\(\int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx\) [25]

Optimal result

Integrand size = 20, antiderivative size = 84 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx=-\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 84, 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} \[ \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx=-\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

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

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^5 c^4}{x^8}-\frac {3 a^4 b c^4}{x^7}+\frac {2 a^3 b^2 c^4}{x^6}+\frac {2 a^2 b^3 c^4}{x^5}-\frac {3 a b^4 c^4}{x^4}+\frac {b^5 c^4}{x^3}\right ) \, dx \\ & = -\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx=-\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73

method	result	size
gosper	\(-\frac {c^{4} \left (35 b^{5} x^{5}-70 a \,b^{4} x^{4}+35 a^{2} b^{3} x^{3}+28 a^{3} b^{2} x^{2}-35 a^{4} b x +10 a^{5}\right )}{70 x^{7}}\)	\(61\)
default	\(c^{4} \left (\frac {a^{4} b}{2 x^{6}}-\frac {a^{5}}{7 x^{7}}+\frac {a \,b^{4}}{x^{3}}-\frac {b^{5}}{2 x^{2}}-\frac {a^{2} b^{3}}{2 x^{4}}-\frac {2 a^{3} b^{2}}{5 x^{5}}\right )\)	\(61\)
norman	\(\frac {a \,b^{4} c^{4} x^{4}-\frac {1}{7} a^{5} c^{4}-\frac {1}{2} b^{5} c^{4} x^{5}-\frac {1}{2} a^{2} b^{3} c^{4} x^{3}-\frac {2}{5} a^{3} b^{2} c^{4} x^{2}+\frac {1}{2} a^{4} b \,c^{4} x}{x^{7}}\)	\(74\)
risch	\(\frac {a \,b^{4} c^{4} x^{4}-\frac {1}{7} a^{5} c^{4}-\frac {1}{2} b^{5} c^{4} x^{5}-\frac {1}{2} a^{2} b^{3} c^{4} x^{3}-\frac {2}{5} a^{3} b^{2} c^{4} x^{2}+\frac {1}{2} a^{4} b \,c^{4} x}{x^{7}}\)	\(74\)
parallelrisch	\(\frac {-35 b^{5} c^{4} x^{5}+70 a \,b^{4} c^{4} x^{4}-35 a^{2} b^{3} c^{4} x^{3}-28 a^{3} b^{2} c^{4} x^{2}+35 a^{4} b \,c^{4} x -10 a^{5} c^{4}}{70 x^{7}}\)	\(76\)

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/70*c^4*(35*b^5*x^5-70*a*b^4*x^4+35*a^2*b^3*x^3+28*a^3*b^2*x^2-35*a^4*b*x+10*a^5)/x^7

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx=-\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="fricas")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx=\frac {- 10 a^{5} c^{4} + 35 a^{4} b c^{4} x - 28 a^{3} b^{2} c^{4} x^{2} - 35 a^{2} b^{3} c^{4} x^{3} + 70 a b^{4} c^{4} x^{4} - 35 b^{5} c^{4} x^{5}}{70 x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**8,x)

[Out]

(-10*a**5*c**4 + 35*a**4*b*c**4*x - 28*a**3*b**2*c**4*x**2 - 35*a**2*b**3*c**4*x**3 + 70*a*b**4*c**4*x**4 - 35
*b**5*c**4*x**5)/(70*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx=-\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="maxima")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx=-\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="giac")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx=-\frac {\frac {a^5\,c^4}{7}-\frac {a^4\,b\,c^4\,x}{2}+\frac {2\,a^3\,b^2\,c^4\,x^2}{5}+\frac {a^2\,b^3\,c^4\,x^3}{2}-a\,b^4\,c^4\,x^4+\frac {b^5\,c^4\,x^5}{2}}{x^7} \]

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^8,x)

[Out]

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