\(\int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx\) [20]

Optimal result

Integrand size = 20, antiderivative size = 78 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx=-\frac {a^5 c^4}{2 x^2}+\frac {3 a^4 b c^4}{x}+2 a^2 b^3 c^4 x-\frac {3}{2} a b^4 c^4 x^2+\frac {1}{3} b^5 c^4 x^3+2 a^3 b^2 c^4 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 78, 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} \[ \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx=-\frac {a^5 c^4}{2 x^2}+\frac {3 a^4 b c^4}{x}+2 a^3 b^2 c^4 \log (x)+2 a^2 b^3 c^4 x-\frac {3}{2} a b^4 c^4 x^2+\frac {1}{3} b^5 c^4 x^3 \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx=-\frac {a^5 c^4}{2 x^2}+\frac {3 a^4 b c^4}{x}+2 a^2 b^3 c^4 x-\frac {3}{2} a b^4 c^4 x^2+\frac {1}{3} b^5 c^4 x^3+2 a^3 b^2 c^4 \log (x) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.76

method	result	size
default	\(c^{4} \left (\frac {b^{5} x^{3}}{3}-\frac {3 a \,b^{4} x^{2}}{2}+2 a^{2} b^{3} x +2 a^{3} b^{2} \ln \left (x \right )+\frac {3 a^{4} b}{x}-\frac {a^{5}}{2 x^{2}}\right )\)	\(59\)
risch	\(\frac {b^{5} c^{4} x^{3}}{3}-\frac {3 a \,b^{4} c^{4} x^{2}}{2}+2 a^{2} b^{3} c^{4} x +\frac {3 a^{4} b \,c^{4} x -\frac {1}{2} a^{5} c^{4}}{x^{2}}+2 a^{3} b^{2} c^{4} \ln \left (x \right )\)	\(73\)
norman	\(\frac {-\frac {1}{2} a^{5} c^{4}+\frac {1}{3} b^{5} c^{4} x^{5}-\frac {3}{2} a \,b^{4} c^{4} x^{4}+2 a^{2} b^{3} c^{4} x^{3}+3 a^{4} b \,c^{4} x}{x^{2}}+2 a^{3} b^{2} c^{4} \ln \left (x \right )\)	\(75\)
parallelrisch	\(\frac {2 b^{5} c^{4} x^{5}-9 a \,b^{4} c^{4} x^{4}+12 a^{3} c^{4} b^{2} \ln \left (x \right ) x^{2}+12 a^{2} b^{3} c^{4} x^{3}+18 a^{4} b \,c^{4} x -3 a^{5} c^{4}}{6 x^{2}}\)	\(78\)

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx=\frac {2 \, b^{5} c^{4} x^{5} - 9 \, a b^{4} c^{4} x^{4} + 12 \, a^{2} b^{3} c^{4} x^{3} + 12 \, a^{3} b^{2} c^{4} x^{2} \log \left (x\right ) + 18 \, a^{4} b c^{4} x - 3 \, a^{5} c^{4}}{6 \, x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*c^4*x^5 - 9*a*b^4*c^4*x^4 + 12*a^2*b^3*c^4*x^3 + 12*a^3*b^2*c^4*x^2*log(x) + 18*a^4*b*c^4*x - 3*a^5
*c^4)/x^2

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx=2 a^{3} b^{2} c^{4} \log {\left (x \right )} + 2 a^{2} b^{3} c^{4} x - \frac {3 a b^{4} c^{4} x^{2}}{2} + \frac {b^{5} c^{4} x^{3}}{3} + \frac {- a^{5} c^{4} + 6 a^{4} b c^{4} x}{2 x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx=\frac {1}{3} \, b^{5} c^{4} x^{3} - \frac {3}{2} \, a b^{4} c^{4} x^{2} + 2 \, a^{2} b^{3} c^{4} x + 2 \, a^{3} b^{2} c^{4} \log \left (x\right ) + \frac {6 \, a^{4} b c^{4} x - a^{5} c^{4}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx=\frac {1}{3} \, b^{5} c^{4} x^{3} - \frac {3}{2} \, a b^{4} c^{4} x^{2} + 2 \, a^{2} b^{3} c^{4} x + 2 \, a^{3} b^{2} c^{4} \log \left ({\left | x \right |}\right ) + \frac {6 \, a^{4} b c^{4} x - a^{5} c^{4}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^3} \, dx=\frac {b^5\,c^4\,x^3}{3}-\frac {\frac {a^5\,c^4}{2}-3\,a^4\,b\,c^4\,x}{x^2}+2\,a^2\,b^3\,c^4\,x-\frac {3\,a\,b^4\,c^4\,x^2}{2}+2\,a^3\,b^2\,c^4\,\ln \left (x\right ) \]

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^3,x)

[Out]

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