b*c**3*x + 21*a*b**3*c**3*x**3 - 14*b**4*c**3*x**4)/(42*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx=\frac {14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^8,x, algorithm="maxima")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx=\frac {14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^8,x, algorithm="giac")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx=-\frac {\frac {a^4\,c^3}{7}-\frac {a^3\,b\,c^3\,x}{3}+\frac {a\,b^3\,c^3\,x^3}{2}-\frac {b^4\,c^3\,x^4}{3}}{x^7} \]

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x^8,x)

[Out]

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