3.48 \(\int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx\)

Optimal. Leaf size=114 \[ -\frac {a^7 c^6}{11 x^{11}}+\frac {a^6 b c^6}{2 x^{10}}-\frac {a^5 b^2 c^6}{x^9}+\frac {5 a^4 b^3 c^6}{8 x^8}+\frac {5 a^3 b^4 c^6}{7 x^7}-\frac {3 a^2 b^5 c^6}{2 x^6}+\frac {a b^6 c^6}{x^5}-\frac {b^7 c^6}{4 x^4} \]

[Out]

-1/11*a^7*c^6/x^11+1/2*a^6*b*c^6/x^10-a^5*b^2*c^6/x^9+5/8*a^4*b^3*c^6/x^8+5/7*a^3*b^4*c^6/x^7-3/2*a^2*b^5*c^6/
x^6+a*b^6*c^6/x^5-1/4*b^7*c^6/x^4

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Rubi [A]  time = 0.05, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \[ -\frac {a^5 b^2 c^6}{x^9}+\frac {5 a^4 b^3 c^6}{8 x^8}+\frac {5 a^3 b^4 c^6}{7 x^7}-\frac {3 a^2 b^5 c^6}{2 x^6}+\frac {a^6 b c^6}{2 x^{10}}-\frac {a^7 c^6}{11 x^{11}}+\frac {a b^6 c^6}{x^5}-\frac {b^7 c^6}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^12,x]

[Out]

-(a^7*c^6)/(11*x^11) + (a^6*b*c^6)/(2*x^10) - (a^5*b^2*c^6)/x^9 + (5*a^4*b^3*c^6)/(8*x^8) + (5*a^3*b^4*c^6)/(7
*x^7) - (3*a^2*b^5*c^6)/(2*x^6) + (a*b^6*c^6)/x^5 - (b^7*c^6)/(4*x^4)

Rule 75

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*} \int \frac {(a+b x) (a c-b c x)^6}{x^{12}} \, dx &=\int \left (\frac {a^7 c^6}{x^{12}}-\frac {5 a^6 b c^6}{x^{11}}+\frac {9 a^5 b^2 c^6}{x^{10}}-\frac {5 a^4 b^3 c^6}{x^9}-\frac {5 a^3 b^4 c^6}{x^8}+\frac {9 a^2 b^5 c^6}{x^7}-\frac {5 a b^6 c^6}{x^6}+\frac {b^7 c^6}{x^5}\right ) \, dx\\ &=-\frac {a^7 c^6}{11 x^{11}}+\frac {a^6 b c^6}{2 x^{10}}-\frac {a^5 b^2 c^6}{x^9}+\frac {5 a^4 b^3 c^6}{8 x^8}+\frac {5 a^3 b^4 c^6}{7 x^7}-\frac {3 a^2 b^5 c^6}{2 x^6}+\frac {a b^6 c^6}{x^5}-\frac {b^7 c^6}{4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 114, normalized size = 1.00 \[ -\frac {a^7 c^6}{11 x^{11}}+\frac {a^6 b c^6}{2 x^{10}}-\frac {a^5 b^2 c^6}{x^9}+\frac {5 a^4 b^3 c^6}{8 x^8}+\frac {5 a^3 b^4 c^6}{7 x^7}-\frac {3 a^2 b^5 c^6}{2 x^6}+\frac {a b^6 c^6}{x^5}-\frac {b^7 c^6}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^12,x]

[Out]

-1/11*(a^7*c^6)/x^11 + (a^6*b*c^6)/(2*x^10) - (a^5*b^2*c^6)/x^9 + (5*a^4*b^3*c^6)/(8*x^8) + (5*a^3*b^4*c^6)/(7
*x^7) - (3*a^2*b^5*c^6)/(2*x^6) + (a*b^6*c^6)/x^5 - (b^7*c^6)/(4*x^4)

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fricas [A]  time = 0.67, size = 103, normalized size = 0.90 \[ -\frac {154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^12,x, algorithm="fricas")

[Out]

