\(\int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx\) [46]

Optimal result

Integrand size = 20, antiderivative size = 65 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx=-\frac {c^6 (a-b x)^7}{9 x^9}-\frac {11 b c^6 (a-b x)^7}{72 a x^8}-\frac {11 b^2 c^6 (a-b x)^7}{504 a^2 x^7} \]

[Out]

-1/9*c^6*(-b*x+a)^7/x^9-11/72*b*c^6*(-b*x+a)^7/a/x^8-11/504*b^2*c^6*(-b*x+a)^7/a^2/x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx=-\frac {11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}-\frac {c^6 (a-b x)^7}{9 x^9}-\frac {11 b c^6 (a-b x)^7}{72 a x^8} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

-1/9*(c^6*(a - b*x)^7)/x^9 - (11*b*c^6*(a - b*x)^7)/(72*a*x^8) - (11*b^2*c^6*(a - b*x)^7)/(504*a^2*x^7)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = -\frac {c^6 (a-b x)^7}{9 x^9}+\frac {1}{9} (11 b) \int \frac {(a c-b c x)^6}{x^9} \, dx \\ & = -\frac {c^6 (a-b x)^7}{9 x^9}-\frac {11 b c^6 (a-b x)^7}{72 a x^8}+\frac {\left (11 b^2\right ) \int \frac {(a c-b c x)^6}{x^8} \, dx}{72 a} \\ & = -\frac {c^6 (a-b x)^7}{9 x^9}-\frac {11 b c^6 (a-b x)^7}{72 a x^8}-\frac {11 b^2 c^6 (a-b x)^7}{504 a^2 x^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx=-\frac {a^7 c^6}{9 x^9}+\frac {5 a^6 b c^6}{8 x^8}-\frac {9 a^5 b^2 c^6}{7 x^7}+\frac {5 a^4 b^3 c^6}{6 x^6}+\frac {a^3 b^4 c^6}{x^5}-\frac {9 a^2 b^5 c^6}{4 x^4}+\frac {5 a b^6 c^6}{3 x^3}-\frac {b^7 c^6}{2 x^2} \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

-1/9*(a^7*c^6)/x^9 + (5*a^6*b*c^6)/(8*x^8) - (9*a^5*b^2*c^6)/(7*x^7) + (5*a^4*b^3*c^6)/(6*x^6) + (a^3*b^4*c^6)
/x^5 - (9*a^2*b^5*c^6)/(4*x^4) + (5*a*b^6*c^6)/(3*x^3) - (b^7*c^6)/(2*x^2)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.28

method	result	size
gosper	\(-\frac {c^{6} \left (252 b^{7} x^{7}-840 a \,b^{6} x^{6}+1134 a^{2} b^{5} x^{5}-504 a^{3} b^{4} x^{4}-420 a^{4} b^{3} x^{3}+648 a^{5} b^{2} x^{2}-315 a^{6} b x +56 a^{7}\right )}{504 x^{9}}\)	\(83\)
default	\(c^{6} \left (\frac {5 a^{4} b^{3}}{6 x^{6}}-\frac {9 a^{5} b^{2}}{7 x^{7}}+\frac {5 a^{6} b}{8 x^{8}}+\frac {5 a \,b^{6}}{3 x^{3}}-\frac {b^{7}}{2 x^{2}}-\frac {9 a^{2} b^{5}}{4 x^{4}}+\frac {a^{3} b^{4}}{x^{5}}-\frac {a^{7}}{9 x^{9}}\right )\)	\(83\)
norman	\(\frac {a^{3} b^{4} c^{6} x^{4}-\frac {1}{9} a^{7} c^{6}-\frac {1}{2} b^{7} c^{6} x^{7}+\frac {5}{3} a \,b^{6} c^{6} x^{6}-\frac {9}{4} a^{2} b^{5} c^{6} x^{5}+\frac {5}{6} a^{4} b^{3} c^{6} x^{3}-\frac {9}{7} a^{5} b^{2} c^{6} x^{2}+\frac {5}{8} a^{6} b \,c^{6} x}{x^{9}}\)	\(102\)
risch	\(\frac {a^{3} b^{4} c^{6} x^{4}-\frac {1}{9} a^{7} c^{6}-\frac {1}{2} b^{7} c^{6} x^{7}+\frac {5}{3} a \,b^{6} c^{6} x^{6}-\frac {9}{4} a^{2} b^{5} c^{6} x^{5}+\frac {5}{6} a^{4} b^{3} c^{6} x^{3}-\frac {9}{7} a^{5} b^{2} c^{6} x^{2}+\frac {5}{8} a^{6} b \,c^{6} x}{x^{9}}\)	\(102\)
parallelrisch	\(\frac {-252 b^{7} c^{6} x^{7}+840 a \,b^{6} c^{6} x^{6}-1134 a^{2} b^{5} c^{6} x^{5}+504 a^{3} b^{4} c^{6} x^{4}+420 a^{4} b^{3} c^{6} x^{3}-648 a^{5} b^{2} c^{6} x^{2}+315 a^{6} b \,c^{6} x -56 a^{7} c^{6}}{504 x^{9}}\)	\(104\)

[In]

int((b*x+a)*(-b*c*x+a*c)^6/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/504*c^6*(252*b^7*x^7-840*a*b^6*x^6+1134*a^2*b^5*x^5-504*a^3*b^4*x^4-420*a^4*b^3*x^3+648*a^5*b^2*x^2-315*a^6
*b*x+56*a^7)/x^9

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx=-\frac {252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="fricas")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx=\frac {- 56 a^{7} c^{6} + 315 a^{6} b c^{6} x - 648 a^{5} b^{2} c^{6} x^{2} + 420 a^{4} b^{3} c^{6} x^{3} + 504 a^{3} b^{4} c^{6} x^{4} - 1134 a^{2} b^{5} c^{6} x^{5} + 840 a b^{6} c^{6} x^{6} - 252 b^{7} c^{6} x^{7}}{504 x^{9}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**10,x)

[Out]

(-56*a**7*c**6 + 315*a**6*b*c**6*x - 648*a**5*b**2*c**6*x**2 + 420*a**4*b**3*c**6*x**3 + 504*a**3*b**4*c**6*x*
*4 - 1134*a**2*b**5*c**6*x**5 + 840*a*b**6*c**6*x**6 - 252*b**7*c**6*x**7)/(504*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx=-\frac {252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="maxima")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx=-\frac {252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="giac")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx=-\frac {\frac {a^7\,c^6}{9}-\frac {5\,a^6\,b\,c^6\,x}{8}+\frac {9\,a^5\,b^2\,c^6\,x^2}{7}-\frac {5\,a^4\,b^3\,c^6\,x^3}{6}-a^3\,b^4\,c^6\,x^4+\frac {9\,a^2\,b^5\,c^6\,x^5}{4}-\frac {5\,a\,b^6\,c^6\,x^6}{3}+\frac {b^7\,c^6\,x^7}{2}}{x^9} \]

[In]

int(((a*c - b*c*x)^6*(a + b*x))/x^10,x)

[Out]

-((a^7*c^6)/9 + (b^7*c^6*x^7)/2 - (5*a*b^6*c^6*x^6)/3 + (9*a^5*b^2*c^6*x^2)/7 - (5*a^4*b^3*c^6*x^3)/6 - a^3*b^
4*c^6*x^4 + (9*a^2*b^5*c^6*x^5)/4 - (5*a^6*b*c^6*x)/8)/x^9