\(\int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx\) [11]

Optimal result

Integrand size = 20, antiderivative size = 55 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx=-\frac {a^4 c^3}{6 x^6}+\frac {2 a^3 b c^3}{5 x^5}-\frac {2 a b^3 c^3}{3 x^3}+\frac {b^4 c^3}{2 x^2} \]

[Out]

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

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

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx=c^3 \left (-\frac {a^4}{6 x^6}+\frac {2 a^3 b}{5 x^5}-\frac {2 a b^3}{3 x^3}+\frac {b^4}{2 x^2}\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

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

Maple [A] (verified)

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

method	result	size
gosper	\(-\frac {c^{3} \left (-15 b^{4} x^{4}+20 a \,b^{3} x^{3}-12 a^{3} b x +5 a^{4}\right )}{30 x^{6}}\)	\(39\)
default	\(c^{3} \left (-\frac {a^{4}}{6 x^{6}}-\frac {2 a \,b^{3}}{3 x^{3}}+\frac {b^{4}}{2 x^{2}}+\frac {2 a^{3} b}{5 x^{5}}\right )\)	\(40\)
norman	\(\frac {-\frac {1}{6} a^{4} c^{3}+\frac {1}{2} b^{4} c^{3} x^{4}-\frac {2}{3} a \,b^{3} c^{3} x^{3}+\frac {2}{5} a^{3} b \,c^{3} x}{x^{6}}\)	\(47\)
risch	\(\frac {-\frac {1}{6} a^{4} c^{3}+\frac {1}{2} b^{4} c^{3} x^{4}-\frac {2}{3} a \,b^{3} c^{3} x^{3}+\frac {2}{5} a^{3} b \,c^{3} x}{x^{6}}\)	\(47\)
parallelrisch	\(\frac {15 b^{4} c^{3} x^{4}-20 a \,b^{3} c^{3} x^{3}+12 a^{3} b \,c^{3} x -5 a^{4} c^{3}}{30 x^{6}}\)	\(48\)

[In]

int((b*x+a)*(-b*c*x+a*c)^3/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/30*c^3*(-15*b^4*x^4+20*a*b^3*x^3-12*a^3*b*x+5*a^4)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx=\frac {15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="fricas")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx=- \frac {5 a^{4} c^{3} - 12 a^{3} b c^{3} x + 20 a b^{3} c^{3} x^{3} - 15 b^{4} c^{3} x^{4}}{30 x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**7,x)

[Out]

-(5*a**4*c**3 - 12*a**3*b*c**3*x + 20*a*b**3*c**3*x**3 - 15*b**4*c**3*x**4)/(30*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx=\frac {15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="maxima")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx=\frac {15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="giac")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^7} \, dx=-\frac {\frac {a^4\,c^3}{6}-\frac {2\,a^3\,b\,c^3\,x}{5}+\frac {2\,a\,b^3\,c^3\,x^3}{3}-\frac {b^4\,c^3\,x^4}{2}}{x^6} \]

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x^7,x)

[Out]

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