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

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]

-(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)

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Rubi [A]  time = 0.141146, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^7 c^6}{11 x^{11}}+\frac{a^6 b c^6}{2 x^{10}}-\frac{a^5 b^2 c^6}{x^9}+\frac{5 a^4 b^3 c^6}{8 x^8}+\frac{5 a^3 b^4 c^6}{7 x^7}-\frac{3 a^2 b^5 c^6}{2 x^6}+\frac{a b^6 c^6}{x^5}-\frac{b^7 c^6}{4 x^4} \]

Antiderivative was successfully verified.

[In]  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)

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Rubi in Sympy [A]  time = 43.7732, size = 112, normalized size = 0.98 \[ - \frac{a^{7} c^{6}}{11 x^{11}} + \frac{a^{6} b c^{6}}{2 x^{10}} - \frac{a^{5} b^{2} c^{6}}{x^{9}} + \frac{5 a^{4} b^{3} c^{6}}{8 x^{8}} + \frac{5 a^{3} b^{4} c^{6}}{7 x^{7}} - \frac{3 a^{2} b^{5} c^{6}}{2 x^{6}} + \frac{a b^{6} c^{6}}{x^{5}} - \frac{b^{7} c^{6}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**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)

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Mathematica [A]  time = 0.0124432, size = 114, normalized size = 1. \[ -\frac{a^7 c^6}{11 x^{11}}+\frac{a^6 b c^6}{2 x^{10}}-\frac{a^5 b^2 c^6}{x^9}+\frac{5 a^4 b^3 c^6}{8 x^8}+\frac{5 a^3 b^4 c^6}{7 x^7}-\frac{3 a^2 b^5 c^6}{2 x^6}+\frac{a b^6 c^6}{x^5}-\frac{b^7 c^6}{4 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(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)

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Maple [A]  time = 0.007, size = 83, normalized size = 0.7 \[{c}^{6} \left ({\frac{5\,{a}^{4}{b}^{3}}{8\,{x}^{8}}}-{\frac{{a}^{7}}{11\,{x}^{11}}}+{\frac{{a}^{6}b}{2\,{x}^{10}}}+{\frac{5\,{a}^{3}{b}^{4}}{7\,{x}^{7}}}-{\frac{{a}^{5}{b}^{2}}{{x}^{9}}}+{\frac{a{b}^{6}}{{x}^{5}}}-{\frac{{b}^{7}}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}{b}^{5}}{2\,{x}^{6}}} \right ) \]

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

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Maxima [A]  time = 1.34935, size = 139, normalized size = 1.22 \[ -\frac{154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/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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Fricas [A]  time = 0.197285, size = 139, normalized size = 1.22 \[ -\frac{154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/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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Sympy [A]  time = 3.71153, size = 112, normalized size = 0.98 \[ - \frac{56 a^{7} c^{6} - 308 a^{6} b c^{6} x + 616 a^{5} b^{2} c^{6} x^{2} - 385 a^{4} b^{3} c^{6} x^{3} - 440 a^{3} b^{4} c^{6} x^{4} + 924 a^{2} b^{5} c^{6} x^{5} - 616 a b^{6} c^{6} x^{6} + 154 b^{7} c^{6} x^{7}}{616 x^{11}} \]

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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GIAC/XCAS [A]  time = 0.248457, size = 139, normalized size = 1.22 \[ -\frac{154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/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