\(\int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx\) [33]

Optimal result

Integrand size = 20, antiderivative size = 75 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx=-\frac {a^6 c^5}{x}+5 a^4 b^2 c^5 x-\frac {5}{3} a^2 b^4 c^5 x^3+a b^5 c^5 x^4-\frac {1}{5} b^6 c^5 x^5-4 a^5 b c^5 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, 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} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx=-\frac {a^6 c^5}{x}-4 a^5 b c^5 \log (x)+5 a^4 b^2 c^5 x-\frac {5}{3} a^2 b^4 c^5 x^3+a b^5 c^5 x^4-\frac {1}{5} b^6 c^5 x^5 \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx=c^5 \left (-\frac {a^6}{x}+5 a^4 b^2 x-\frac {5}{3} a^2 b^4 x^3+a b^5 x^4-\frac {b^6 x^5}{5}-4 a^5 b \log (x)\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77

method	result	size
default	\(c^{5} \left (-\frac {b^{6} x^{5}}{5}+a \,b^{5} x^{4}-\frac {5 a^{2} b^{4} x^{3}}{3}+5 a^{4} b^{2} x -4 a^{5} b \ln \left (x \right )-\frac {a^{6}}{x}\right )\)	\(58\)
risch	\(-\frac {a^{6} c^{5}}{x}+5 a^{4} b^{2} c^{5} x -\frac {5 a^{2} b^{4} c^{5} x^{3}}{3}+a \,b^{5} c^{5} x^{4}-\frac {b^{6} c^{5} x^{5}}{5}-4 a^{5} b \,c^{5} \ln \left (x \right )\)	\(72\)
norman	\(\frac {a \,b^{5} c^{5} x^{5}-a^{6} c^{5}-\frac {1}{5} b^{6} c^{5} x^{6}-\frac {5}{3} a^{2} b^{4} c^{5} x^{4}+5 a^{4} b^{2} c^{5} x^{2}}{x}-4 a^{5} b \,c^{5} \ln \left (x \right )\)	\(76\)
parallelrisch	\(-\frac {3 b^{6} c^{5} x^{6}-15 a \,b^{5} c^{5} x^{5}+25 a^{2} b^{4} c^{5} x^{4}+60 a^{5} c^{5} b \ln \left (x \right ) x -75 a^{4} b^{2} c^{5} x^{2}+15 a^{6} c^{5}}{15 x}\)	\(78\)

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx=-\frac {3 \, b^{6} c^{5} x^{6} - 15 \, a b^{5} c^{5} x^{5} + 25 \, a^{2} b^{4} c^{5} x^{4} - 75 \, a^{4} b^{2} c^{5} x^{2} + 60 \, a^{5} b c^{5} x \log \left (x\right ) + 15 \, a^{6} c^{5}}{15 \, x} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^2,x, algorithm="fricas")

[Out]

-1/15*(3*b^6*c^5*x^6 - 15*a*b^5*c^5*x^5 + 25*a^2*b^4*c^5*x^4 - 75*a^4*b^2*c^5*x^2 + 60*a^5*b*c^5*x*log(x) + 15
*a^6*c^5)/x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx=- \frac {a^{6} c^{5}}{x} - 4 a^{5} b c^{5} \log {\left (x \right )} + 5 a^{4} b^{2} c^{5} x - \frac {5 a^{2} b^{4} c^{5} x^{3}}{3} + a b^{5} c^{5} x^{4} - \frac {b^{6} c^{5} x^{5}}{5} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx=-\frac {1}{5} \, b^{6} c^{5} x^{5} + a b^{5} c^{5} x^{4} - \frac {5}{3} \, a^{2} b^{4} c^{5} x^{3} + 5 \, a^{4} b^{2} c^{5} x - 4 \, a^{5} b c^{5} \log \left (x\right ) - \frac {a^{6} c^{5}}{x} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx=-\frac {1}{5} \, b^{6} c^{5} x^{5} + a b^{5} c^{5} x^{4} - \frac {5}{3} \, a^{2} b^{4} c^{5} x^{3} + 5 \, a^{4} b^{2} c^{5} x - 4 \, a^{5} b c^{5} \log \left ({\left | x \right |}\right ) - \frac {a^{6} c^{5}}{x} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^2} \, dx=5\,a^4\,b^2\,c^5\,x-\frac {b^6\,c^5\,x^5}{5}-\frac {a^6\,c^5}{x}+a\,b^5\,c^5\,x^4-4\,a^5\,b\,c^5\,\ln \left (x\right )-\frac {5\,a^2\,b^4\,c^5\,x^3}{3} \]

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^2,x)

[Out]

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