\(\int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx\) [44]

Optimal result

Integrand size = 20, antiderivative size = 113 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx=-\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

[Out]

-1/7*a^7*c^6/x^7+5/6*a^6*b*c^6/x^6-9/5*a^5*b^2*c^6/x^5+5/4*a^4*b^3*c^6/x^4+5/3*a^3*b^4*c^6/x^3-9/2*a^2*b^5*c^6
/x^2+5*a*b^6*c^6/x+b^7*c^6*ln(x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 113, 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} \[ \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx=-\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^8,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx=-\frac {a^7 c^6}{7 x^7}+\frac {5 a^6 b c^6}{6 x^6}-\frac {9 a^5 b^2 c^6}{5 x^5}+\frac {5 a^4 b^3 c^6}{4 x^4}+\frac {5 a^3 b^4 c^6}{3 x^3}-\frac {9 a^2 b^5 c^6}{2 x^2}+\frac {5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^8,x]

[Out]

-1/7*(a^7*c^6)/x^7 + (5*a^6*b*c^6)/(6*x^6) - (9*a^5*b^2*c^6)/(5*x^5) + (5*a^4*b^3*c^6)/(4*x^4) + (5*a^3*b^4*c^
6)/(3*x^3) - (9*a^2*b^5*c^6)/(2*x^2) + (5*a*b^6*c^6)/x + b^7*c^6*Log[x]

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.73

method	result	size
default	\(c^{6} \left (b^{7} \ln \left (x \right )+\frac {5 a^{6} b}{6 x^{6}}-\frac {a^{7}}{7 x^{7}}+\frac {5 a^{3} b^{4}}{3 x^{3}}+\frac {5 a \,b^{6}}{x}-\frac {9 a^{2} b^{5}}{2 x^{2}}+\frac {5 a^{4} b^{3}}{4 x^{4}}-\frac {9 a^{5} b^{2}}{5 x^{5}}\right )\)	\(82\)
norman	\(\frac {-\frac {1}{7} a^{7} c^{6}+5 a \,b^{6} c^{6} x^{6}-\frac {9}{2} a^{2} b^{5} c^{6} x^{5}+\frac {5}{3} a^{3} b^{4} c^{6} x^{4}+\frac {5}{4} a^{4} b^{3} c^{6} x^{3}-\frac {9}{5} a^{5} b^{2} c^{6} x^{2}+\frac {5}{6} a^{6} b \,c^{6} x}{x^{7}}+b^{7} c^{6} \ln \left (x \right )\)	\(102\)
risch	\(\frac {-\frac {1}{7} a^{7} c^{6}+5 a \,b^{6} c^{6} x^{6}-\frac {9}{2} a^{2} b^{5} c^{6} x^{5}+\frac {5}{3} a^{3} b^{4} c^{6} x^{4}+\frac {5}{4} a^{4} b^{3} c^{6} x^{3}-\frac {9}{5} a^{5} b^{2} c^{6} x^{2}+\frac {5}{6} a^{6} b \,c^{6} x}{x^{7}}+b^{7} c^{6} \ln \left (x \right )\)	\(102\)
parallelrisch	\(\frac {420 b^{7} c^{6} \ln \left (x \right ) x^{7}+2100 a \,b^{6} c^{6} x^{6}-1890 a^{2} b^{5} c^{6} x^{5}+700 a^{3} b^{4} c^{6} x^{4}+525 a^{4} b^{3} c^{6} x^{3}-756 a^{5} b^{2} c^{6} x^{2}+350 a^{6} b \,c^{6} x -60 a^{7} c^{6}}{420 x^{7}}\)	\(106\)

[In]

int((b*x+a)*(-b*c*x+a*c)^6/x^8,x,method=_RETURNVERBOSE)

[Out]

c^6*(b^7*ln(x)+5/6*a^6*b/x^6-1/7*a^7/x^7+5/3*a^3*b^4/x^3+5*a*b^6/x-9/2*a^2*b^5/x^2+5/4*a^4*b^3/x^4-9/5*a^5*b^2
/x^5)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx=\frac {420 \, b^{7} c^{6} x^{7} \log \left (x\right ) + 2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^8,x, algorithm="fricas")

[Out]

1/420*(420*b^7*c^6*x^7*log(x) + 2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*a^3*b^4*c^6*x^4 + 525*a^4*b^3*
c^6*x^3 - 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x - 60*a^7*c^6)/x^7

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx=b^{7} c^{6} \log {\left (x \right )} + \frac {- 60 a^{7} c^{6} + 350 a^{6} b c^{6} x - 756 a^{5} b^{2} c^{6} x^{2} + 525 a^{4} b^{3} c^{6} x^{3} + 700 a^{3} b^{4} c^{6} x^{4} - 1890 a^{2} b^{5} c^{6} x^{5} + 2100 a b^{6} c^{6} x^{6}}{420 x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**8,x)

[Out]

b**7*c**6*log(x) + (-60*a**7*c**6 + 350*a**6*b*c**6*x - 756*a**5*b**2*c**6*x**2 + 525*a**4*b**3*c**6*x**3 + 70
0*a**3*b**4*c**6*x**4 - 1890*a**2*b**5*c**6*x**5 + 2100*a*b**6*c**6*x**6)/(420*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx=b^{7} c^{6} \log \left (x\right ) + \frac {2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^8,x, algorithm="maxima")

[Out]

b^7*c^6*log(x) + 1/420*(2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*a^3*b^4*c^6*x^4 + 525*a^4*b^3*c^6*x^3
- 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x - 60*a^7*c^6)/x^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx=b^{7} c^{6} \log \left ({\left | x \right |}\right ) + \frac {2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^8,x, algorithm="giac")

[Out]

b^7*c^6*log(abs(x)) + 1/420*(2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*a^3*b^4*c^6*x^4 + 525*a^4*b^3*c^6
*x^3 - 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x - 60*a^7*c^6)/x^7

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^8} \, dx=\frac {c^6\,\left (5\,a\,b^6\,x^6-\frac {a^7}{7}-\frac {9\,a^5\,b^2\,x^2}{5}+\frac {5\,a^4\,b^3\,x^3}{4}+\frac {5\,a^3\,b^4\,x^4}{3}-\frac {9\,a^2\,b^5\,x^5}{2}+b^7\,x^7\,\ln \left (x\right )+\frac {5\,a^6\,b\,x}{6}\right )}{x^7} \]

[In]

int(((a*c - b*c*x)^6*(a + b*x))/x^8,x)

[Out]

(c^6*(5*a*b^6*x^6 - a^7/7 - (9*a^5*b^2*x^2)/5 + (5*a^4*b^3*x^3)/4 + (5*a^3*b^4*x^4)/3 - (9*a^2*b^5*x^5)/2 + b^
7*x^7*log(x) + (5*a^6*b*x)/6))/x^7