\(\int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx\) [26]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx=-\frac {a^5 c^4}{8 x^8}+\frac {3 a^4 b c^4}{7 x^7}-\frac {a^3 b^2 c^4}{3 x^6}-\frac {2 a^2 b^3 c^4}{5 x^5}+\frac {3 a b^4 c^4}{4 x^4}-\frac {b^5 c^4}{3 x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, 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} \[ \int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx=-\frac {a^5 c^4}{8 x^8}+\frac {3 a^4 b c^4}{7 x^7}-\frac {a^3 b^2 c^4}{3 x^6}-\frac {2 a^2 b^3 c^4}{5 x^5}+\frac {3 a b^4 c^4}{4 x^4}-\frac {b^5 c^4}{3 x^3} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^9,x]

[Out]

-1/8*(a^5*c^4)/x^8 + (3*a^4*b*c^4)/(7*x^7) - (a^3*b^2*c^4)/(3*x^6) - (2*a^2*b^3*c^4)/(5*x^5) + (3*a*b^4*c^4)/(
4*x^4) - (b^5*c^4)/(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^5 c^4}{x^9}-\frac {3 a^4 b c^4}{x^8}+\frac {2 a^3 b^2 c^4}{x^7}+\frac {2 a^2 b^3 c^4}{x^6}-\frac {3 a b^4 c^4}{x^5}+\frac {b^5 c^4}{x^4}\right ) \, dx \\ & = -\frac {a^5 c^4}{8 x^8}+\frac {3 a^4 b c^4}{7 x^7}-\frac {a^3 b^2 c^4}{3 x^6}-\frac {2 a^2 b^3 c^4}{5 x^5}+\frac {3 a b^4 c^4}{4 x^4}-\frac {b^5 c^4}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx=-\frac {a^5 c^4}{8 x^8}+\frac {3 a^4 b c^4}{7 x^7}-\frac {a^3 b^2 c^4}{3 x^6}-\frac {2 a^2 b^3 c^4}{5 x^5}+\frac {3 a b^4 c^4}{4 x^4}-\frac {b^5 c^4}{3 x^3} \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^9,x]

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70

method	result	size
gosper	\(-\frac {c^{4} \left (280 b^{5} x^{5}-630 a \,b^{4} x^{4}+336 a^{2} b^{3} x^{3}+280 a^{3} b^{2} x^{2}-360 a^{4} b x +105 a^{5}\right )}{840 x^{8}}\)	\(61\)
default	\(c^{4} \left (-\frac {a^{3} b^{2}}{3 x^{6}}+\frac {3 a^{4} b}{7 x^{7}}-\frac {a^{5}}{8 x^{8}}-\frac {b^{5}}{3 x^{3}}+\frac {3 a \,b^{4}}{4 x^{4}}-\frac {2 a^{2} b^{3}}{5 x^{5}}\right )\)	\(62\)
norman	\(\frac {-\frac {1}{8} a^{5} c^{4}-\frac {1}{3} b^{5} c^{4} x^{5}+\frac {3}{4} a \,b^{4} c^{4} x^{4}-\frac {2}{5} a^{2} b^{3} c^{4} x^{3}-\frac {1}{3} a^{3} b^{2} c^{4} x^{2}+\frac {3}{7} a^{4} b \,c^{4} x}{x^{8}}\)	\(75\)
risch	\(\frac {-\frac {1}{8} a^{5} c^{4}-\frac {1}{3} b^{5} c^{4} x^{5}+\frac {3}{4} a \,b^{4} c^{4} x^{4}-\frac {2}{5} a^{2} b^{3} c^{4} x^{3}-\frac {1}{3} a^{3} b^{2} c^{4} x^{2}+\frac {3}{7} a^{4} b \,c^{4} x}{x^{8}}\)	\(75\)
parallelrisch	\(\frac {-280 b^{5} c^{4} x^{5}+630 a \,b^{4} c^{4} x^{4}-336 a^{2} b^{3} c^{4} x^{3}-280 a^{3} b^{2} c^{4} x^{2}+360 a^{4} b \,c^{4} x -105 a^{5} c^{4}}{840 x^{8}}\)	\(76\)

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/840*c^4*(280*b^5*x^5-630*a*b^4*x^4+336*a^2*b^3*x^3+280*a^3*b^2*x^2-360*a^4*b*x+105*a^5)/x^8

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx=-\frac {280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="fricas")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx=\frac {- 105 a^{5} c^{4} + 360 a^{4} b c^{4} x - 280 a^{3} b^{2} c^{4} x^{2} - 336 a^{2} b^{3} c^{4} x^{3} + 630 a b^{4} c^{4} x^{4} - 280 b^{5} c^{4} x^{5}}{840 x^{8}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**9,x)

[Out]

(-105*a**5*c**4 + 360*a**4*b*c**4*x - 280*a**3*b**2*c**4*x**2 - 336*a**2*b**3*c**4*x**3 + 630*a*b**4*c**4*x**4
- 280*b**5*c**4*x**5)/(840*x**8)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx=-\frac {280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="maxima")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx=-\frac {280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="giac")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx=-\frac {\frac {a^5\,c^4}{8}-\frac {3\,a^4\,b\,c^4\,x}{7}+\frac {a^3\,b^2\,c^4\,x^2}{3}+\frac {2\,a^2\,b^3\,c^4\,x^3}{5}-\frac {3\,a\,b^4\,c^4\,x^4}{4}+\frac {b^5\,c^4\,x^5}{3}}{x^8} \]

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^9,x)

[Out]

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