\(\int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx\) [10]

Optimal result

Integrand size = 20, antiderivative size = 50 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx=-\frac {a^4 c^3}{5 x^5}+\frac {a^3 b c^3}{2 x^4}-\frac {a b^3 c^3}{x^2}+\frac {b^4 c^3}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, 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} \[ \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx=-\frac {a^4 c^3}{5 x^5}+\frac {a^3 b c^3}{2 x^4}-\frac {a b^3 c^3}{x^2}+\frac {b^4 c^3}{x} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^6,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx=c^3 \left (-\frac {a^4}{5 x^5}+\frac {a^3 b}{2 x^4}-\frac {a b^3}{x^2}+\frac {b^4}{x}\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^6,x]

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.78

method	result	size
gosper	\(-\frac {c^{3} \left (-10 b^{4} x^{4}+10 a \,b^{3} x^{3}-5 a^{3} b x +2 a^{4}\right )}{10 x^{5}}\)	\(39\)
default	\(c^{3} \left (\frac {b^{4}}{x}-\frac {a \,b^{3}}{x^{2}}+\frac {a^{3} b}{2 x^{4}}-\frac {a^{4}}{5 x^{5}}\right )\)	\(39\)
norman	\(\frac {b^{4} c^{3} x^{4}-\frac {1}{5} a^{4} c^{3}-a \,b^{3} c^{3} x^{3}+\frac {1}{2} a^{3} b \,c^{3} x}{x^{5}}\)	\(46\)
risch	\(\frac {b^{4} c^{3} x^{4}-\frac {1}{5} a^{4} c^{3}-a \,b^{3} c^{3} x^{3}+\frac {1}{2} a^{3} b \,c^{3} x}{x^{5}}\)	\(46\)
parallelrisch	\(\frac {10 b^{4} c^{3} x^{4}-10 a \,b^{3} c^{3} x^{3}+5 a^{3} b \,c^{3} x -2 a^{4} c^{3}}{10 x^{5}}\)	\(48\)

[In]

int((b*x+a)*(-b*c*x+a*c)^3/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx=\frac {10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^6,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx=- \frac {2 a^{4} c^{3} - 5 a^{3} b c^{3} x + 10 a b^{3} c^{3} x^{3} - 10 b^{4} c^{3} x^{4}}{10 x^{5}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**6,x)

[Out]

-(2*a**4*c**3 - 5*a**3*b*c**3*x + 10*a*b**3*c**3*x**3 - 10*b**4*c**3*x**4)/(10*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx=\frac {10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^6,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx=\frac {10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^6,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^6} \, dx=-\frac {\frac {a^4\,c^3}{5}-\frac {a^3\,b\,c^3\,x}{2}+a\,b^3\,c^3\,x^3-b^4\,c^3\,x^4}{x^5} \]

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x^6,x)

[Out]

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