-1/616*(154*b^7*c^6*x^7 - 616*a*b^6*c^6*x^6 + 924*a^2*b^5*c^6*x^5 - 440*a^3*b^4*c^6*x^4 - 385*a^4*b^3*c^6*x^3
+ 616*a^5*b^2*c^6*x^2 - 308*a^6*b*c^6*x + 56*a^7*c^6)/x^11

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giac [A]  time = 0.98, size = 103, normalized size = 0.90 \[ -\frac {154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^12,x, algorithm="giac")

[Out]

-1/616*(154*b^7*c^6*x^7 - 616*a*b^6*c^6*x^6 + 924*a^2*b^5*c^6*x^5 - 440*a^3*b^4*c^6*x^4 - 385*a^4*b^3*c^6*x^3
+ 616*a^5*b^2*c^6*x^2 - 308*a^6*b*c^6*x + 56*a^7*c^6)/x^11

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maple [A]  time = 0.01, size = 83, normalized size = 0.73 \[ \left (-\frac {b^{7}}{4 x^{4}}+\frac {a \,b^{6}}{x^{5}}-\frac {3 a^{2} b^{5}}{2 x^{6}}+\frac {5 a^{3} b^{4}}{7 x^{7}}+\frac {5 a^{4} b^{3}}{8 x^{8}}-\frac {a^{5} b^{2}}{x^{9}}+\frac {a^{6} b}{2 x^{10}}-\frac {a^{7}}{11 x^{11}}\right ) c^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^6/x^12,x)

[Out]

c^6*(-3/2*a^2*b^5/x^6+a*b^6/x^5-a^5*b^2/x^9+5/7*a^3*b^4/x^7+1/2*a^6*b/x^10+5/8*a^4*b^3/x^8-1/11*a^7/x^11-1/4*b
^7/x^4)

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maxima [A]  time = 1.02, size = 103, normalized size = 0.90 \[ -\frac {154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^12,x, algorithm="maxima")

[Out]

-1/616*(154*b^7*c^6*x^7 - 616*a*b^6*c^6*x^6 + 924*a^2*b^5*c^6*x^5 - 440*a^3*b^4*c^6*x^4 - 385*a^4*b^3*c^6*x^3
+ 616*a^5*b^2*c^6*x^2 - 308*a^6*b*c^6*x + 56*a^7*c^6)/x^11

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mupad [B]  time = 0.06, size = 102, normalized size = 0.89 \[ -\frac {\frac {a^7\,c^6}{11}-\frac {a^6\,b\,c^6\,x}{2}+a^5\,b^2\,c^6\,x^2-\frac {5\,a^4\,b^3\,c^6\,x^3}{8}-\frac {5\,a^3\,b^4\,c^6\,x^4}{7}+\frac {3\,a^2\,b^5\,c^6\,x^5}{2}-a\,b^6\,c^6\,x^6+\frac {b^7\,c^6\,x^7}{4}}{x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^6*(a + b*x))/x^12,x)

[Out]

-((a^7*c^6)/11 + (b^7*c^6*x^7)/4 - a*b^6*c^6*x^6 + a^5*b^2*c^6*x^2 - (5*a^4*b^3*c^6*x^3)/8 - (5*a^3*b^4*c^6*x^
4)/7 + (3*a^2*b^5*c^6*x^5)/2 - (a^6*b*c^6*x)/2)/x^11

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sympy [A]  time = 0.80, size = 110, normalized size = 0.96 \[ \frac {- 56 a^{7} c^{6} + 308 a^{6} b c^{6} x - 616 a^{5} b^{2} c^{6} x^{2} + 385 a^{4} b^{3} c^{6} x^{3} + 440 a^{3} b^{4} c^{6} x^{4} - 924 a^{2} b^{5} c^{6} x^{5} + 616 a b^{6} c^{6} x^{6} - 154 b^{7} c^{6} x^{7}}{616 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**12,x)

[Out]

(-56*a**7*c**6 + 308*a**6*b*c**6*x - 616*a**5*b**2*c**6*x**2 + 385*a**4*b**3*c**6*x**3 + 440*a**3*b**4*c**6*x*
*4 - 924*a**2*b**5*c**6*x**5 + 616*a*b**6*c**6*x**6 - 154*b**7*c**6*x**7)/(616*x**11)

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